What are operator spaces?

Operator algebra group G. Wittstock, B. Betz, H.-J. Fischer, A. Lambert, K. Louis, M. Neufang, I. Zimmermann Universität des Saarlandes

January 6, 2001

Contents

1 Short History 3

2 Operator Spaces and Completely Bounded Maps 3 2.1 Basic facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Ruan’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Elementary constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 The space CB(X, Y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.5 The dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.6 Direct sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.7 MIN and MAX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.8 Injective operator spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.8.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.8.2 Examples and elementary constructions . . . . . . . . . . . . . . . . 11 2.8.3 Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Operator Systems and Completely Positive Maps 12 3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 Matrix order unit norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.4 Injective operator systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Hilbertian Operator Spaces 15 4.1 The spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2 The morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.3 The column Hilbert space C_{H} . . . . . . . . . . . . . . . . . . . . . . . . . . 17

What are operator spaces ? January 6, 2001 http://www.math.uni-sb.de/∼wittck/ 2

4.3.1 Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.4 Column Hilbert space factorization . . . . . . . . . . . . . . . . . . . . . . . 18 5 Multiplicative Structures 19

5.1 Operator modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.2 Completely bounded module homomorphisms . . . . . . . . . . . . . . . . . 21 5.3 Operator algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6 Tensor Products 25 6.1 Operator space tensor products . . . . . . . . . . . . . . . . . . . . . . . . . 26 6.2 Injective operator space tensor product . . . . . . . . . . . . . . . . . . . . . 29

6.2.1 Exact operator spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 30 6.3 Projective operator space tensor product . . . . . . . . . . . . . . . . . . . . 31 6.4 The Haagerup tensor product . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6.5 Completely bounded bilinear mappings . . . . . . . . . . . . . . . . . . . . . 37 6.6 Module tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6.6.1 Module Haagerup tensor product . . . . . . . . . . . . . . . . . . . . 41

7 Complete Local Reflexivity 42

8 Completely Bounded Multilinear Mappings 43

9 Automatic Complete Boundedness 45

10 Convexity 47

10.1 Matrix convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 10.1.1 Separation theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 10.1.2 Bipolar theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

10.2 Matrix extreme points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 10.3 C^{∗}-convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

10.3.1 Separation theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 10.4 C^{∗}-extreme points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

11 Mapping Spaces 54 11.1 Completely nuclear mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 55 11.2 Completely integral mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 56

12 Appendix 57

12.1 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 12.1.1 Tensor products of operator matrices . . . . . . . . . . . . . . . . . . 57 12.1.2 Joint amplification of a duality . . . . . . . . . . . . . . . . . . . . . 59 12.1.3 Tensor matrix multiplication . . . . . . . . . . . . . . . . . . . . . . 59

12.2 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

13 Symbols 61

1 Short History

The theory of operator spaces grew out of the analysis of completely positive and completely bounded mappings. These maps were first studied on C^{∗}-algebras, and later on suitable subspaces of C^{∗}-algebras. For such maps taking values in B(H) representation and extension theorems were proved [Sti55], [Arv69], [Haa80], [Wit81], [Pau82]. Many of the properties shared by completely positive mappings can be taken over to the framework of operator systems [CE77]. Operator systems provide an abstract description of the order structure of selfadjoint unital subspaces of C^{∗}-algebras. Paulsen’s monograph [Pau86] presents many applications of the theory of completely bounded maps to operator theory. The extension and representation theorems for completely bounded maps show that subspaces of C^{∗}-algebras carry an intrinsic metric structure which is preserved by complete isometries. This structure has been characterized by Ruan in terms of the axioms of an operator space [Rua88]. Just as the theory of C^{∗}-algebras can be viewed as noncommutative topology and the theory of von Neumann algebras as noncommutative measure theory, one can think of the theory of operator spaces as noncommutative functional analysis.

This program has been presented to the mathematical community by E.G. Effros [Eff87] in his address to the ICM in 1986. The following survey articles give a fairly complete account of the development of the theory: [CS89], [MP94], [Pis97].

2 Operator Spaces and Completely Bounded Maps

2.1 Basic facts

The spaces

Let X be a complex vector space. A matrix seminorm [EW97b] is a family of mappings ∥ · ∥ : M_{n}(X) → ℝ, one on each matrix level^{1} M_{n}(X) = M_{n} ⊗ X for n ∈ ℕ, such that

(R1) ∥αxβ∥ ≤ ∥α∥∥x∥∥β∥ for all x ∈ M_{n}(X), α ∈ M_{m,n}, β ∈ M_{n,m}
(R2) ∥x ⊕ y∥ = max{∥x∥, ∥y∥} for all x ∈ M_{n}(X), y ∈ M_{m}(X).^{2}

Then every one of these mappings ∥ · ∥ : M_{n}(X) → ℝ is a seminorm. If one (and then every one) of them is definite, the operator space seminorm is called a matrix norm.

1 The term matrix level is to be found for instance in []

2 It suffices to show one of the following two weaker conditions:

(R1′) ∥αxβ∥ ≤ ∥α∥∥x∥∥β∥ for all x ∈ M_{n}(X), α ∈ M_{n}, β ∈ M_{n}, (R2) ∥x ⊕ y∥ = max{∥x∥, ∥y∥} for all x ∈ M_{n}(X), y ∈ M_{m}(X), which is often found in the literature, or

(R1) ∥αxβ∥ ≤ ∥α∥∥x∥∥β∥ for all x ∈ M_{n}(X), α ∈ M_{m,n}, β ∈ M_{n,m},

(R2′) ∥x ⊕ y∥ ≤ max{∥x∥, ∥y∥} for all x ∈ M_{n}(X), y ∈ M_{m}(X), which seems to be appropriate in convexity theory.

A matricially normed space is a complex vector space with a matrix norm. It can be defined equivalently, and is usually defined in the literature, as a complex vector space with a family of norms with (R1) and (R2) on its matrix levels.

If M_{n}(X) with this norm is complete for one n (and then for all n), then X is called an operator space^{3}([Rua88], cf. [Wit84a]).

For a matricially normed space (operator space) X the spaces M_{n}(X) are normed spaces (Banach spaces).^{4} These are called the matrix levels of X (first matrix level, second level …).

The operator space norms on a fixed vector space X are partially ordered by the pointwise order on each matrix level M_{n}(X). One says that a greater operator space norm dominates a smaller one.

The mappings

A linear mapping Φ between vector spaces X and Y induces a linear mapping Φ^{(n)} = id_{M_{n}} ⊗ Φ ,

Φ^{(n)} : M_{n}(X) → M_{n}(Y)
[x_{ij}] ↦ [Φ(x_{ij})] ,

the nth amplification of Φ.

For matricially normed X and Y, one defines

∥Φ∥_{cb} := sup {∥Φ^{(n)}∥ | n ∈ ℕ} .

Φ is called completely bounded if ∥Φ∥_{cb} < ∞ and completely contractive if ∥Φ∥_{cb} ≤ 1.

Among the complete contractions, the complete isometries and the complete quotient mappings play a special role. Φ is called completely isometric if all Φ^{(n)} are isometric,^{5} and a complete quotient mapping if all Φ^{(n)} are quotient mappings.^{6}

The set of all completely bounded mappings from X to Y is denoted by CB(X, Y) [Pau86, Chap. 7].

An operator space X is called homogeneous if each bounded operator Φ ∈ B(M_{1}(X)) is completely bounded with the same norm: Φ ∈ CB(X), and ∥Φ∥_{cb} = ∥Φ∥ [Pis96].

Notations

Using the disjoint union

M(X) := ˙ ⋃_{n∈ℕ}M_{n}(X),

3 In the literature, the terminology is not conseqent. We propose this distinction between matricially normed space and operator space in analogy with normed space and Banach space.

4 In the literature, the normed space M_{1}(X) usually is denoted also by X. We found that a more distinctive notation is sometimes usefull.

5 I. e.: ∥x∥ = ∥Φ^{(n)}(x)∥ for all n ∈ ℕ, x ∈ M_{n}(X).

6 I. e.: ∥y∥ = inf{∥x∥ | x ∈ Φ^{(n)}^{−1}(y)} for alle n ∈ ℕ and y ∈ M_{n}(Y), or equivalently Φ^{(n)}(Ball^{∘}M_{n}(X)) = Ball^{∘}M_{n}(Y) for all n ∈ ℕ, where Ball^{∘}M_{n}(X) = {x ∈ M_{n}(X) | ∥x∥ < 1}.

the notation becomes simpler.^{7}

Examples

B(H) is an operator space by the identification M_{n}(B(H)) = B(H^{n}). Generally, each C^{∗}-algebra A is an operator space if M_{n}(A) is equipped with its unique C^{∗}-norm. Closed subspaces of C^{∗}-algebras are called concrete operator spaces. Each concrete operator space is an operator space. Conversely, by the theorem of Ruan, each operator space is completely isometrically isomorphic to a concrete operator space.

Commutative C^{∗}-algebras are homogeneous operator spaces.

The transposition Φ on l_{2}(I) has norm ∥Φ∥ = 1, but ∥Φ∥_{cb} = dim l_{2}(I). If I is infinite, then Φ is bounded, but not completely bounded.

If dim H ≥ 2, then B(H) is not homogeneous [Pau86, p. 6].

Smith’s lemma

For a matricially normed space X and a linear operator Φ : X → M_{n}, we have ∥Φ∥_{cb} = ∥Φ^{(n)}∥. In particular, Φ is completely bounded if and only if Φ^{(n)} is bounded [Smi83, Thm. 2.10].

Rectangular matrices

For a matricially normed space X, the spaces M_{n,m}(X) = M_{n,m} ⊗X of n×m-matrices over X are normed by adding zeros so that one obtains a square matrix, no matter of which size.

Then

M_{n,m}(B(H)) = B(H^{m}, H^{n})

holds isometrically.

2.2 Ruan’s theorem

Each concrete operator space is an operator space. The converse is given by Ruan’s theorem: Each (abstract) operator space is completely isometrically isomorphic to a concrete operator space [Rua88].

7 The norms on the matrix levels M_{n}(X) are then one mapping M(X) → ℝ. The amplifications of Φ : X → Y can be described as one mapping Φ : M(X) → M(Y). We have

∥Φ∥_{cb} = sup{∥Φ(x)∥ | x ∈ M(X), ∥x∥ ≤ 1}.

Φ is completely isometric if ∥x∥ = ∥Φ(x)∥ for all x ∈ M(X), and Φ is a complete quotient mapping if ∥y∥ = inf{∥x∥ | x ∈ Φ^{−1}(y)} for all y ∈ M(Y) or Φ(Ball^{∘}X) = Ball^{∘}Y, where Ball^{∘}X = {x ∈ M(X) | ∥x∥ < 1}.

More concretely, for a matricially normed space X let S_{n} be the set of all complete contractions from X to M_{n}. Then the mapping

Φ : X → ⨁_{n∈ℕ} ⨁_{Φ∈S_{n}} M_{n}
x ↦ (Φ(x))_{Φ}

is a completely isometric embedding of X into a C^{∗}-algebra [ER93].

A proof relies on the separation theorem for absolutely matrix convex sets.

This theorem can be used to show that many constructions with concrete operator spaces yield again concrete operator spaces (up to complete isometry).

2.3 Elementary constructions

Subspaces and quotients

Let X be a matricially normed space and X_{0} ⊂ X a linear subspace. Then M_{n}(X_{0}) ⊂ M_{n}(X), and X_{0} together with the restriction of the operator space norm again is a matricially normed space. The embedding X_{0} ↪ X is completely isometric. If X is an operator space and X_{0} ⊂ X is a closed subspace, then M_{n}(X_{0}) ⊂ M_{n}(X) is closed and X_{0} is an operator space.

Algebraically we have M_{n}(X/X_{0}) = M_{n}(X)/M_{n}(X_{0}). If X_{0} is closed, then X/X_{0} together with the quotient norm on each matrix level is matricially normed (an operator space if X is one). The quotient mapping X ↠ X/X_{0} is a complete quotient mapping.

More generally, a subspace of a matricially normed space (operator space) X is a matricially normed space (operator space) Y together with a completely isometric operator Y → X. A quotient of X is a matricially normed space (operator space) Y together with a complete quotient mapping X → Y.

Matrices over an operator space

The vector space M_{p}(X) of matrices over a matricially normed space X itself is matricially normed in a natural manner: The norm on the nth level M_{n}(M_{p}(X)) is given by the identification

M_{n}(M_{p}(X)) = M_{np}(X)

[BP91, p. 265]. We^{8} write

𝕄_{p}(X)

for M_{p}(X) with this operator space structure. In particular,

M_{1}(𝕄_{p}(X)) = M_{p}(X)

8 In the literature, the symbol M_{p}(X) stands for both the operator space with first matrix level M_{p}(X)

and for the pth level of the operator space X. We found that the distinction between 𝕄_{p}(X) and M_{p}(X)

clarifies for instance the definition of the operator space structure of CB(X, Y).

holds isometrically. Analogously M_{p,q}(X) becomes a matricially normed space 𝕄_{p,q}(X) by the identification

M_{n}(M_{p,q}(X)) = M_{np,nq}(X).

By adding zeros it is a subspace of 𝕄_{r}(X) for r ≥ p, q.

Examples: For a C^{∗}-algebra A, 𝕄_{p}(A) is the C^{∗}-Algebra of p × p-matrices over A with its natural operator space structure.

The Banach space M_{p}(A) is the first matrix level of the operator space 𝕄_{p}(A).

The complex numbers have a unique operator space structure which on the first matrix level is isometric to ℂ, and for this M_{p}(ℂ) = M_{p} holds isometrically. We write

𝕄_{p} := 𝕄_{p}(ℂ).

Then 𝕄_{p} always stands for the C^{∗}-algebra of p×p-matrices with its operator space structure. The Banach space M_{p} is the first matrix level of the operator space 𝕄_{p}.

Columns and rows of an operator space

The space X^{p} of p-tupels over an operator space X can be made into an operator space for instance by reading the p-tupels as p × 1- or as 1 × p-matrices. This leads to the frequently used columns and rows of an operator space X:

C_{p}(X) := 𝕄_{p,1}(X) and R_{p}(X) := 𝕄_{1,p}(X).

The first matrix level of these spaces are

M_{1}(C_{p}(X)) = M_{p,1}(X) and M_{1}(R_{p}(X)) = M_{1,p}(X), respectively.

If X ≠ {0}, the spaces C_{p}(X) and R_{p}(X) are not completely isometric. In general even the first matrix levels M_{p,1}(X) and M_{1,p}(X) are not isometric.

C_{p} := C_{p}(ℂ) is called the p-dimensional column space and R_{p} := R_{p}(ℂ) the pdimensional row space.

The first matrix levels of C_{p} and R_{p} are isometric to l_{2}^{p}, but C_{p} and R_{p} are not completely isometric.

2.4 The space CB(X, Y)

Let X and Y be matricially normed spaces. A matrix [T_{ij}] ∈ M_{n}(CB(X, Y)) determines a completely bounded operator

T : X → 𝕄_{n}(Y)
x ↦ [T_{ij}(x)] .

Defining ∥[T_{ij}]∥ = ∥T ∥_{cb}, CB(X, Y) becomes a matricially normed space. It is an operator space, if Y is one. The equation

𝕄_{p}(CB(X, Y)) =^{cb} CB(X, 𝕄_{p}(Y))

holds completely isometrically.

2.5 The dual

The dual^{9} of a matricially normed space X is defined as X^{∗} = CB(X, ℂ) [Ble92a].^{10} Its first matrix level is the dual of the first matrix level of X: M_{1}(X^{∗}) = (M_{1}(X))^{∗}.

The canonical embedding X ↪ X^{∗∗} is completely isometric [BP91, Thm. 2.11].

Some formulae

For matricially normed spaces X and Y, m ∈ ℕ, y ∈ M_{m}(Y) and T ∈ CB(X, Y) we have

∥y∥ = sup {∥Φ(y)∥ | n ∈ ℕ, Φ ∈ CB(Y, 𝕄_{n}), ∥Φ∥_{cb} ≤ 1}

and

∥T ∥_{cb} = sup{∥Φ^{(n)} ∘ T ∥_{cb} | n ∈ ℕ, Φ ∈ CB(Y, 𝕄_{n}), ∥Φ∥_{cb} ≤ 1}.

A matrix [T_{ij}] ∈ M_{n}(X^{∗}) defines an operator

T : M_{n}(X) → ℂ
[x_{ij}] ↦ ∑_{i,j} T_{ij}x_{ij}.

Thus we have an algebraic identification of M_{n}(X^{∗}) and M_{n}(X)^{∗} and further of M_{n}(X^{∗∗}) and M_{n}(X)^{∗∗}. The latter even is a complete isometry ([Ble92b, Cor. 2.14]):

𝕄_{n}(X^{∗∗}) =^{cb} 𝕄_{n}(X)^{∗∗}.

The isometry on the first matrix level is shown in [Ble92a, Thm. 2.5]. This already implies^{11} the complete isometry. More generally we have^{12}

CB(X^{∗}, 𝕄_{n}(Y)) =^{cb} CB(𝕄_{n}(X)^{∗}, Y).

X is called reflexive, if X =^{cb} X^{∗∗}. An operator space X is reflexive if and only if its first matrix level M_{1}(X) is a reflexive Banach space.

9In the literature, this dual was originally called standard dual [Ble92a].

10 The norm of a matricially normed space X is given by the unit ball BallX ⊂ M(X). Here, BallX^{∗} = {Φ : X → M_{n} | n ∈ ℕ, Φ completely contractive}.

11

M_{k}(𝕄_{n}(X)^{∗∗}) = M_{k}(M_{n}(X))^{∗∗} = M_{kn}(X)^{∗∗} = M_{kn}(X^{∗∗}) = M_{k}(𝕄_{n}(X^{∗∗})).

12 This follows from 𝕄_{n}(X^{∗∗}) =^{cb} 𝕄_{n}(X)^{∗∗} and the above mentioned formula

∥T ∥_{cb} = sup{∥Φ^{(n)} ∘ T ∥_{cb} | n ∈ ℕ, Φ ∈ CB(Y, 𝕄_{n}), ∥Φ∥_{cb} ≤ 1}.

The adjoint operator

For T ∈ CB(X, Y), the adjoint operator T^{∗} is defined as usual. We have: T^{∗} ∈ CB(Y^{∗}, X^{∗}), and ∥T ∥_{cb} = ∥T^{∗}∥_{cb}. The mapping

∗ : CB(X, Y) → CB(Y^{∗}, X^{∗})
T ↦ T^{∗}

even is completely isometric [Ble92b, Lemma 1.1].^{13}

T^{∗} is a complete quotient mapping if and only if T is completely isometric; T^{∗} is completely isometric if T is a complete quotient mapping. Especially for a subspace X_{0} ⊂ X we have [Ble92a]:

X_{0}^{∗} =^{cb} X^{∗}/X_{0}^{⊥}

and, if X_{0} is closed,

(X/X_{0})^{∗} =^{cb} X_{0}^{⊥}.

2.6 Direct sums

∞-direct sums

Let I be an index set and X_{i} for each i ∈ I an operator space. Then there are an operator space X and complete contractions π_{i} : X → X_{i} with the following universal mapping property: For each family of complete contractions φ_{i} : Z → X_{i} there is exactly one complete contraction φ : Z → X such that φ_{i} = π_{i} ∘ φ for all i. X is called ∞-direct sum of the X_{i} and is denoted by ⨁_{∞}(X_{i} | i ∈ I). The π_{i} are complete quotient mappings.

One can construct a ∞-direct sum for instance as the linear subspace X = {(x_{i}) ∈ ∏_{i∈I} X_{i} | sup{∥x_{i}∥ | i ∈ I} < ∞} of the cartesian product of the X_{i}, the π_{i} being the projections on the components. We have M_{n}(X) = {(x_{i}) ∈ ∏_{i∈I} M_{n}(X_{i}) | sup{∥x_{i}∥ | i ∈ I} < ∞}, and the operator space norm is given by ∥(x_{i})∥ = sup{∥x_{i}∥ | i ∈ I}.

1-direct sums

Let I be an index set and X_{i} for each i ∈ I an operator space. Then there are an operator space X and complete contractions ι_{i} : X_{i} → X with the following universal mapping property: For each family of complete contractions φ_{i} : X_{i} → Z there is exactly one

↪ M_{1}(CB(𝕄_{n}(Y)^{∗}, X^{∗}))
= M_{1}(CB(Y^{∗}, 𝕄_{n}(X^{∗})))
= M_{n}(CB(Y^{∗}, X^{∗})).

13 The isometry on the matrix levels follows from the isometry on the first matrix level using the above mentioned formula CB(𝕄_{n}(X)^{∗}, Y) =^{cb} CB(X^{∗}, 𝕄_{n}(Y)):

M_{n}(CB(X, Y)) = M_{1}(CB(X, 𝕄_{n}(Y)))

complete contraction φ : X → Z such that φ_{i} = φ ∘ ι_{i} for all i. X is called 1-direct sum of the X_{i} and is denoted by ⨁_{1}(X_{i} | i ∈ I).The ι_{i} are completely isometric.

One can construct a 1-direct sum for instance as the closure of the sums of the images

of the mappings X_{i} ↪ X_{i}^{∗∗} →^{π_{i}^{∗}} (⨁_{∞}(X_{i}^{∗} | i ∈ I))^{∗}, where π_{i} is the projection from ⨁_{∞}(X_{i}^{∗} | i ∈ I) onto X_{i}^{∗}.

The equation

(⨁_{1}(X_{i} | i ∈ I))^{∗} = ⨁_{∞}(X_{i}^{∗} | i ∈ I)

holds isometrically.

p-direct sums

p-direct sums of operator spaces for 1 < p < ∞ can be obtained by interpolation between the ∞- and the 1-direct sum.

2.7 MIN and MAX

Let E be a normed space. Among all operator space norms on E which coincide on the first matrix level with the given norm, there is a greatest and a smallest. The matricially normed spaces given by these are called MAX (E) and MIN (E). They are characterized by the following universal mapping property:^{14} For a matricially normed space X,

M_{1}(CB(MAX (E), X)) = B(E, M_{1}(X))

and

M_{1}(CB(X, MIN (E))) = B(M_{1}(X), E)

holds isometrically.

We have [Ble92a]

MIN (E)^{∗} =^{cb} MAX (E^{∗}),
MAX (E)^{∗} =^{cb} MIN (E^{∗}).

For dim(E) = ∞,

id_{E} : MIN (E) → MAX (E)

is not completely bounded[Pau92, Cor. 2.13].^{15}

Subspaces of MIN -spaces are MIN -spaces: For each isometric mapping φ : E_{0} → E, the mapping φ : MIN (E_{0}) → MIN (E) is completely isometric.

Quotients of MAX -spaces are MAX -spaces: For each quotient mapping φ : E → E_{0}, the mapping φ : MAX (E) → MAX (E_{0}) is a complete quotient mapping.

14 MAX is the left adjoint and MIN the right adjoint of the forgetfull functor which maps an operator space X to the Banach space M_{1}(X).

15 Paulsen uses in his proof a false estimation for the projection constant of the finite dimensional Hilbert spaces; the converse estimation is correct [Woj91, p. 120], but here useless. The gap can be filled [Lam97, Thm. 2.2.15] using the famous theorem of Kadets-Snobar: The projection constant of an n-dimensional Banach space is less or equal than \sqrt{n} [KS71].

Construction of MIN :

For a commutative C^{∗}-algebra A = C(K), each bounded linear mapping Φ : M_{1}(X) → A is automatically completely bounded with ∥Φ∥_{cb} = ∥Φ∥ [Loe75].^{16}

Each normed space E is isometric to a subspace of the commutative C^{∗}-algebra l_{∞}(Ball(E^{∗})). Thus the operator space MIN (E) is given as a subspace of l_{∞}(Ball(E^{∗})).

For x ∈ M_{n}(MIN (E)) we have

∥x∥ = sup {∥f^{(n)}(x)∥| f ∈ Ball(E^{∗})}.

The unit ball of MIN (E) is given as the absolute matrix polar of Ball(E^{∗}).

Construction of MAX :

For a index set I, l_{1}(I) = c_{0}(I)^{∗}. l_{1}(I) is an operator space as dual of the commutative C^{∗}-algebra c_{0}(I), and each bounded linear mapping Φ : l_{1}(I) → M_{1}(X) is automatically completely bounded with ∥Φ∥_{cb} = ∥Φ∥.^{17}

Each Banach space^{18} E is isometric to a quotient of l_{1}(Ball(E)). Thus the operator space MAX (E) is given as a quotient of l_{1}(Ball(E)).

For x ∈ M_{n}(MAX (E)) we have

∥x∥ = sup{∥φ^{(n)}(x)∥ | n ∈ ℕ, φ : E → M_{n}, ∥φ∥ ≤ 1}.

The unit ball of MAX (E) is given as the absolute matrix bipolar of Ball(E^{)}.

2.8 Injective operator spaces

2.8.1 Definition

A matricially normed space X is called injective if completely bounded mappings into X can be extended with the same norm. More exactly:

For all matricially normed spaces Y_{0} and Y, each complete contraction φ : Y_{0} → X and each complete isometry ι : Y_{0} → Y there is a complete contraction φ̃ : Y → X such that φ̃ι = φ.

It suffices to consider only operator spaces Y_{0} and Y. Injective matricially normed spaces are automatically comlete, so they are also called injective operator spaces.

2.8.2 Examples and elementary constructions

B(H) is injective [].

Completely contractively projectable subspaces of injective operator spaces are injective. L^{∞}-direct sums of injective operator spaces are injective. Injective operator systems, injective C^{∗}-algebras and injective von Neumann-algebras

are injective operator spaces.

16I. e. A is a MIN -space.

17I. e. l_{1}(I) is a MAX -space.

18 A similar construction is possible for normed spaces.

2.8.3 Characterizations

For a matricially normed space X the following conditions are equivalent:

a) X is injective.

b) For each complete isometry ι : X → Z there is a complete contraction π : Z → X such that πι = id_{X}. I. e. X is completely contractively projectable in each space containing it as a subspace.

c) For each complete isometry ι : X → Z and each complete contraction φ : X → Y there is a complete contraction φ̃ : Z → Y such that φ̃ι = φ. I. e. Complete contractions from X can be extended completely contractively to any space conaining X as a subspace.^{19}

d) X is completely isometric to a completely contractively projectable subspace of B(H) for some Hilbert space H.

e) X is completely isometric to pAq, where A is an injective C^{∗}-algebra and p and q are projections in A.[]

Robertson[] characterized the infinite dimensional injective subspaces of B(l_{2}) up to isometry (not complete isometry!). They are B(l_{2}), l_{∞}, l_{2}, l_{∞}⊕l_{2} and ⨁_{n∈ℕ}^{L^{∞}} l_{2}. (Countable L^{∞}-direct sums of such are again comletely isometric to one of these.) If an injective subspace of B(l_{2}) is isometric to l_{2}, it is completely isometric to R_{l_{2}} or C_{l_{2}}.[]

Injective envelopes

Let X be a matricially normed space.

An operator space Z together with a completely isometric mapping ι : X → Z is called an injective envelope of X if Z is injective, and if id_{Z} is the unique extension of ι onto Z.

This is the case if and only if Z is the only injective subspace of Z which contains the image of X.[]

Each matricially normed space has an injective envelope. It is unique up to a canonical isomorphism.

A matricially normed space Z together with a completely isometric mapping ι : X → Z is called an essential extension of X if a complete contraction φ : Z → Y is completely isometric if only φ ∘ ι is completely isometric.

ι : X → Z is an injective envelope if and only if Z is injective and ι : X → Z is an essential extension.

Every injective envelope ι : X → Z is a maximal essential extension, i. e. for each essential extension ι̃: X → Z̃, there is a completely isometric mapping φ : Z̃ → Z such that φ ∘ ι̃ = ι.

3 Operator Systems and Completely Positive Maps

19Equivalently: Completely bounded mappings from X can be extended with the same norm.

3.1 Definitions

Let V be a complex vector space. An involution on V is a conjugate linear map ∗ : V → V, v ↦ v^{∗}, such that v^{∗∗} = v. A complex vector space is an involutive vector space if there is an involution on V. Let V be an involutive vector space. Then V_{sa} is the real vector space of selfadjoint elements of V, i.e. those elements of V, such that v^{∗} = v. An involutive vector space is an ordered vector space if there is a proper cone^{20} V^{+} ⊂ V_{sa}. The elements of V^{+} are called positive and there is an order on V_{sa} defined by v ≤ w if w − v ∈ V^{+} for v, w ∈ V_{sa}.

An element 𝟙 ∈ V^{+} is an order unit if for any v ∈ V_{sa} there is a real number t > 0, such that −t𝟙 ≤ v ≤ t𝟙. If V_{+} has an order unit then V_{sa} = V_{+} − V_{+}.

The cone V^{+} is called Archimedian if w ∈ −V^{+} whenever there exists v ∈ V_{sa} such that tw ≤ v for all t > 0. If there is an order unit 𝟙 then V_{+} is Archimedian if w ∈ −V^{+} whenever tw ≤ 𝟙 for all t > 0.

Let V be an ordered vector space. If V_{+} is Archimedian and contains a distinguished order unit 𝟙 then (V, 𝟙) is called an ordered unit space.

Let V, W be involutive vector spaces. We define an involution ⋆ on the space L(V, W) of all linear mappings from V → W by φ^{⋆}(v) = φ(v^{∗})^{∗}, φ ∈ L(V, W). If moreover V, W are ordered vector spaces, then φ is positive if φ^{⋆} = φ and φ(V^{+}) ⊂ W^{+}. If (V, 𝟙) and (W, 𝟙′) are ordered unit spaces a positve map φ : V → W is called unital if φ(𝟙) = 𝟙′.

Let V be an involutive vector space. Then M_{n}(V) is also an involutive vector space by [v_{ij}]^{∗} = [v_{ji}^{∗}]. V is a matrix ordered vector space if there are proper cones M_{n}(V)^{+} ⊂ M_{n}(V)_{sa} for all n ∈ ℕ, such that α^{∗}M_{p}(V)^{+}α ⊂ M_{q}(V)^{+} for all α ∈ M_{pq} and p, q ∈ ℕ holds^{21}. This means that (M_{n}(V)_{+})_{n∈ℕ} is a matrix cone.

Let V, W be matrix ordered vector spaces. A linear mapping ϕ : V → W is completely positive if ϕ^{(n)} : M_{n}(V) → M_{n}(W) is positive for all n ∈ ℕ. A complete order isomorphism^{22} from V to W is a completely positive map from V → W that is bijective, such that the inverse map is completely positive.

The well-known Stinespring theorem for completely positive maps reads [Pau86, Theorem 4.1]:

Let A be a unital C^{∗}-algebra and let H be a Hilbert space. If ψ : A → B(H) is completely positive then there are a Hilbert space H_{π}, a unital ∗-homomorphism π : A → B(H_{π}) and a linear mapping V : H → H_{π}, such that ψ(a) = V^{∗}π(a)V for all a ∈ A.

Let V be an involutive vector space. Then V is called an operator system if it is a matrix ordered ordered unit space, such that M_{n}(V)^{+} is Archimedian for all n ∈ ℕ. In this case M_{n}(V) is an ordered unit space with order unit 𝟙_{n} = 𝟙 ⊗ id_{M_{n}} for all n ∈ ℕ, where 𝟙 ∈ V_{+} is the distinguished order unit of V and id_{M_{n}} is the unit of 𝕄_{n}.

20A cone K is a subset of a vector space, such that K+K ⊂ K and ℝ_{+}K ⊂ K. If moreover (−K)∩K = {0} holds, K is a proper cone.

21If V^{+} = M_{1}(V)^{+} is a proper cone, by the matrix condition all the cones M_{n}(V)^{+} will be proper.

22Note that some authors don’t include surjectivity in the definition.

Example

Let H be a Hilbert space. Then, obviously, B(H) is an ordered unit space with order unit the identity operator. Using the identification M_{n}(B(H)) = B(H^{n}) we let M_{n}(B(H))_{+} = B(H^{n})_{+}. So we see that B(H) is an operator system.

Let L be an operator system. Then any subspace S ⊂ L that is selfadjoint, i.e. S^{∗} ⊂ S, and contains the order unit of L is again an operator system with the induced matrix order. So unital C^{∗}-algebras and selfadjoint subspaces of unital C^{∗}-algebras containig the identity are operator systems.

Note that a unital complete order isomorphism between unital C^{∗}-algebras must be a ∗-isomorphism [Cho74, Corollary 3.2]. So unital C^{∗}-algebras are completely characterized by their matrix order. They are not characterized by their order. For instance take the opposite algebra A^{op} of a unital C^{∗}-algebra A. Then A_{+} = A_{+}^{op} but A and A^{op} are not ∗-isomorphic. Obviously M_{2}(A)_{+} ≠ M_{2}(A^{op})_{+}.

3.2 Characterization

Choi and Effros [CE77, Theorem 4.4] showed the following characterization theorem:

Let V be an operator system. Then there are a Hilbert space H and a unital complete order isomorphism from V to a selfadjoint subspace of B(H).

A unital complete order isomorphism is obtained by

Φ : V → ⨁_{n∈ℕ} ⨁_{φ∈S_{n}} 𝕄_{n}
x ↦ (φ(x))_{φ},

where S_{n} is the set of all unital completely positive maps φ : V → 𝕄_{n}.

3.3 Matrix order unit norm

Let L be an operator system. We define norms by

∥x∥_{n} := inf \{r ∈ ℝ| (\begin{matrix}r𝟙_{n} & x \\ x^{∗} & r𝟙_{n} \end{matrix}) ∈ M_{2n}(L)_{+}\}  (1)

for all n ∈ ℕ and x ∈ M_{n}(L). With these norms L becomes an operator space.

If Φ is any unital completely positve embedding from L into some B(H) (cf. section 3.2) then ∥Φ^{(n)}(x)∥ = ∥x∥_{n} for all n ∈ ℕ and x ∈ M_{n}(L). This holds because ∥y∥ ≤ 1 if and only if

0 ≤ (\begin{matrix}𝟙 & y \\ y^{∗} & 𝟙 \end{matrix})

for all y ∈ B(H) and all Hilbert spaces H.

Let L and S be operator systems and let ψ : L → S be completely positive. We supply L and S with the norms from equation (1). Then ψ is completely bounded and ∥ψ(𝟙)∥ = ∥ψ∥ = ∥ψ∥_{cb} (cf. [Pau86, Proposition 3.5]).

3.4 Injective operator systems

An operator system R is called injectiv if given operator systems N ⊂ M each completely positive map φ : N → R has a completely positive extension ψ : M → R.

If an operator system R is injective then there is a unital complete order isomorphism from R onto a unital C^{∗}-algebra. The latter is conditionally complete^{23}. (cf. [CE77, Theorem 3.1])

4 Hilbertian Operator Spaces

4.1 The spaces

An operator space X is called hilbertian, if M_{1}(X) is a Hilbert space H. An operator space X is called homogeneous, if each bounded operator T : M_{1}(X) → M_{1}(X) is completely bounded and ∥T ∥_{cb} = ∥T ∥ [Pis96].

Examples The minimal hilbertian operator space MIN_{H} := MIN (H) and the maximal hilbertian operator space MAX_{H} := MAX (H), the column Hilbert space C_{H} := B(ℂ, H) and the row Hilbert space R_{H} := B(\overline{H}, ℂ) are homogeneous hilbertian operator spaces on the Hilbert space H.

Furthermore, for two Hilbert spaces H and K, we have completely isometric isomorphisms [ER91, Thm. 4.1] [Ble92b, Prop. 2.2]

CB(C_{H}, C_{K}) =^{cb} B(H, K) and CB(R_{H}, R_{K}) =^{cb} B(\overline{K}, \overline{H}).

These spaces satisfy the following dualities [Ble92b, Prop. 2.2] [Ble92a, Cor. 2.8]:

C_{H}^{∗} =^{cb} R\overline{H}
R_{H}^{∗} =^{cb} C\overline{H}
MIN _{H}^{∗} =^{cb} MAX \overline{H}
MAX _{H}^{∗} =^{cb} MIN \overline{H}.

For each Hilbert space H there is a unique completely self dual homogeneous operator space, the operator Hilbert space OH_{H} [Pis96, §1]:

OH _{H}^{∗} =^{cb} OH \overline{H}.

The intersection and the sum of two homogeneous hilbertian operator spaces are again homogeneous hilbertian operator spaces [Pis96].

23An ordered vector space V is conditionally complete if any upward directed subset of V_{sa} that is bounded above has a supremum in V_{sa}.

4.2 The morphisms

The space CB(X, Y) of completely bounded mappings between two homogeneous hilbertian operator spaces X, Y enjoys the following properties (cf. [MP95, Prop. 1.2]):

1. (CB(X, Y), ∥ · ∥_{cb}) is a Banach space.

2. ∥AT B∥_{cb} ≤ ∥A∥∥T ∥_{cb}∥B∥ for all A, B ∈ B(H), T ∈ CB(X, Y).

3. ∥T ∥_{cb} = ∥T ∥ for all T with rank(T) = 1.

Consequently, CB(X, Y) is a symmetrically normed ideal (s.n. ideal) in the sense of Calkin, Schatten [Sch70] and Gohberg [GK69].

The classical examples for s.n. ideals are the famous Schatten ideals:

S_{p} := {T ∈ B(H) | the sequence of singular values of T is in ℓ_{p} } (1 ≤ p < ∞).

Many, but not all s.n. ideals can be represented as spaces of completely bounded maps between suitable homogeneous hilbertian operator spaces.

The first result in this direction was

CB(R_{H}, C_{H}) = S_{2}(H) = HS(H)

isometrically [ER91, Cor. 4.5].

We have the following characterizations isometrically or only isomorphically (≃) [Mat94], [MP95], [Lam97]:

CB(↓, →) MIN_{H}  C_{H}  OH_{H}  R_{H}  MAX_{H}
MIN_{H}  B(H) HS(H) ≃ HS(H) HS(H) ≃ N(H)
C_{H}  B(H) B(H)  S_{4}(H) HS(H) HS(H)
OH_{H}  B(H) S_{4}(H)  B(H)  S_{4}(H) ≃ HS(H)
R_{H}  B(H) HS(H) S_{4}(H)  B(H) HS(H)
MAX_{H} B(H) B(H)  B(H)  B(H)  B(H)

As a unique completely isometric isomorphism, we get CB(C_{H}) =^{cb} B(H) (cf. [Ble95, Thm. 3.4]). The result

CB(MIN_{H}, MAX_{H}) ≃ N(H)

is of special interest. Here, we have a new quite natural norm on the nuclear operators, which is not equal to the canonical one.

Even in the finite dimensional case, we only know

\frac{n}{2} ≤ ∥id : MIN (ℓ_{2}^{n}) → MAX (ℓ_{2}^{n})∥_{cb} ≤ \frac{n}{\sqrt{2}}

[Pau92, Thm. 2.16]. To compute the exact constant is still an open problem. Paulsen conjectured that the upper bound is sharp [Pau92, p. 121].

Let E be a Banach space and H a Hilbert space. An operator T ∈ B(E, H) is completely bounded from MIN (E) to C_{H}, if and only if T is 2-summing [Pie67] from E to H (cf. [ER91, Thm. 5.7]):

M_{1}(CB(MIN (E), C_{H})) = Π_{2}(E, H)

with ∥T ∥_{cb} =π_{2}(T).

4.3 The column Hilbert space C_{H}

For the Hilbert space H = ℓ_{2}, we can realize the column Hilbert space C_{H} as a column, by the embedding

\begin{matrix}H & ↪ & B(H), \\ ( ⋮ ; ξ_{i} ; ⋮ ; ⋮ ) & ↦ & [ \begin{matrix}⋮ & 0 & ⋯ & 0 & ⋯ \\ ξ_{i} & ⋮ & & ⋮ \\ ⋮ & 0 & ⋯ & 0 \\ ⋮ & ⋮ & & & ... \end{matrix} ]. \end{matrix}

Via this identification, C_{ℓ_{2}^{n}} is the n-dimensional column space C_{n}.

C_{H} is a homogeneous hilbertian operator space: All bounded maps on H are completely bounded with the same norm on C_{H}. Actually we have CB(C_{H}) =^{cb} B(H) completely isometrically [ER91, Thm. 4.1].

C_{H} is a injective operator space (cf. [Rob91]).

Tensor products

Let X be an operator space. We have complete isometries [ER91, Thm. 4.3 (a)(c)] [Ble92b, Prop. 2.3 (i)(ii)]:

C_{H} ⊗_{h} X =^{cb} C_{H} ⊗^{∨} X

and

X ⊗_{h} C_{H} =^{cb} X ⊗^{∧} C_{H}.

Herein, ⊗_{h} is the Haagerup tensor product, ⊗^{∨} the injective tensor product and ⊗^{∧} the projective tensor product.

For Hilbert spaces H and K, we have complete isometries [ER91, Cor. 4.4.(a)] [Ble92b, Prop. 2.3(iv)]

C_{H} ⊗_{h} C_{K} =^{cb} C_{H} ⊗^{∨} C_{K} =^{cb} C_{H} ⊗^{∧} C_{K} =^{cb} C_{H⊗_{2}K}.

4.3.1 Characterizations

In connection with the column Hilbert space, it is enough to calculate the row norm

∥T ∥_{row} := sup_{n∈ℕ} sup_{∥[x_{1}...x_{n}]∥_{M_{1,n}(X)}≤1} ∥[T x_{1} … T x_{n}]∥_{M_{1,n}(Y)}

or the column norm

∥T ∥_{col} := sup_{n∈ℕ} sup_{∥[ ⋮_{x_{n}}^{x_{1}} ]∥_{M_{n,1}(Y )}≤1} ∥[ T x_{1} ; ⋮ ; T x_{n} ]∥_{M_{n,1}(X)}

of an operator T, instead of the cb-norm, to ascertain the complete boundedness.

Let X be an operator space and S : C_{H} → X bzw. T : X → C_{H}. Then we have ∥S∥_{cb} = ∥S∥_{row} resp. ∥T ∥_{cb} = ∥T ∥_{col} ([Mat94, Prop. 4] resp. [Mat94, Prop. 2]).

The column Hilbert space is characterized as follows,

(A) as a hilbertian operator space [Mat94, Thm. 8]:

For an operator space X on an Hilbert space H, we have the following equivalences:

1. X is completely isometric to C_{H}.

2. For all operator spaces Y and all T : X → Y we have ∥T ∥_{cb} = ∥T ∥_{row}, and for all S : Y → X we have ∥S∥ = ∥S∥_{row}. For all operator spaces Y and all T : Y → X we have ∥T ∥_{cb} = ∥T ∥_{col}, and for all S : X → Y we have ∥S∥ = ∥S∥_{col}.

3. X coincides with the maximal hilbertian operator space on columns and with the minimal hilbertian operator space on rows. That means isometrically

M_{n,1}(X) = M_{n,1}(MAX (H))
M_{1,n}(X) = M_{1,n}(MIN (H))

(B) as an operator space: For an operator space X TFAE:

1. There is a Hilbert space H, such that X =^{cb} C_{H} completely isometrically.

2. We have

M_{n,1}(X) = ⊕_{2}M_{1}(X)

and

M_{1,n}(X) = M_{1,n}(MIN (M_{1}(X)))

isometrically. [Mat94, Thm. 10].

3. CB(X) with the composition as multiplication is an operator algebra [Ble95, Thm. 3.4].

4.4 Column Hilbert space factorization

Let X, Y be operator spaces. We say that a linear map T : M_{1}(X) → M_{1}(Y) factors through a column Hilbert space, if there is a Hilbert space H and completely bounded maps T_{2} : X → C_{H}, T_{1} : C_{H} → Y with T = T_{1} ∘ T_{2}. We define

γ_{2}(T) := inf ∥T_{1}∥_{cb}∥T_{2}∥_{cb},

where the infimum runs over all possible factorizations. If no such factorisation exists we say γ_{2}(T) := ∞. Γ_{2}(X, Y) is the Banach space of all linear maps T : X → Y with γ_{2}(T) < ∞ [ER91, Chap. 5],[Ble92b, p. 83].

Let X_{1} and Y_{1} be operator spaces and T ∈ Γ_{2}(X, Y), S ∈ CB(X_{1}, X), R ∈ CB(Y, Y_{1}). Then we have the CB ideal property

γ_{2}(RT S) ≤ ∥R∥_{cb}γ_{2}(T)∥S∥_{cb}.

We interpret a matrix T = [T_{ij}] ∈ M_{n}(Γ_{2}(X, Y)) as a mapping from X to M_{n}(Y): [T_{ij}](x) := [T_{ij}(x)]. T has a factorization in completely bounded mappings

X →^{T_{2}} M_{1,n}(C_{H}) →^{T_{1}} M_{n}(Y).

Again, we define

γ_{2}(T) := inf ∥T_{1}∥_{cb}∥T_{2}∥_{cb},

where the infimum is taken over all factorizations. So we get an operator space structure on Γ_{2}(X, Y) [ER91, Cor. 5.4].

Let X, Y be operator spaces and Y_{0} an operator subspace of Y. Then the inclusion Γ_{2}(X, Y_{0}) ↪ Γ_{2}(X, Y) is completely isometric [ER91, Prop. 5.2].

Let X, Y be operator spaces. It is well known that every linear map

T : X → Y^{∗}

defines a linear functional

f_{T} : Y ⊗ X → ℂ

via

⟨f_{T} , y ⊗ x⟩ := ⟨T (x), y⟩.

This identification determines the complete isometry [ER91, Thm. 5.3] [Ble92b, Thm. 2.11]

Γ_{2}(X, Y^{∗}) =^{cb} (Y ⊗_{h} X)^{∗}.

Let X, Y, Z be operator spaces. We get a complete isometry

Γ_{2}(Y ⊗_{h} X, Z) =^{cb} Γ_{2}(X, Γ_{2}(Y, Z))

via the mapping

T ↦ T̃
T̃(x)(y) := T (y ⊗ x)

[ER91, Cor. 5.5].

5 Multiplicative Structures

For an abstract C^{∗}-algebra the GNS construction provides a concrete representation of its elements as bounded operators on a Hilbert space. For non-selfadjoint algebras there is, hitherto, no analogue in the framework of classical functional analysis. But endowed with an operator space structure (which is compatible with the multiplicative structure), these non-selfadjoint algebras do have a representation in some B(H) (theorem of Ruan type for operator algebras).

The so-called operator modules (over algebras) are also characterized by Axioms of Ruan type; here, matrices whose entries are algebra elements take the place of the scalar ones. The corresponding morphisms are the completely bounded module homomorphisms, the most important properties of which make their appearance in Representation, Decomposition and Extension Theorems.

5.1 Operator modules

Let A_{1}, A_{2} ⊂ B(H) be C^{∗}-algebras with 𝟙_{H} ∈ A_{1}, A_{2}. A closed subspace X of B(H) is called a concrete (A_{1}, A_{2})-operator module , if A_{1}X ⊂ X and XA_{2} ⊂ X. Whenever A_{1} = A_{2} = A, we call X a concrete A-operator bimodule (cf. [ER88, p. 137]).

In analogy to the operator space or the operator algebra situation, there is an abstract characterization of operator modules (cf. [Pop00, Déf. 4.1]): Consider, as above, two unital C^{∗}-algebras A_{1}, A_{2} ⊂ B(H) with 𝟙_{H} ∈ A_{1}, A_{2}, and an (algebraic) (A_{1}, A_{2})-module X. We call X an abstract (A_{1}, A_{2})-operator module, if it carries an operator space structure satisfying the following axioms (of Ruan type):

\begin{aligned}(R1) & ∥axb∥_{m} & ≤ & ∥a∥∥x∥_{m}∥b∥ \\ (R2) & ∥( \begin{matrix}x & 0 \\ 0 & y \end{matrix} )∥_{m+n} & = & max\{∥x∥_{m}, ∥y∥_{n}\}, \end{aligned}

where m, n ∈ ℕ, a ∈ M_{m}(A_{1}), x ∈ M_{m}(X), y ∈ M_{n}(Y), b ∈ M_{m}(A_{2}).

For abstract operator modules holds a representation theorem of Ruan type (cf. [Pop00, Thm. 4.7]): Let V be an abstract (A_{1}, A_{2})-operator module. Then there exist a Hilbert space K, a complete isometry Θ : X ↪ B(K) and ^{∗}-representations π_{1}, π_{2} of A_{1} resp. A_{2} in B(K) such that:

Θ(axb) = π_{1}(a)Θ(x)π_{2}(b),

where x ∈ X, a ∈ A_{1}, b ∈ A_{2}. In case A_{1} = A_{2}, one can even choose π_{1} = π_{2}.

Basic examples of operator modules

Let A be a unital C^{∗}-algebra, X a normed space, and Y an operator space. Then B(X, A) resp. CB(Y, A) are operator spaces via the identifications M_{n}(B(X, A)) = B(X, M_{n}(A)) resp. M_{n}(CB(Y, A)) = CB(Y, M_{n}(A)). These become A-operator bimodules, when endowed with the natural module operations as follows ([ER88, p. 140]):

(a · φ · b)(x) = aφ(x)b

for all a, b ∈ A, φ ∈ B(X, A) resp. CB(Y, A), x ∈ X resp. x ∈ Y.

In the category of operator modules, the morphisms are the completely bounded module homomorphisms. For these we have a representation and an extension theorem. Representation theorem (cf. [Hof95, Kor. 1.4]): Let H be a Hilbert space, M a C^{∗}-algebra in B(H), and A, B C^{∗}-subalgebras of M. Then the following hold true:

(a) (cf. [Pau86, Thm. 7.4]) For each completely bounded (A, B)-module homomorphism Φ : M → B(H), there exist a Hilbert space K, a ^{∗}-representation π : M → B(K) and linear operators v, w ∈ B(H, K) sharing the following properties:

(a1) Φ(x) = v^{∗}π(x)w for all x ∈ M, i.e. (K; π; v^{∗}; w) is a representation of Φ

(a2) ∥Φ∥_{cb} = ∥v∥∥w∥

(a3) \overline{lin}(π(M)vH) = \overline{lin}(π(M)wH) = K

(a4) v^{∗}π(a) = av^{∗} for all a ∈ A and π(b)w = wb for all b ∈ B.

(b) (cf. also [Smi91, Thm.3.1]) If, in addition, M ⊂ B(H) is a von Neumann algebra and Φ : M → B(H) is a normal completely bounded (A, B)-module homomorphism, one can require the ^{∗}-representation π of part (a) to be normal. There exist families (a_{i})_{i∈I} and (b_{i})_{i∈I} in the commutant of A and B, respectively, with the following properties (the sums are to be taken in the WOT topology):

(b1) Φ(x) = ∑_{i∈I} a_{i}xb_{i} for all x ∈ M

(b2) ∑_{i∈I} a_{i}a_{i}^{∗} ∈ B(H), ∑_{i∈I} b_{i}^{∗}b_{i} ∈ B(H) and ∥Φ∥_{cb} = ∥ ∑_{i∈I} a_{i}a_{i}^{∗}∥^{\frac{1}{2}} ∥ ∑_{i∈I} b_{i}^{∗}b_{i}∥^{\frac{1}{2}} .

Extension theorem ([Wit84a, Thm. 3.1], cf. also [MN94, Thm. 3.4] and [Pau86, Thm. 7.2]): Let F be an injective C^{∗}-algebra, and let A, B ⊂ F be two unital C^{∗}-subalgebras. Consider furthermore two (A, B)-operator modules E_{0} and E with E_{0} ⊂ E. Then for each ϕ_{0} ∈ CB_{(A,B)}(E_{0}, F), there exists an extension ϕ ∈ CB_{(A,B)}(E, F) with ϕ|_{E_{0}} = ϕ_{0} and ∥ϕ∥_{cb} = ∥ϕ_{0}∥_{cb}.

The decomposition theorem for completely bounded module homomorphisms can be found in the corresponding chapter.

5.2 Completely bounded module homomorphisms

Let A, B ⊂ B(H) be C^{∗}-algebras with 𝟙_{H} ∈ A, B, and let E and F be two (A, B)-operator modules, i.e. (algebraic) A-left-B-right-modules. A mapping ϕ ∈ L(E, F) is called (A, B)- module homomorphism (in case A = B A-bimodule homomorphism) if

ϕ(axb) = aϕ(x)b

for all a ∈ A, b ∈ B, x ∈ E. Furthermore we will write CB_{(A,B)}(E, F) for the set of all completely bounded (A, B)- module homomorphisms between E and F. The space CB_{(A,B)}(E) with the composition of operators as multiplication is a Banach algebra.

Let A_{1}, A_{2} ⊂ B(H) be C^{∗}-algebras such that 𝟙_{H} ∈ A_{1}, A_{2}. Let further A ⊂ A_{1} ∩ A_{2} be a unital ^{∗}-subalgebra of A_{1} and A_{2} with 𝟙_{H} ∈ A. An A-bimodule homomorphism

Φ : A_{1} → A_{2}

is called self-adjoint if

Φ(x)^{∗} = Φ(x^{∗})

for all x ∈ A_{1}.

Dealing with completely bounded module homomorphisms, we have at our disposal a representation theorem, an extension theorem and the following decomposition theorem of Wittstock ([Wit81, Satz 4.5] and cf. [Pau86, Thm. 7.5]):

Let A, E and F be unital C^{∗}-algebras. Let moreover F be injective, and A be a subalgebra of E and F with 𝟙_{E} = 𝟙_{F} = 𝟙_{A}. Then for each self-adjoint completely bounded A-bimodule homomorphism ϕ : E → F, there exist two completely positive A-bimodule homomorphisms ϕ_{1} and ϕ_{2} sharing the properties ϕ = ϕ_{1} − ϕ_{2} and ∥ϕ∥_{cb} = ∥ϕ_{1} + ϕ_{2}∥_{cb} .

Consider two von Neumann algebras M and N, and two C^{∗}-algebras A_{1}, A_{2} ⊂ B(H), where 𝟙_{H} ∈ A_{1}, A_{2} and A_{1} ∪A_{2} ⊂ M ∩N. We then have the decomposition theorem of Tomiyama-Takesaki (cf. [Tak79, Def. 2.15]): Each operator ϕ ∈ CB_{(A_{1},A_{2})}(M, N) has a unique decomposition ϕ = ϕ^{σ} +ϕ^{s}, ϕ^{σ}, ϕ^{s} ∈ CB_{(A_{1},A_{2})}(M, N) normal resp. singular, where ∥ϕ^{σ}∥_{cb}, ∥ϕ^{s}∥_{cb} ≤ ∥ϕ∥_{cb}. We thus obtain the algebraically direct sum decomposition:

CB_{(A_{1},A_{2})}(M, N) = CB_{(A_{1},A_{2})}^{σ}(M, N) ⊕ CB_{(A_{1},A_{2})}^{s}(M, N).  (2)

Here, the notions ”normal” and ”singular”, repectively, are built in analogy to the framework of linear functionals on a von Neumann algebra M.^{24}

We list some basic facts about the spaces and mappings mentioned in (2):

(a) In case M = N, all the spaces in (2) are Banach algebras.

(b) The following properties of ϕ are hereditary for the normal part ϕ^{σ} and the singular part ϕ^{s} : completely positive, homomorphism, ^{∗}-homomorphism.

(c) If α ∈ Aut(M) and β ∈ Aut(N) are ^{∗}-automorphisms, we have (βϕα)^{σ} = βϕ^{σ}α and (βϕα)^{s} = βϕ^{s}α .

(d) For ϕ ∈ CB(B(H)), H a Hilbert space, we have: ϕ ∈ CB^{s}(B(H)) ⇔ ϕ|_{K(H)} ≡ 0 .

Let H be a Hilbert space, and let A_{1}, A_{2} ⊂ B(H) be two C^{∗}-algebras with 𝟙_{H} ∈ A_{1}, A_{2}. Then we obtain [Pet97, Prop. 4.2.5]:

CB_{(A_{1},A_{2})}^{σ}(B(H)) =^{cb} CB_{(A_{1},A_{2})}(K(H), B(H))  (3)
CB_{(A_{1},A_{2})}^{s}(B(H)) =^{cb} CB_{(A_{1},A_{2})}(Q(H), B(H))  (4)

completely isometrically, where Q(H) = B(H)/K(H) denotes the Calkin algebra.

Let X be an arbitrary operator space. Then the space of all completely bounded (A_{1}, A_{2})-module homomorphisms between X and B(H) can be identified with the dual of a module Haagerup tensor product in the following way ([Pet97, p. 67], cf. also [ER91, Cor. 4.6], [Ble92b, Prop. 2.3]):

24Let M_{∗} denote the (unique) predual of M. Then we have the ℓ_{1}-direct sum decomposition

M^{∗} = M_{∗} ⊕_{ℓ_{1}} (M^{∗})^{s} of M^{∗} into normal (i.e. w^{∗}-continuous) and singular functionals. [In the literature, one usually writes M_{∗}^{⊥} instead of M^{∗s}, corresponding to M_{∗}(= M^{∗σ}).] Analogously, an operator ϕ ∈ B(M, N), M, N von Neumann algebras, is called normal (i.e. w^{∗}-w^{∗}-continuous), if ϕ^{∗}(N_{∗}) ⊂ M_{∗}, and it is called singular, if ϕ^{∗}(N_{∗}) ⊂ M^{∗s}.

CB_{(A_{1},A_{2})}(X, B(H)) =^{cb} (R\overline{H} ⊗_{hA_{1}} X ⊗_{hA_{2}} C_{H})^{∗}

completely isometrically. Hence we see that CB(B(H)) itself and (looking at (3), (4)), just so, CB_{(A_{1},A_{2})}^{σ}(B(H)) and CB_{(A_{1},A_{2})}^{s}(B(H)) are dual operator spaces [Pet97, p. 70].

5.3 Operator algebras

In analogy to concrete operator spaces we define (cf. [BRS90, Def. 1.1]): An operator algebra is a closed, not necessarily self-adjoint subalgebra X of B(H) (H a Hilbert space). Example: For selfadjoint X we have the theory of C^{∗}-algebras. As in the operator space situation, one can also adopt an abstract point of view: here, this leads to considering Banach algebras which are operator spaces and are equipped with a multiplication compatible with the operator space structure. These provide an abstract characterization of the (concrete) operator algebras (cf. below: analogue of Ruan’s theorem for operator algebras).

If X, Y, Z are operator spaces, and if Φ : X × Y → Z is bilinear, we can define another bilinear map in the following way (cf.: Amplification of bilinear mappings):

Φ^{(n,l)} : M_{n,l}(X) × M_{l,n}(Y) → M_{n}(Z)
([x_{ij}] , [y_{jk}]) ↦ [∑_{j=1}^{l} Φ(x_{ij}, y_{jk})] (l, n ∈ ℕ).

This is called the bilinear amplification^{25} of Φ.

Φ is called completely bounded if ∥Φ∥_{cb} := sup_{n}∥Φ^{(n,n)}∥ < ∞, and completely contractive if ∥Φ∥_{cb} ≤ 1.^{26} Compare this definition with the approach presented in Completely bounded bilinear Mappings.

[In the sequel, for Banach algebras with unit e, we will require ∥e∥ = 1.] An operator space (X, ∥·∥_{n}) with a bilinear, associative and completely contractive map m : X×X → X, the multiplication, is called an abstract operator algebra (cf. [BRS90, Def. 1.4]). Here, the multiplication on M_{n}(X) is just the matrix multiplication m_{n}. In the unital case m is automatically associative [BRS90, Cor. 2.4].

We have an analogue of Ruan’s theorem ([Ble95, Thm. 2.1], cf. also [BRS90, Thm. 3.1]): Let A be a unital Banach algebra and an operator space. Then A is completely isometrically isomorphic to an operator algebra if and only if the multiplication on A is completely contractive. This yields the following stability result:

25In the literature, e.g. in [BRS90, p. 190], the bilinear amplification Φ^{(n,n)} is often referred to as the amplification and is noted by Φ^{(n)}.

26In order to define the notion of complete boundedness of bilinear mappings, it suffices to consider only the Φ^{(n,n)} instead of all the Φ^{(n,l)}; this definition is usually chosen in the literature about completely bounded bi- and, analogously, multilinear maps [BRS90, p. 190], [CES87, p. 281].

1.) The quotient of an operator algebra with a closed ideal is again an operator algebra [BRS90, Cor. 3.2]. With this at hand, one deduces another important theorem on hereditary properties of operator algebras: 2.) The class of operator algebras is stable under complex interpolation [BLM95, (1.12), p. 320].

Adopting a more general point of view than in the Ruan type Representation Theorem above, one obtains the following [Ble95, Thm. 2.2]: Let A be a Banach algebra and an operator space. Then A is completely isomorphic to an operator algebra if and only if the multiplication on A is completely bounded. (cf. the chapter Examples!)

Basic examples of operator algebras are provided by the completely bounded maps on some suitable operator spaces. More precisely, for an operator space X, one obtains the following [Ble95, Thm. 3.4]: CB(X) with the composition as multiplication, is completely isomorphic to an operator algebra if and only if X is completely isomorphic to a column Hilbert space. – An analogue statement holds for the isometric case.

In the following result, for operator algebras A and B, the assumption that A and B be (norm-) closed, is essential (in contrast to the whole rest) [ER90b, Prop. 3.1]: A unital complete isometry φ between A ⊂ B(H) and B ⊂ B(K) (H, K Hilbert spaces), where 𝟙_{B(H)} ∈ A, 𝟙_{B(K)} ∈ B, is already an algebra homomorphism.

Examples

In the sequel we will equip the spaces ℓ_{p} (1 ≤ p ≤ ∞) with the pointwise product and consider them as Banach algebras. We will further consider the Schatten classes S_{p} (1 ≤ p ≤ ∞) endowed with either the usual multiplication or the Schur product.

1. The space ℓ_{2} [BLM95, Thm. 2.1]

With the following operator space structures, ℓ_{2} is completely isometrically isomorphic to an operator algebra: R_{ℓ_{2}}, C_{ℓ_{2}}, OH_{ℓ_{2}}, R_{ℓ_{2}} ∩ C_{ℓ_{2}}, MAX_{ℓ_{2}}. More generally: The space ℓ_{2} is completely isometrically isomorphic to an operator algebra, if endowed with an operator space structure which dominate dominates both R_{ℓ_{2}} and C_{ℓ_{2}}.

With the following operator space structures, ℓ_{2} is not completely isomorphic to an operator algebra: R_{ℓ_{2}} + C_{ℓ_{2}}, MIN_{ℓ_{2}}. More generally: The space ℓ_{2} is not completely isomorphic to an operator algebra, if endowed with an operator space structure which is dominate dominated by both R_{ℓ_{2}} and C_{ℓ_{2}}.

2. The spaces^{27} MIN (ℓ_{p}), MAX (ℓ_{p}) and Oℓ_{p} = (MIN (ℓ_{∞}), MAX (ℓ_{1}))_{\frac{1}{p}}

In the extreme cases p = 1 resp. p = ∞, we have two opposite results [BLM95, Thm. 3.1]:

27For the construction of the operator spaces Oℓ_{p} compare the chapter on complex interpolation.

(a) Equipped with any operator space structure, ℓ_{1} is completely isometrically isomorphic to an operator algebra.

(b) MIN (ℓ_{∞}) is, up to complete isomorphy, the only operator algebra structure on ℓ_{∞}.

For 1 ≤ p ≤ ∞ the following holds true (cf. [BLM95, Thm. 3.4]):

(a) MIN (ℓ_{p}) is completely isomorphic to an operator algebra if and only if p = 1 or p = ∞.

(b) MAX (ℓ_{p}), in case 1 ≤ p ≤ 2, is completely isometrically isomorphic to an operator algebra. In all the other cases MAX (ℓ_{p}) is not completely isomorphic to an operator algebra.

On the contrary, the operator space structure on the ℓ_{p} spaces obtained via complex interpolation always defines an operator algebra structure. More precisely [BLM95, Cor. 3.3]: For each 1 ≤ p ≤ ∞, Oℓ_{p} is completely isometrically isomorphic to an operator algebra.

3. The Schatten classes S_{p}

We write OS_{p} for the operator space structure defined on S_{p} by G. Pisier. This operator space structure is obtained by complex interpolation between S_{∞} = K(ℓ_{2}) and S_{1} = K(ℓ_{2})^{∗}.

(a) Let us first consider the usual product on the Schatten classes S_{p}. Here we have the following negative result [BLM95, Thm. 6.3]: For each 1 ≤ p < ∞, the operator space OS_{p} with the usual product is not completely isomorphic to an operator algebra.

(b) Consider now the Schur product on the Schatten classes S_{p}. Here we obtain positive results, even for different operator space structures: (b1) MAX (S_{p}) with the Schur product is, in case 1 ≤ p ≤ 2, completely isomet-

rically isomorphic to an operator algebra [BLM95, Thm. 6.1]. (b2) OS_{p} with the Schur product is, in case 2 ≤ p ≤ ∞, completely isometrically

isomorphic to an operator algebra [BLM95, Cor. 6.4].

Caution is advised: The space OS_{1} (and likewise OS_{1}^{op}), whether endowed with the usual or the Schur product, is not completely isomorphic to an operator algebra [BLM95, Thm. 6.3].

6 Tensor Products

An operator space tensor product is an operator space whose structure is deduced from the operator space structure of the factors. Operator space tensor products are defined for all operator spaces and have functorial properties. On the tensor product of two fixed operator spaces one usually considers operator space norms which are cross norms.

A lot of spaces, especially spaces of mappings, may be considered as operator space tensor products of simpler ones. The theory of operator space tensor products follows the lines of the theory of tensor products of Banach spaces. But at some points tensor products of operator spaces have new properties not found for tensor products of Banach spaces or even better properties as their counterparts. So in some cases the theory of operator space tensor products gives solutions to problems not solvable within the theory of Banach spaces (cp. [ER90a, Thm. 3.2]).

The Haagerup tensor product ⊗_{h} has a variety of applications in the theory of operator spaces and completely bounded operators.

The injective operator space tensor product ⊗^{∨} is the least^{28} and the projective operator space tensor product ⊗^{∧} is the greatest ^{29}among all operator space tensor products. [BP91, Prop. 5.10].

On the algebraic tensor product X ⊗ Y of operator spaces X, Y one can compare an operator space tensor norm ∥ · ∥_{α} with the injective tensor norm ∥ · ∥_{λ} and the projective tensor norm ∥ · ∥_{γ} of normed spaces:

∥ · ∥_{λ} ≤ ∥ · ∥_{∨,1} ≤ ∥ · ∥_{α,1} ≤ ∥ · ∥_{∧,1} ≤ ∥ · ∥_{γ}.

6.1 Operator space tensor products

An operator space tensor product is the completion of the algebraic tensor product with respect to an operator space tensor norm.

An operator space tensor norm ∥ · ∥_{α} is defined for each pair (X, Y) of operator spaces and endows their algebraic tensor product X ⊗ Y with the structure of an matrix normed space (X ⊗ Y, ∥ · ∥_{α}) thuch that the following two properties 1 and 2 [BP91, Def. 5.9]. hold.

The completion is called the α-operator space tensor product of X and Y and is denoted by X ⊗_{α} Y.

1. For the complex numbers holds

ℂ ⊗_{α} ℂ = ℂ.

2. For all S ∈ CB(X_{1}, X_{2}) and T ∈ CB(Y_{1}, Y_{2}) the operator S ⊗ T : X_{1} ⊗ Y_{1} → X_{2} ⊗ Y_{2} has a continuous extension

S ⊗_{α} T ∈ CB(X_{1} ⊗_{α} Y_{1}, X_{2} ⊗_{α} Y_{2}).

The bilinear mapping

⊗_{α} : CB(X_{1}, X_{2}) × CB(Y_{1}, Y_{2}) → CB(X_{1} ⊗_{α} Y_{1}, X_{2} ⊗_{α} Y_{2})
(S, T) ↦ S ⊗_{α} T

28i.e. the injective operator space tensor norm is minimal among all operator space tensor norms.

29i.e. the projective operator space tensor norm is maximal among all operator space tensor norms.

is jointly completely contractive.^{30}

Property 2 may be replaced by the assumptions 3 and 4.^{31}

3. An operator space tensor product ⊗_{α} is functorial: For all S ∈ CB(X_{1}, X_{2}) and T ∈ CB(Y_{1}, Y_{2}) the operator S ⊗ T : X_{1} ⊗ Y_{1} → X_{2} ⊗ Y_{2} has a continuous extension

S ⊗_{α} T ∈ CB(X_{1} ⊗_{α} Y_{1}, X_{2} ⊗_{α} Y_{2}),

and

∥S ⊗_{α} T ∥_{cb} ≤ ∥S∥_{cb}∥T ∥_{cb}.

Remark: This is indeed an equality ∥S ⊗_{α} T ∥_{cb} = ∥S∥_{cb}∥T ∥_{cb}.

4. The algebraic shuffle isomorphism M_{p}(X) ⊗ M_{q}(Y) ≅ M_{pq}(X ⊗ Y) has a continuous extension to a complete contraction:

𝕄_{p}(X) ⊗_{α} 𝕄_{q}(Y) → 𝕄_{pq}(X ⊗_{α} Y).

This complete contraction is called the shuffle map of the α-operator space tensor product.

30i.e., let [S_{ij}] ∈ M_{p}(CB(X_{1}, X_{2})), [T_{kl}] ∈ M_{q}(CB(Y_{1}, Y_{2})), p, q ∈ ℕ, then the norm of the linear operator

[S_{ij} ⊗_{α} T_{kl}] ∈ M_{pq}(CB(X_{1} ⊗_{α} Y_{1}, X_{2} ⊗_{α} Y_{2}))

is estimated by

∥[S_{ij} ⊗ T_{kl}]∥_{cb} ≤ ∥[S_{ij}]∥_{cb}∥[T_{kl}]∥_{cb}.

Remark: This is indeed an equality.

31(2) ⇒(3),(4): Condition 3 is a special case of 2. Let I ∈ M_{p}(CB(𝕄_{p}(X), X)), J ∈ M_{q}(CB(𝕄_{q}(Y), Y)) be matrices, which are algebraically the the identical mappings of the vector spaces M_{p}(X) respectivly M_{q}(Y). By assumption 2 we have

I ⊗_{α} J ∈ M_{pq}(CB(𝕄_{p}(X) ⊗_{α} 𝕄_{q}(Y), X ⊗_{α} Y))
= M_{1}(CB(𝕄_{p}(X) ⊗_{α} 𝕄_{q}(Y), 𝕄_{pq}(X ⊗_{α} Y))),
∥I ⊗_{α} J∥_{cb} ≤ ∥I∥_{cb}∥J∥_{cb} = 1.

Now I ⊗_{α} J : 𝕄_{p}(X) ⊗_{α} 𝕄_{q}(Y) → 𝕄_{pq}(X ⊗_{α} Y) is the shuffle-map in 4.

(3),(4)⇒(2): Let [S_{ij}] ∈ M_{p}(CB(X_{1}, X_{2})), [T_{kl}] ∈ M_{q}(CB(Y_{1}, Y_{2})), p, q ∈ ℕ, and S ∈ CB(X_{1}, 𝕄_{p}(X_{2})), T ∈ CB(Y_{1}, 𝕄_{q}(Y_{2})) the corresponding operators. By 3 holds

S ⊗_{α} T ∈ CB(X_{1} ⊗_{α} Y_{1}, 𝕄_{p}(X_{2}) ⊗_{α} 𝕄_{q}(Y_{2})),
∥S ⊗_{α} T ∥_{cb} ≤ ∥S∥_{cb}∥T ∥_{cb} = ∥[S_{ij}]∥_{cb}∥[T_{kl}]∥_{cb}.

We apply the shuffle map A : 𝕄_{p}(X_{2}) ⊗_{α} 𝕄_{q}(Y_{2}) → 𝕄_{pq}(X_{2} ⊗_{α} Y_{2}) and obtain from 4

[S_{ij} ⊗_{α} T_{kl}] = A(S ⊗_{α} T) ∈ M_{1}(CB(X_{1} ⊗_{α} Y_{1}, 𝕄_{pq}(X_{2} ⊗_{α} Y_{2}))) = M_{pq}(CB(X_{1} ⊗_{α} Y_{1}, X_{2} ⊗_{α} Y_{2}),
∥[S_{ij} ⊗_{α} T_{kl}]∥_{cb} ≤ ∥S ⊗_{α} T ∥_{cb} ≤ ∥[S_{ij}]∥_{cb}∥[T_{kl}]∥_{cb}.

Hence ⊗_{α} is jointly completely bounded.

Condition 4 is equivalent to the following two conditions: The shuffle mappings

𝕄_{p}(X) ⊗_{α} Y → 𝕄_{p}(X ⊗_{α} Y),
X ⊗_{α} 𝕄_{q}(Y) → 𝕄_{q}(X ⊗_{α} Y)

are completely contractive.

Operator space tensor producte may have further special properties:

An operator space tensor product ⊗_{α} is called

symmetric, if X ⊗_{α} Y =^{cb} Y ⊗_{α} X is a complete isometry;

associative, if (X ⊗_{α} Y) ⊗_{α} Z =^{cb} X ⊗_{α} (Y ⊗_{α} Z) is acomplete isometry;

injective, if for all subspaces X_{1} ⊂ X, Y_{1} ⊂ Y the map X_{1} ⊗_{α} Y_{1} ↪ X ⊗_{α} Y is acomplete isometry;

projective, if for all subspaces X_{1} ⊂ X, Y_{1} ⊂ Y the map X ⊗_{α} Y → X/X_{1} ⊗_{α} Y/Y_{1} is a complete quotient map;

self dual, if the algebraic embedding X^{∗} ⊗ Y^{∗} ⊂ (X ⊗_{α} Y)^{∗} has a completely isometric extension X^{∗} ⊗_{α} Y^{∗} ⊂ (X ⊗_{α} Y)^{∗}.

In many applications one finds the Haagerup-tensor product. It is not symmetric, but associative, injective, projective and self dual.

cross norms

Sometimes one considers an operator space norm on the algebraic tensor product of two fixed operator spaces. Then one usually demands that this norm and its dual norm are at least cross norms. Operator space tensor norms always have these properties.

An operator space norm ∥ · ∥_{α} on the algebraic tensor product X ⊗ Y of two operator spaces X and Y is said to be a cross norm, if

∥x ⊗ y∥_{α,pq} = ∥x∥_{p} ∥y∥_{q}

for all p, q ∈ ℕ, x ∈ M_{p}(X), y ∈ M_{q}(Y) holds.

For cross norms ℂ ⊗_{α} X =^{cb} X is completely isometric.

For an operator space norm ∥ · ∥_{α} on the algebraic tensor product of two fixed operator spaces X and Y one usually asks for the following three properties (i)–(iii).^{32} (i) ∥ · ∥_{α} is a cross norm.

32The conditions (i)-(iii) are equivalent to the following: The bilinear maps

X × Y → X ⊗_{α} Y,  (x, y) ↦ x ⊗ y X^{∗} × Y^{∗} → (X ⊗_{α} Y)^{∗},  (φ, ψ) ↦ φ ⊗ ψ are jointly completely bounded.

(ii) Let φ ∈ X^{∗}, ψ ∈ Y^{∗} be linear functionals and

φ ⊗ ψ : X ⊗ Y → ℂ
⟨x ⊗ y, φ ⊗ ψ⟩ := ⟨x, φ⟩⟨y, ψ⟩

where x ∈ X, y ∈ Y. their tensorproduct. The tensor product φ ⊗ ψ has a continuous linear extension to X ⊗_{α} Y.

Then the dual operator space norm ∥ · ∥_{α^{∗}} is defined on the algebraic tensor product X^{∗} ⊗ Y^{∗} by the algebraic embedding

X ⊗ Y ⊂ (X^{∗} ⊗_{α} Y^{∗})^{∗}.

(iii) The dual operator space norm ∥ · ∥_{α^{∗}} is a cross norm.

There is a smallest operator space norms among the operator space norms on X ⊗ Y, for which ∥ · ∥_{α} and the dual norm ∥ · ∥_{α^{∗}} are cross norms. This is the injective operator space tensor norm ∥ · ∥_{∨}. [BP91, Prop. 5.10].

There is a greatest operator space norm among the operator space norms on X ⊗ Y, for which ∥ · ∥_{α} and the dual norm ∥ · ∥_{α^{∗}} are cross norms. This is the projective operator space tensor norm ∥ · ∥_{∧} [BP91, Prop. 5.10].

On the algebraic tensor product X ⊗Y one can compare the operator space norms ∥·∥_{α} for which ∥ · ∥_{α} and the dual norm ∥ · ∥_{α^{∗}} are cross norms with the injective tensor norm ∥ · ∥_{λ} and the projective tensor norm ∥ · ∥_{γ} of normed spaces:

∥ · ∥_{λ} ≤ ∥ · ∥_{∨,1} ≤ ∥ · ∥_{α,1} ≤ ∥ · ∥_{∧,1} ≤ ∥ · ∥_{γ}.

6.2 Injective operator space tensor product

The representations of two operator spaces X in B(H) and Y in B(K) yield a representation of the algebraic tensor product of X and Y in B(H ⊗_{2} K). The operator space structure obtained in this way turns out to be independent of the representations chosen. It is called the injectiveoperator space tensor product of X and Y and is denoted by X ⊗^{∨} Y [BP91, p. 285]. Hence, in the case of C^{∗}–algebras, the injective operator space tensor product and the minimal C^{∗}–tensor product coincide.^{33}

By means of the duality of tensor products we obtain a formula [BP91, Thm. 5.1] for the injective operator space tensor norm of an element u ∈ M_{n}(X ⊗ Y) which is representation free:

∥u∥_{∨} = sup ∥⟨u, φ ⊗ ψ⟩∥_{M_{nkl}},

33For this reason the injective operator space tensor product is also called spatial tensor product of operator spaces and denoted by X ⊗_{min} Y .

where k, l ∈ ℕ, φ ∈ Ball(M_{k}(X^{∗})) and ψ ∈ Ball(M_{l}(Y^{∗})).

Interpreting, as is usual, the elements of the algebraic tensor product as finite rank operators we have the completely isometric embeddings [BP91, Cor. 5.2]

X ⊗^{∨} Y ↪ CB(X^{∗}, Y) resp. X ⊗^{∨} Y ↪ CB(Y^{∗}, X).

The injective operator space tensor norm is the least cross norm whose dual norm again is a cross norm.

The injective operator space tensor product is symmetric, associative and injective. But it is not projective [BP91, Cor. 5.2].

The injective norm is the dual norm of the projective operator space tensor norm [BP91, Thm. 5.6]; but the projective operator space tensor norm is not in general the dual of the injective operator space tensor norm even if one of the two spaces involved is finite dimensional [ER90a, p. 168], [ER91, p. 264].

Some formulae for the injective operator space tensor product

If E and F are normed spaces, then [BP91, Prop. 4.1]

MIN(E) ⊗^{∨} MIN(F) =^{cb} MIN(E ⊗_{λ} F)

holds completely isometrically.

6.2.1 Exact operator spaces

We consider now the exact sequence

0 ↪ K(ℓ_{2}) ↪ B(ℓ_{2}) → Q(ℓ_{2}) → 0.

An operator space X is said to be exact [Pis95, §1], if the short sequence of injective tensor products

0 ↪ X ⊗^{∨} K(ℓ_{2}) ↪ X ⊗^{∨} B(ℓ_{2}) → X ⊗^{∨} Q(ℓ_{2}) → 0

is again exact. Then tensorizing with such an operator space preserves the exactness of arbitrary exact sequences of C^{∗}-algebras (for C^{∗}-algebras cf [Kir83]). Obviously, all finite dimensional operator spaces are exact.

Exactness is inherited by arbitrary subspaces. The injective tensor product of two exact operator spaces is again exact. For an exact space X we are given a degree of exactness by the quantity

ex(X) = ∥X ⊗^{∨} Q(H) → (X ⊗^{∨} B(H))/(X ⊗^{∨} K(H))∥.

We have 1 ≤ ex(X) < ∞ [Pis95, §1], because the mapping

(X ⊗^{∨} B(H))/(X ⊗^{∨} K(H)) → X ⊗^{∨} Q(H)

is a complete contraction. For non-exact operator spaces X we put ex(X) = ∞. For an exact C^{∗}-algebra ^{34} A we have ex(A) = 1. For an operator space X we have:

ex(X) = sup{ex(L) : L ⊂ X, dim L < ∞}.

so we can confine our examinations to finite dimensional spaces. From this it is also immediate that: ex(X_{0}) ≤ ex(X) if X_{0} ⊂ X. One has for finite dimensional operator spaces X_{1} and X_{2} the complete variant of the Banach-Mazur distance

d_{CB}(X_{1}, X_{2}) = inf{∥φ∥_{cb}∥φ^{−1}∥_{cb}}

( the infimum is taken over all isomorphisms φ from X_{1} to X_{2} ). Via this Banach-Mazur distance we can define the quantity

d_{SK}(X) := inf{d_{CB}(X, L), dim(L) = dim(X), L ⊂ M_{n}, n ∈ ℕ}^{35}.

According to [Pis95, Thm. 1] ex(X) = d_{SK}(X) holds, and ex(X) ≤ \sqrt{dim(X)}.

6.3 Projective operator space tensor product

The projective operator space tensor product X ⊗^{∧} Y of two operator spaces X and Y is characterized by the following complete isometry:

(X ⊗^{∧} Y)^{∗} =^{cb} CB(X, Y^{∗}) =^{cb} CB(Y, X^{∗}).

where linear mappings are identified in the usual way with bilinear forms.

One can also characterize the projective operator space tensor product by the following universal property [BP91, Def. 5.3]

CB(X ⊗^{∧} Y, Z) =^{cb} JCB(X × Y; Z),

where Z is an operator space.

Here, JCB(X ×Y; Z) denotes the operator space of jointly completely bounded bilinear mappings.

One also has an explicit expression for the projective operator space tensor norm of an element u ∈ M_{n}(X ⊗ Y): (cf. [ER91, Formel (2.10)])

∥u∥_{∧} = inf {∥α∥∥x∥_{p}∥y∥_{q}∥β∥ : u = α(x ⊗ y)β} ,

where p, q ∈ ℕ, x ∈ M_{p}(X), y ∈ M_{q}(Y) and α ∈ M_{n,pq}, β ∈ M_{pq,n}.

The projective operator space tensor norm is symmetric, associative and projective [ER91, p. 262]. But it is not injective.

The projective operator space tensor norm is the greatest operator space tensor norm which is a cross norm [BP91, Thm. 5.5].

Its dual norm is the injective operator space tensor norm [BP91, Thm. 5.6]; but the projective operator space tensor norm is not in general the dual of the injective operator space tensor norm even if one of the two spaces involved is finite dimensional [ER90a, p. 168], [ER91, p. 264].

34 A characterization of exact C^{∗}-algebras is given in [Kir94] and [Kir95].

35 The abbreviation SK stands for Subspaces of K(H)

Some formulae for the projective operator space tensor product

1. If E and F are normed spaces, we have [BP91, Prop. 4.1]

MAX(E) ⊗^{∧} MAX(F) =^{cb} MAX(E ⊗_{γ} F)

completely isometrically.

2. Taking the projective operator space tensor product of various combinations of the column Hilbert spaces C, the row Hilbert spaces R and an arbitrary operator space X one obtains the following completely isometric identifications: [ER91], [Ble92b, Prop. 2.3]

(i) X ⊗^{∧} C_{H} =^{cb} X ⊗_{h} C_{H}

(ii) R_{H} ⊗^{∧} X =^{cb} R_{H} ⊗_{h} X

(iii) C_{H} ⊗^{∧} C_{K} =^{cb} C_{H} ⊗^{∨} C_{K} =^{cb} C_{H} ⊗_{h} C_{K} =^{cb} C_{H⊗_{2}K}

(iv) R_{H} ⊗^{∧} R_{K} =^{cb} R_{H} ⊗^{∨} R_{K} =^{cb} R_{H} ⊗_{h} R_{K} =^{cb} R_{H⊗_{2}K}

(v) R\overline{H} ⊗^{∧} C_{K} =^{cb} R\overline{H} ⊗_{h} C_{K} =^{cb} T (H, K)

(vi) X ⊗^{∧} T (H, K) =^{cb} R\overline{H} ⊗^{∧} X ⊗^{∧} C_{K} =^{cb} R\overline{H} ⊗_{h} X ⊗_{h} C_{K}

(vii) CB(X, B(K, H)) =^{cb} (R\overline{H} ⊗^{∧} X ⊗^{∧} C_{K})^{∗} =^{cb} (R\overline{H} ⊗_{h} X ⊗_{h} C_{K})^{∗},

where the space of trace class operators T (H, K) is endowed with its natural operator space structure: T (H, K) :=^{cb} K(K, H)^{∗}.

3. Let M and N be von Neumann algebras and denote by M\overline{⊗}N the von Neumann tensor product^{36}. For the preduals one has

M_{∗} ⊗^{∧} N_{∗} =^{cb} (M\overline{⊗}N)_{∗}

completely isometrically [ER90a].

Let G and H be locally compact topological groups and denote by VN (G), VN (H) the corresponding group von Neumann algebras.^{37} It is well-known that VN (G) \overline{⊗} VN (H) = V N(G × H). Since the Fourier algebra^{38} A(G) can be identified with the predual of the group von Neumann algebra VN (G) [Eym64], this implies that

A(G) ⊗^{∧} A(H) =^{cb} A(G × H)

36If M ⊂ B(H), N ⊂ B(K) are von Neumann algebras, M\overline{⊗}N is defined to be the closure in the weak operator topology of the algebraic tensor product M ⊗ N ⊂ B(H ⊗_{2} K).

37The group von Neumann algebra VN (G) of a locally compact group G is defined to be the von Neumann algebra generated by the left regular representation of G in B(L_{2}(G)).

38For a locally compact group G, the set {f ∗ ǧ|f, g ∈ L_{2}(G)} ⊂ C_{0}(G) turns out to be a linear space and even an algebra (with pointwise multiplication). Its completion with respect to the norm ∥u∥ = inf{∥f∥_{2}∥g∥_{2} | u = f ∗ ǧ} is a Banach algebra and is called the Fourier algebra of G.

holds completely isometrically^{39} [ER90a].

6.4 The Haagerup tensor product

The Haagerup tensor product was first introduced by Effros and Kishimoto [EK87] for C^{∗}-algebras generalizing the original work of U. Haagerup [Haa80].

The Haagerup tensor product X ⊗_{h} Y of two operator spaces X and Y is characterized by the complete isometry

(X ⊗_{h} Y)^{∗} =^{cb} CB(X × Y; ℂ),

where bilinear forms are identified with linear maps in the usual fashion.

We can also characterize the Haagerup tensor product by the following universal property: For an operator space Z we have

CB(X ⊗_{h} Y; Z) =^{cb} CB(X × Y; Z)

completely isometrically.

Here, CB(X × Y; Z) denotes the operator space of completely bounded bilinear mappings.

For operator spaces X and Y the Haagerup operator space tensor norm of u ∈ M_{n}(X ⊗ Y) is explicitly given by (cf. [ER91, Formel (2.11)], [BP91, Lemma 3.2])

∥u∥_{h} = inf ∥x∥_{n,l} ∥y∥_{l,n},

where l ∈ ℕ, x ∈ M_{n,l}(X), y ∈ M_{l,n}(Y) and u is the tensor matrix product u = x ⊙ y. The Haagerup tensor product X ⊗_{h} Y then of course is the completion of the algebraic tensor product X ⊗ Y with respect to this operator space tensor norm. There are several other useful formulae 2 3, 4 for the Haagerup norm.

The Haagerup tensor product is not symmetric as shown by concrete examples. But it is associative, injective [PS87, p. 272; Thm. 4.4], [BP91, Thm. 3.6], projektiv [ER91, Thm. 3.1] and selfdual [ER91, Thm. 3.2]. Thus the embedding

X^{∗} ⊗_{h} Y^{∗} ↪ (X ⊗_{h} Y)^{∗}

is a complete isometry.

The extension of the identity mapping on the algebraic tensor product of two operator spaces X, Y from the Haagerup tensor product into the injective tensor product is injective. One therefore obtains a canonical embedding

X ⊗_{h} Y ⊂ X ⊗^{∨} Y.

39If G is a locally compact abelian group, then A(G) is identified – via the Fourier transform – with L_{1}(Ĝ), where Ĝ denotes the dual group of G. Thus, for locally compact groups G and H, the identification A(G) ⊗^{∧} A(H) =^{cb} A(G × H) can be thought of as a non commutative analogue of the well-known classical identification L_{1}(G) ⊗_{γ} L_{1}(H) = L_{1}(G × H).

The complex interpolation of operator spaces and the Haagerup tensor product commute [Pis96, Thm. 2.3]. Let (X_{0}, X_{1}) and (Y_{0}, Y_{1}) be compatible pairs of operator spaces. Then (X_{0} ⊗_{h} Y_{0}, X_{1} ⊗_{h} Y_{1}) is a compatible pair of operator spaces and we have completely isometrically

(X_{0} ⊗_{h} Y_{0}, X_{1} ⊗_{h} Y_{1})_{ϑ} =^{cb} (X_{0}, X_{1})_{ϑ} ⊗_{h} (Y_{0}, Y_{1})_{ϑ}

for 0 ≤ ϑ ≤ 1.

On normed spaces there is no tensor norm which at the same time is associative, injective, projective and selfdual. The Haagerup tensor product can be interpreted as a generalization of the H-tensor product introduced by Grothendieck^{40} for normed spaces E and F [BP91, pp. 277-279, Prop. 4.1]. In fact, on the first matrix level we have:

MIN (E) ⊗_{h} MIN (F) = E ⊗_{H} F,
MAX (E) ⊗_{h} MAX (F) = E ⊗_{H^{∗}} F

isometrically. The non-associativity of the H-tensor product is reflected by the fact that in general MIN (E) ⊗_{h} MIN (F) and MIN (E ⊗_{H} F) are not completely isometric.

Some formulae for the Haagerup tensor product

1. Let A and B be C^{∗}-algebras in B(H). Then on the algebraic tensor product A ⊗ B the Haagerup tensor norm is explicitly given by

∥u∥_{h} := inf \{∥∑_{ν=1}^{n} a_{ν}a_{ν}^{∗}∥^{\frac{1}{2}} ∥∑_{ν=1}^{n} b_{ν}^{∗}b_{ν}∥^{\frac{1}{2}} : u = ∑_{ν=1}^{n} a_{ν} ⊗ b_{ν}\},

where n ∈ ℕ, a_{ν} ∈ A, b_{ν} ∈ B.

The Haagerup norm of ∑_{ν=1}^{n} a_{ν} ⊗ b_{ν} ∈ A ⊗ B equals the cb–norm of the elementary operator B(H) ∋ x ↦ ∑_{ν=1}^{n} a_{ν}xb_{ν}. The Haagerup tensor product A ⊗_{h} B is the completion of the algebraic tensor product A ⊗ B with respect to the above norm. The following more general definition in particular yields a completely isometric embedding A ⊗_{h} B ↪ CB(B(H)).

2. We have

∥u∥_{h} = inf{∑_{κ=1}^{k} ∥x_{κ}∥∥y_{κ}∥ : u = ∑_{κ=1}^{k} x_{κ} ⊙ y_{κ}}

where k, l ∈ ℕ, x_{κ} ∈ M_{n,l}(X), y_{κ} ∈ M_{l,n}(Y). In fact, one summand suffices [BP91, Lemma 3.2].

3. For elements u ∈ M_{n}(X ⊗ Y) in the algebraic tensor product there is an l ∈ ℕ and elements x ∈ M_{n,l}(X), y ∈ M_{l,n}(Y) such that

u = x ⊙ y
∥u∥_{h} = ∥x∥∥y∥.

40This tensor norm also is known as γ_{2} [Pis86].

The infimum occuring in the formula describing the norm in this case is actually a minimum [ER91, Prop. 3.5].

4. The Haagerup norm of an element u ∈ M_{n}(X ⊗_{h} Y) can also be expressed using a supremum:^{41}

∥u∥_{h} = sup ∥⟨u, φ ⊙ ψ⟩∥_{M_{n^{2}}} ,

where l ∈ ℕ, φ ∈ M_{n,l}(X^{∗}), ψ ∈ M_{l,n}(Y^{∗}), ∥φ∥ = ∥ψ∥ = 1 [ER91, Prop. 3.4].

5. From the definition of the Haagerup norm one easily deduces that the shuffle-map

𝕄_{p}(X) ⊗_{h} 𝕄_{q}(Y) → 𝕄_{pq}(X ⊗_{h} Y)

is a complete contraction. Hence the Haagerup tensor product enjoys property 2 of an operator space tensor product.

But the shuffle-map is not an isometry in general as shown by the following example:^{42}

𝕄_{n}(C_{l}) ⊗_{h} 𝕄_{n}(R_{l}) =^{cb} 𝕄_{l}(𝕄_{n}((T_{n})) ≠ 𝕄_{l}(𝕄_{n}(𝕄_{n})) =^{cb} 𝕄_{n}(𝕄_{n}(C_{l} ⊗_{h} R_{l})).

Since the bilinear mapping

⊗_{h} : CB(X_{1}, X_{2}) × CB(Y_{1}, Y_{2}) → CB(X_{1} ⊗_{h} X_{2}, Y_{1} ⊗_{h} Y_{2})
(S, T) ↦ S ⊗_{h} T

is contractive, it is jointly completely contractive.^{43}

In fact, using 8, we see that it is even completely contractive.

(φ ⊙ ψ)(x ⊗ y) = ⟨x ⊗ y, φ ⊙ ψ⟩ = [∑_{j=1}^{l}⟨x ⊗ y, φ_{ij} ⊗ ψ_{jk}⟩] = φ(x)ψ(y) ∈ M_{n}.

𝕄_{n}(C_{l}) ⊗_{h} 𝕄_{n}(R_{l}) =^{cb} C_{l}(𝕄_{n}) ⊗_{h} R_{l}(𝕄_{n}) =^{cb} 𝕄_{l}(𝕄_{n} ⊗_{h} 𝕄_{n})

= 𝕄_{l}(C_{n} ⊗_{h} R_{n} ⊗_{h} C_{n} ⊗_{h} R_{n}) = 𝕄_{l}(C_{n} ⊗_{h} T_{n} ⊗_{h} R_{n}) =^{cb} 𝕄_{l}(M_{n}(T_{n})).

𝕄_{n}(𝕄_{n}(C_{l} ⊗_{h} R_{l}) = 𝕄_{n}(𝕄_{n}(𝕄_{l})) =^{cb} 𝕄_{l}(𝕄_{n}(𝕄_{n})).

41For x ∈ X, y ∈ Y, we have:

Here, we used the definitions of two fundamental notions in operator space theory: the tensor matrix multiplication φ ⊙ ψ of mappings φ, ψ and the joint amplification of the duality of tensor products.

42Algebraically, we have on both sides the same spaces of matrices. But on the left side one obtains the finer operator space structure T_{n} := 𝕄_{n}^{∗} of the trace class:

On the right side, we get the coarser operator space structure of the matrices M_{n}:

43This follows from the equivalence of the property 2 and the properties 3 and 4 of operator space tensor products.

6. For the row and column structure the shuffle-map even is a complete isometry. We have the Lemma of Blecher and Paulsen [BP91, Prop. 3.5]:

C_{n}(X) ⊗_{h} R_{n}(Y) =^{cb} 𝕄_{n}(X ⊗_{h} Y).

In many cases it suffices to prove a statement about the Haagerup tensor product on the first matrix level and then to deduce it for all matrix levels using the above formula.^{44} Here we list some special cases of the Lemma of Blecher and Paulsen:

C_{n} ⊗_{h} R_{n} =^{cb} 𝕄_{n},
C_{n} ⊗_{h} X =^{cb} C_{n}(X),
X ⊗_{h} R_{n} =^{cb} R_{n}(X),
C_{n} ⊗_{h} X ⊗_{h} R_{n} =^{cb} 𝕄_{n}(X).

7. In contrast to 6, for R_{n} ⊗_{h} C_{n} one obtains the finer operator space structure of the

trace class

T_{n} := 𝕄_{n}^{∗} =^{cb} R_{n} ⊗^{∧} C_{n} =^{cb} R_{n} ⊗_{h} C_{n}.

For an operator space X we have [Ble92b, Prop. 2.3]:

R_{n} ⊗_{h} X =^{cb} R_{n} ⊗^{∧} X,
X ⊗_{h} C_{n} =^{cb} C_{n} ⊗^{∧} X,
R_{n} ⊗_{h} X ⊗_{h} C_{n} =^{cb} T_{n} ⊗^{∧} X,
R_{n} ⊗_{h} X^{∗} ⊗_{h} C_{n} =^{cb} 𝕄_{n}(X)^{∗}.

8. By the very construction the bilinear mapping X × Y → X ⊗_{h} Y, (x, y) ↦ x ⊗ y is a complete contraction. Hence its amplification, the tensor matrix product

⊙_{h} : 𝕄_{n,l}(X) × 𝕄_{l,n}(Y) → 𝕄_{n}(X ⊗_{h} Y)
(x, y) ↦ x ⊙ y,

also is a complete contraction. The linearization of the tensor matrix product gives the complete contraction

𝕄_{n,l}(X) ⊗_{h} 𝕄_{l,n}(Y) → 𝕄_{n}(X ⊗_{h} Y).

44This method can be applied to obtain this complete isometry itself. It is easy to see that on the first matrix level we have

M_{1}(C_{n}(X) ⊗_{h} R_{n}(Y)) = M_{n}(X ⊗_{h} Y)

isometrically. From this the complete isometry follows – for all p ∈ ℕ we have:

M_{p}(C_{n}(X) ⊗_{h} R_{n}(Y)) = M_{1}(C_{p}(C_{n}(X)) ⊗_{h} R_{p}(R_{n}(Y))) = M_{1}(C_{pn}(X) ⊗_{h} R_{pn}(Y))

= M_{pn}(X ⊗_{h} Y) = M_{p}(𝕄_{n}(X ⊗_{h} Y))

isometrically .

9. The bilinear mapping^{45}

⊗_{h} : CB(X_{1}, X_{2}) × CB(Y_{1}, Y_{2}) → CB(X_{1} ⊗_{h} Y_{1}, X_{2} ⊗_{h} Y_{2})
(S, T) ↦ S ⊗_{h} T

is completely contractive^{46}and gives rise to a complete contraction

CB(X_{1}, X_{2}) ⊗_{h} CB(Y_{1}, Y_{2}) → CB(X_{1} ⊗_{h} Y_{1}, X_{2} ⊗_{h} Y_{2}).

10. Let H and K be Hilbert spaces. Taking the Haagerup tensor product of the column Hilbert space C and the row Hilbert space R one obtains completely isometrically the space of compact operators K resp. of trace class^{47} operators T [ER91, Cor. 4.4]:

R\overline{H} ⊗_{h} C_{K} =^{cb} T (H, K),
C_{K} ⊗_{h} R\overline{H} =^{cb} K(H, K).

This example also shows that the Haagerup tensor product is not symmetric.

6.5 Completely bounded bilinear mappings

In the case of bilinear mappings between operator spaces one has to distinguish between two different notions of complete boundedness: on one hand we have the jointly completely bounded [BP91, Def. 5.3 (jointly completely bounded)] and, on the other hand, the completely bounded bilinear mappings [CS87, Def. 1.1].

The class of completely bounded bilinear maps is is contained in the first one. These notions are in perfect analogy to those of bounded bilinear forms on normed spaces. For completely bounded bilinear mappings, we have at our disposal similar representation and extension theorems^{48} as in the case of completely bounded linear maps. There are two tensor products corresponding to the above two classes of bilinear mappings, namely the projective and the Haagerup tensor product. Depending on the class of bilinear maps, one uses different methods to define the amplification of a bilinear mapping Φ : X × Y → Z.

45The amplification of ⊗_{h} is nothing but the tensor matrix multiplication ⊙_{h} of operator matrices.

46Let S ∈ M_{n}(CB(X_{1}, X_{2})), T ∈ M_{n}(CB(Y_{1}, Y_{2})). For x ∈ M_{p,q}(X_{1}), y ∈ M_{q,p}(Y_{1}) we have

(S ⊙ T)^{(p)}(x ⊙ y) = (S^{(p,q)}(x)) ⊙ (T^{(q,p)}(y)), ∥(S ⊙ T)^{(p)}(x ⊙ y)∥_{M_{pn}(X_{2}⊗_{h}Y_{2})} ≤ ∥S^{(p,q)}(x)∥∥T^{(q,p)}(y)∥ ≤ ∥S∥_{cb}∥T ∥_{cb}∥x∥∥y∥.

By the definition of the Haagerup norm (in X_{1} ⊗_{h} Y_{1}) we obtain

∥(S ⊙ T)^{(p)}(x ⊙ y)∥_{M_{pn}(X_{2}⊗Y_{2})} ≤ ∥S∥_{cb}∥T ∥_{cb}∥x ⊙ y∥_{M_{p}(X_{1}⊗_{h}Y_{1})},

∥S ⊙_{h} T ∥_{cb} ≤ ∥S∥_{cb}∥T ∥_{cb}.

This means that ⊗_{h} is completely contractive.

47T is endowed with its natural operator space structure T (H, K) :=^{cb} K(K, H)^{∗}.

48Extension theorems for completely bounded bilinear (and, more generally, multilinear) maps can be derived from the injectivity of the Haagerup tensor product.

Amplification

In the literature, there are two different notions of an amplification of a bilinear mapping.

We shall call the first kind 1 of amplification the joint amplification. This joint amplification is needed to obtain a matrix duality – which is fundamental in the duality theory of operator spaces –, starting from an ordinary duality ⟨X, X^{∗}⟩.

The notion of joint amplification leads to the jointly completely bounded bilinear maps as well as the projective operator space tensor product.

We will speak of the second kind 2 of an amplification as the amplification of a bilinear mapping. This notion leads to the completely bounded bilinear maps and the Haagerup tensor product.

In the sequel, we will use the notation Φ : X × Y → Z for a bilinear mapping and Φ̃ : X ⊗ Y → Z for its linearization.

Both notions of an amplification of a bilinear map Φ are formulated in terms of the amplification of its linearization:

Φ̃^{(n)} : M_{n}(X ⊗ Y) → M_{n}(Z).

1. The joint amplification of Φ produces the bilinear mapping

Φ^{(p×q)} : (x, y) ↦ Φ̃^{(pq)}(x ⊗ y) = [Φ(x_{ij}, y_{kl})] ∈ M_{p}(M_{q}(Z)) = M_{pq}(Z).

of the operator matrices x = [x_{ij}] ∈ M_{p}(X) and y = [y_{kl}] ∈ M_{q}(Y). Here, the tensor product of operator matrices is defined via

x ⊗ y = [x_{ij}] ⊗ [y_{kl}] := [x_{ij} ⊗ y_{kl}] ∈ M_{pq}(X ⊗ Y) = M_{p}(M_{q}(X ⊗ Y)).

2. In the case of completely bounded bilinear maps one deals with the tensor matrix multiplication [Eff87]

x ⊙ y = [x_{ij}] ⊙ [y_{jk}] := [∑_{j=1}^{l} x_{ij} ⊗ y_{jk}] ∈ M_{n}(X ⊗ Y)

of operator matrices x = [x_{ij}] ∈ M_{n,l}(X) and y = [y_{jk}] ∈ M_{l,n}(Y).

For more formulae, see: tensor matrix multiplication.

The (n,l)-th amplification of a bilinear map Φ : X × Y → Z is defined by

Φ^{(n,l)} : M_{n,l}(X) × M_{l,n}(Y) → M_{n}(Z)
(x, y) ↦ Φ̃^{(n)}(x ⊙ y) = [∑_{j=1}^{l} Φ(x_{ij}, y_{jk})] ∈ M_{n}(Z)

for l, n ∈ ℕ, x = [x_{ij}] ∈ M_{n,l}(X), y = [y_{jk}] ∈ M_{l,n}(Y). In case n = l, we shortly write^{49}

Φ^{(n)} := Φ^{(n,n)} : M_{n}(x) ⊗ M_{n}(Y) → M_{n}(Z)
(x, y) ↦ Φ^{(n)}(x, y) := Φ̃^{(n)}(x ⊙ y).

49In the literature, the amplification of a bilinear map often is defined only for quadratic operator matrices and is called the amplification Φ^{(n)}.

Nevertheless, we will also be dealing with the amplification for rectangular matrices since this permits the formulation of statements where for fixed n ∈ ℕ all amplifications Φ^{(n,l)}, l ∈ ℕ, are considered.

Jointly complete boundedness

Let X, Y, Z be operator spaces. A bilinear mapping Φ : X × Y → Z is called jointly completely bounded [BP91, Def. 5.3 (jointly completely bounded)] if the norms of the joint amplifications of Φ are uniformly bounded:

∥Φ∥_{jcb} := sup ∥Φ^{(p×q)}(x ⊗ y)∥ < ∞,

where p, q ∈ ℕ, x ∈ Ball(M_{p}(X)), y ∈ Ball(M_{q}(Y)) [BP91, Def. 5.3]. The norm ∥Φ∥_{jcb} equals the norm ∥Φ̃∥_{cb} of the linearization

Φ̃ : X ⊗^{∧} Y → Z

on the projective operator space tensor product.

JCB(X ×Y; Z) denotes the operator space consisting of the jointly completely bounded bilinear maps. One obtains a norm on each matrix level by the identification

M_{n}(JCB(X × Y; Z)) = JCB(X × Y; 𝕄_{n}(Z)).

We have

JCB(X × Y; Z) =^{cb} CB(X, CB(Y, Z)) =^{cb} CB(Y, CB(X, Z)).

completely isometrically. By taking the transposition Φ^{t}(y, x) := Φ(x, y) we obtain a complete isometry

JCB(X × Y; Z) =^{cb} JCB(Y × X; Z).

Complete boundedness

For the definition of the completely bounded bilinear maps we need the amplification Φ^{(n)}, the linearization Φ̃ : X ⊗ Y → Z. and the tensor matrix multiplication x ⊙ y of operator matrices x, y.

A bilinear mapping Φ : X × Y → Z, n ∈ ℕ is called completely bounded if

∥Φ∥_{cb} := sup ∥Φ^{(n)}(x, y)∥ < ∞

where n ∈ ℕ, x ∈ Ball(M_{n}(X)), y ∈ Ball(M_{n}(Y)).

The norm ∥Φ∥_{cb} equals the norm ∥Φ̃∥_{cb} of the linearization

Φ̃ : X ⊗_{h} Y → Z

on the Haagerup tensor product.

Furthermore, the norms ∥Φ∥_{n} are obtained using the tensor matrix products of all ^{50} rectangular matrices of n rows resp. n columns:

∥Φ∥_{n} := sup ∥Φ^{(n,l)}(x, y)∥

50Note that the norm of the bilinear map

Φ^{(n)} : M_{n}(x) ⊗ M_{n}(Y) → M_{n}(Z)

in general is smaller than the norm ∥Φ∥_{n}.

where l ∈ ℕ, x ∈ Ball(M_{n,l}(X)), y ∈ Ball(M_{l,n}(Y)).

We have

∥Φ∥_{cb} = sup {∥Φ∥_{n} : n ∈ ℕ} .

The norm ∥Φ∥_{n} equals the norm ∥Φ̃^{(n)}∥ of the amplification of the linearization

Φ̃^{(n)} : M_{n}(X ⊗_{h} Y) → M_{n}(Z)

on the Haagerup tensor product.

A bilinear form Φ is already seen to be completely bounded if ∥Φ∥_{1} < ∞. Then we have ∥Φ∥_{cb} = ∥Φ∥_{1}.^{51}

CB(X × Y; Z) denotes the operator space consisting of completely bounded bilinear maps. One obtains a norm on each matrix level using the identification

M_{n}(CB(X × Y; Z)) = CB(X × Y; 𝕄_{n}(Z)).

Corresponding to the completely bounded bilinear maps we have the linear maps which are completely bounded on the Haagerup tensor product. The identification

CB(X × Y; Z) =^{cb} CB(X ⊗_{h} Y; Z)

holds completely isometrically.

Completely bounded bilinear mappings are in particular jointly completely bounded. The embedding CB(X × Y; Z) ⊂ JCB(X × Y; Z) is a complete contraction.

The transpose Φ^{t}(y, x) := Φ(x, y) of a completely bounded bilinear mapping Φ in general is not completely bounded.^{52}

For completely bounded bilinear (and, more generally, multilinear) maps Φ ∈ CB(A × B; B(H)) we have some generalizations of Stinespring’s representation theorem.

Representation

Completely bounded bilinear forms were first studied on C^{∗}-algebras A, B [EK87]. For a bilinear form Φ : A × B → ℂ the following properties are equivalent:

(1) Φ is completely bounded.

(2) There is a constant c such that

| ∑_{j=1}^{l} Φ(a_{j}, b_{j})| ≤ c ∥ ∑_{j=1}^{l} a_{j}a_{j}^{∗}∥^{\frac{1}{2}} ∥ ∑_{j=1}^{l} b_{j}^{∗}b_{j}∥^{\frac{1}{2}}

for all l ∈ ℕ, a_{j} ∈ A, b_{j} ∈ B.

51More generally, the equation ∥Φ∥_{cb} = ∥Φ∥_{1}

holds for bilinear maps with values in a commutative C^{∗}-algebra A since every bounded linear map taking values in A is automatically completely bounded and ∥Φ∥_{cb} = ∥Φ∥ [Loe75, Lemma 1]. For bilinear maps Φ : X × Y → M_{n}(A) we have ∥Φ∥_{cb} = ∥Φ∥_{n}.

52To this corresponds the fact that the Haagerup tensor product is not symmetric.

(3) There is a constant c and states ω ∈ S(A), ρ ∈ S(B) such that

|Φ(a, b)| ≤ c ω(aa^{∗})^{\frac{1}{2}} ρ(b^{∗}b)^{\frac{1}{2}}

for all a ∈ A, b ∈ B.

(4) There exist ^{∗}-representations π_{ω} : A → B(H_{ω}) and π_{ρ} : A → B(H_{ρ}) with cyclic vectors ξ_{ω} ∈ H_{ω}, ξ_{ρ} ∈ H_{ρ} and an operator T ∈ B(H_{ω}, H_{ρ}) such that Φ(a, b) = ⟨T π_{ω}(a)ξ_{ω}, π_{ρ}(b)ξ_{ρ}⟩ for all a ∈ A, b ∈ B.

One can choose c = ∥T ∥ = ∥Φ∥_{cb} and ∥ξ_{ω}∥ = ∥ξ_{ρ}∥ = 1.^{53}

6.6 Module tensor products

So far the only module tensor product of operator modules that has been studied is the module Haagerup tensor product [Rua89] [BMP].

6.6.1 Module Haagerup tensor product

Let X be a right operator module over a C^{∗}-algebra A, Y a left A-operator module, and W an operator space. A bilinear mapping Ψ : X × Y → W is called balanced, if the equation

Ψ(x · a, y) = Ψ(x, a · y)

obtains for all x ∈ X, y ∈ Y, a ∈ A.

The module Haagerup tensor product is defined to be the operator space X ⊗_{hA} Y (which is unique up to complete isometry) together with a bilinear, completely contractive, balanced mapping

⊗_{hA} : X × Y → X ⊗_{hA} Y,

such that the following holds true: For each bilinear, completely bounded balanced map

Ψ : X × Y → W

there is a unique linear completely bounded map

Ψ̃ : X ⊗_{hA} Y → W

satisfying Ψ̃ ∘ ⊗_{hA} = Ψ and ∥Ψ̃∥_{cb} = ∥Ψ∥_{cb}.

The module Haagerup tensor product can be realized in different ways:

1. Let

N := lin{(x · a) ⊗ y − x ⊗ (a · y) | a ∈ A, x ∈ X, y ∈ Y }.

The quotient space (X ⊗_{h} Y)/\overline{N} with its canonical matrix norms is an operator space which satisfies the defintion of X ⊗_{hA} Y [BMP].

53For further references see [CS89, Sec. 4].

2. Let us denote by X ⊗_{A} Y the algebraic module tensor product, i.e. the quotient space (X ⊗_{alg} Y)/N. For n ∈ ℕ and u ∈ M_{n}(X ⊗_{A} Y), by

p_{n}(u) := inf {∥S∥∥T ∥ |u = [∑_{k=1}^{l} S_{ik} ⊗_{A} T_{kj}] , l ∈ ℕ, S ∈ M_{nl}(X), T ∈ M_{ln}(Y) }

we define a semi-norm on M_{n}(X ⊗_{A} Y). We obtain

Kern(p_{n}) = M_{n}(Kern(p_{1})),

and the semi-norms p_{n} give an operator space norm on (X ⊗_{A} Y)/Kern(p_{1}) [Rua89]. The completion of this space satisfies the definition of X ⊗_{hA} Y [BMP].

Examples

Let X be an operator space. Then the space of completely bounded (A_{1}, A_{2})- module homomorphisms between X and B(H) can be identified with the dual of a module Haagerup tensor product in the following way ([Pet97, p. 67], cf. also [ER91, Cor. 4.6], [Ble92b, Prop. 2.3]):

CB_{(A_{1},A_{2})}(X, B(H)) =^{cb} (R\overline{H} ⊗_{hA_{1}} X ⊗_{hA_{2}} C_{H})^{∗}

completely isometrically.

7 Complete Local Reflexivity

An operator space X is called completely locally reflexive [EJR98, §1], if to each finite dimensional subspace there is L ⊂ X^{∗∗} a net of completely contractive mappings φ_{α} : L → X that converges to the embedding L → X^{∗∗} in the point weak^{∗} topology.^{54}

This property is inherited by arbitrary subspaces. In general it is not preserved by quotients. In the case that for example the kernel is an M-ideal (e.g. the twosided ideal of a C^{∗}-algebra) and that the original space is completely reflexive we have that the quotient space is completely locally reflexive [ER94, Thm. 4.6].

Banach spaces are always locally reflexive (Principle of local reflexivity [Sch70]).

On the contrary, not all operator spaces are completely locally reflexive. For example the full C^{∗}-algebra of the free group on two generators C^{∗}(F_{2}) and B(ℓ_{2}) are not completely locally reflexive [EH85, p. 124-125].

An operator space X is completely locally reflexive, if and only if one (and then every) of the following conditions is satisfied for all finite dimensional operator spaces L [EJR98, §1, 4.4, 5.8]:

1. L ⊗^{∨} X^{∗∗} =^{cb} (L ⊗^{∨} X)^{∗∗}, where ⊗^{∨} denotes the injective operator space tensor product,

54 The difference to the definition of local reflexivity is the fact that the φ_{α} are not only supposed to be contractive, but even completely contractive.

2. CB(L^{∗}, X^{∗∗}) =^{cb} CB(L^{∗}, X)^{∗∗},

3. L^{∗} ⊗^{∧} X^{∗} =^{cb} (L ⊗^{∨} X)^{∗}, where ⊗^{∧} denotes the projective operator space tensor product and ⊗^{∨} denotes the injective operator space tensor product,

4. CN (X, L^{∗}) =^{cb} CI (X, L^{∗}), where CN (·, ·) denotes the completely nuclear and CI (·, ·) the completely integral mappings,

5. ι(φ) = ι(φ^{∗}) for all φ ∈ CI(X, L^{∗}).

In the conditions 1), 2) and 3) it suffices to check the usual isometry to prove the complete isometry.

Examples

The following classes of operator spaces are completely locally reflexive [EJR98, 6.1, 6.2], [EH85, Prop. 5.4]:

1. reflexive operator spaces (e.g. finite dimensional operator spaces L),

2. nuclear C^{∗}- algebras (e.g. K(H) or commutative C^{∗}- algebras),^{55}

3. preduals of von Neumann algebras, especially duals of C^{∗}-algebras (e.g. T (H) = K(H)^{∗} = B(H)_{∗}).

8 Completely Bounded Multilinear Mappings

Going beyond the linear case, one can introduce the concept of complete boundedness for multilinear maps. The motivation mainly lies in the study of higher dimensional Hochschild cohomology over C^{∗}- and von Neumann algebras^{56} [Christensen/Effros/Sinclair ’87].

As in the linear case, the most important properties of the completely bounded multilinear maps make their appearance^{57} in representation, extension and decomposition theorems.

Let X_{1}, …, X_{k}, Y be operator spaces and Φ : X_{1}×⋯×X_{k} → Y a multilinear mapping. We define [Christensen/Effros/Sinclair ’87, p. 281] a multilinear mapping

Φ^{(n)} : M_{n}(X_{1}) × ⋯ × M_{n}(X_{k}) → M_{n}(Y)
(x_{1}, …, x_{k}) ↦ [ ∑_{j_{1},j_{2},...,j_{k−1}=1}^{n} Φ(x_{1}^{l,j_{1}}, x_{2}^{j_{1},j_{2}}, …, x_{k}^{j_{k−1},m})] ,

55a survey of the theory of nuclear C^{∗}-algebras is e.g. to be found in [Mur90] and [Pat88].

56There is a close connection to the long-standing still open derivation problem for C^{∗}-algebras (or, equivalently [Kirchberg ’96, Cor. 1], the similarity problem). One should note that in the framework of operator spaces and completely bounded maps, considerable progress has been made in attacking these problems. For instance, Christensen [Christensen ’82, Thm. 3.1] was able to show that the inner derivations from a C^{∗}-algebra into B(H) are precisely the completely bounded ones.

57possibly footnote!

where n ∈ ℕ, the nth amplification of Φ.

Φ is called completely bounded if ∥Φ∥_{cb} := sup_{n}∥Φ^{(n)}∥ < ∞. It is called completely contractive if ∥Φ∥_{cb} ≤ 1.

Also compare the chapter Completely bounded bilinear maps.

Example: For bilinear forms on commutative C^{∗}-algebras we have the following result on automatic complete boundedness [Christensen/Sinclair ’87, Cor. 5.6]:

Let A be a commutative C^{∗}-algebra. Then each continuous bilinear form Φ : A×A → ℂ is automatically completely bounded and

∥Φ∥ ≤ ∥Φ∥_{cb} ≤ K_{G}∥Φ∥,

where K_{G} denotes the Grothendieck constant. Furthermore, K_{G} is the least such constant.

One often studies completely bounded multilinear maps by considering the linearization on the Haagerup tensor product, where the following relation holds [Paulsen/Smith ’87, Prop. 1.3; cf. also Sinclair/Smith ’95, Prop. 1.5.1]: If X_{1}, …, X_{n} are operator spaces and H is a Hilbert space, then a multilinear mapping Φ : X_{1} × ⋯ × X_{n} → B(H) is completely bounded if and only if its linearization φ is a completely bounded mapping on X_{1} ⊗_{h} ⋯ ⊗_{h} X_{n}. In this case, ∥Φ∥_{cb} = ∥φ∥_{cb}.

Also compare the chapter: Completely bounded bilinear mappings.

Representation theorem [Paulsen/Smith ’87, Thm. 3.2, cf. also Thm. 2.9; Sinclair/Smith, Thm. 1.5.4]: Let A_{1}, …, A_{k} be C^{∗}-algebras, X_{1} ⊂ A_{1}, …, X_{k} ⊂ A_{k} operator spaces and H a Hilbert space. Let further be Φ : X_{1} × ⋯ × X_{k} → B(H) a completely contractive multilinear mapping. Then there exist Hilbert spaces K_{i} (i = 1, …, k), ^{∗}-representations π_{i} : A_{i} → B(K_{i}) (i = 1, …, k), contractions T_{i} : K_{i+1} → K_{i} (i = 1, …, k−1) and two isometries V_{i} : H → K_{i} (i = 1, k) such that

Φ(x_{1}, …, x_{k}) = V_{1}^{∗}π_{1}(x_{1})T_{1}π_{2}(x_{2})T_{2} ⋯ T_{k−1}π_{k}(x_{k})V_{k}.

Following [Ylinen ’90, p. 296; cf. also Christensen/Effros/Sinclair ’87] it is possible to eliminate the “bridging maps” T_{i}. One obtains the following simpler form for the representation theorem:

Let A_{1}, …, A_{k} be C^{∗}-algebras, X_{1} ⊂ A_{1}, …, X_{k} ⊂ A_{k} operator spaces and H a Hilbert space. Let further Φ : X_{1} × ⋯ × X_{k} → B(H) be a completely contractive multilinear mapping. Then there exist a Hilbert space K, ^{∗}-representations π_{i} : A_{i} → B(K) and two operators V_{1}, V_{k} ∈ B(H, K) such that

Φ(x_{1}, …, x_{k}) = V_{1}^{∗}π_{1}(x_{1})π_{2}(x_{2}) ⋯ π_{k}(x_{k})V_{k}.

From this result one can deduce the following:

Extension theorem [cf. Paulsen/Smith ’87, Cor. 3.3 and Sinclair/Smith ’95, Thm. 1.5.5]: Let X_{i} ⊂ Y_{i} (i = 1, …, k) be operator spaces and H a Hilbert space. Let further Φ : X_{1} × … X_{k} → B(H) be a completely contractive multilinear mapping. Then there exists a multilinear mapping Φ̃ : Y_{1} × ⋯ × Y_{k} → B(H) which extends Φ preserving the cb-norm: ∥Φ∥_{cb} = ∥Φ̃∥_{cb}.

Let A and B be C^{∗}-algebras. For a k-linear mapping Φ : A^{k} → B we define [Christensen/Sinclair ’87, pp. 154-155] another k-linear mapping Φ^{∗} : A^{k} → B by

Φ^{∗}(a_{1}, …, a_{k}) := Φ(a_{k}^{∗}, …, a_{2}^{∗}, a_{1}^{∗})^{∗},

where a_{1}, …, a_{k} ∈ A. A k-linear map Φ : A^{k} → B is called symmetric if Φ = Φ^{∗}. In this case, Φ^{(n)}^{∗} = Φ^{(n)} (n ∈ ℕ).

A k-linear map Φ : A^{k} → B is called completely positive if

Φ^{(n)}(A_{1}, … A_{k}) ≥ 0

for all n ∈ ℕ and (A_{1}, …, A_{k}) = (A_{k}^{∗}, …, A_{1}^{∗}) ∈ M_{n}(A)^{k}, where A_{\frac{k+1}{2}} ≥ 0 for odd k.

Caution is advised: In the multilinear case complete positivity does not necessarily imply complete boundedness! For an example (or more precisely a general method of constructing such), cf. Christensen/Sinclair ’87, p. 155.

There is a multilinear version of the decomposition theorem for completely bounded symmetric multilinear mappings:

Decomposition theorem [Christensen/Sinclair ’87, Cor. 4.3]: Let A and B be C^{∗}-algebras, where B is injective, and let further Φ : A^{k} → B be a completely bounded symmetric k-linear mapping. Then there exist completely bounded, completely positive k-linear mappings Φ_{+}, Φ_{−} : A^{k} → B such that Φ = Φ_{+} − Φ_{−} and ∥Φ∥_{cb} = ∥Φ_{+} + Φ_{−}∥_{cb}.

9 Automatic Complete Boundedness

Completely bounded linear and multilinear mappings share strong structural properties. Thanks to the complete boundedness they have a very specific form (cf. the corresponding representation theorems) whence they are much more accessible than arbitrary bounded linear (or multilinear) mappings.^{58} Moreover, even in the multilinear case we have at our disposal an extension theorem for completely bounded mappings – which again is in striking contrast to the situation of arbitrary bounded (multi)linear mappings.

Because of the very nice structure theory of completely bounded linear and multilinear mappings, respectively, it is highly interesting to decide whether or not a given bounded (multi)linear mapping actually is completely bounded. The most elegant way to proceed is of course to check some simple conditions concerning the spaces involved and/or some (purely) algebraic properties of the mapping which automatically imply the complete boundedness of the latter.

In the following we shall collect various such criteria relying

(1) on the initial and/or target space

(2) mainly on algebraic properties of the mapping (being, e.g., a ^{∗}-homomorphism of C^{∗}-algebras or a module homomorphism).

58For example, the representation theorems provide a very useful tool in calculating cohomology groups (cf. for example the monograph [SS95]).

(1) “Criterion: spaces”

(1.1) The linear case

∗ Smith’s lemma: If X is a matricially normed space and Φ : X → M_{n}, where n ∈ ℕ, is a bounded linear operator, then we have ∥Φ∥_{cb} = ∥Φ^{(n)}∥. In particular, Φ is completely bounded if and only if Φ^{(n)} is bounded [Smi83, Thm. 2.10].

∗ Let X be an operator space, A a commutative C^{∗}-algebra and Φ : X → A a bounded linear operator. Then Φ is completely bounded and ∥Φ∥_{cb} = ∥Φ∥ ([Loe75, Lemma 1], cf. also [Arv69, Prop. 1.2.2]). In particular, every bounded linear functional is automatically completely bounded with the same cb-norm.

∗ The following theorem shows that, roughly speaking, the above situations are the only ones where every bounded linear operator between (arbitrary) C^{∗}-algebras is automatically completely bounded. More precisely: Let A and B be C^{∗}-algebras. In order to have the complete boundedness of very bounded linear operator from A to B it is necessary and sufficient that either A is finite dimensional or B is a subalgebra of M_{n} ⊗ C(Ω) for a compact Hausdorff space Ω [HT83, Cor. 4], [Smi83, Thm. 2.8], cf. also [Smi83, p. 163].

(1.2) The bilinear case

∗ Let A be a commutative C^{∗}-algebra. Then every bounded bilinear form Φ : A × A → ℂ is automatically completely bounded and

∥Φ∥ ≤ ∥Φ∥_{cb} ≤ K_{G}∥Φ∥,

where K_{G} denotes the Grothendieck constant; furthermore, K_{G} is the least possible constant [CS87, Cor. 5.6]

(2) “Criterion: mappings”

(2.1) The linear case ∗ Every ^{∗}-homomorphism between C^{∗}-algebras is completely contractive.^{59} ∗ Let A and B be C^{∗}-subalgebras of B(H), where H is a Hilbert space. Let

further E ⊂ B(H) be an (A, B)-operator module and Φ : E → B(H) be a bounded (A, B)-module homomorphism. Let A and B be quasi-cyclic^{60} , i.e., for every finite set of vectors ξ_{i}, η_{j} ∈ H, 1 ≤ i, j ≤ n, there exist ξ, η ∈ H such that ξ_{i} ∈ \overline{Aξ}, η_{j} ∈ \overline{Bη}, 1 ≤ i, j ≤ n. Then Φ is completely bounded and ∥Φ∥_{cb} = ∥Φ∥ ([SS95, Thm. 1.6.1]; cf. also [Smi91, Thm. 2.1] and [Smi91, Remark 2.2], [Sat82, Satz 4.16]). – For special cases of this result obtained earlier see [Haa80], [EK87, Thm. 2.5], [PPS89, Cor. 3.3] and [DP91, Thm. 2.4].

59This is evident because π^{(n)} : M_{n}(A) → M_{n}(B) is a ^{∗}-homomorphism for every n ∈ ℕ.

60For example, cyclic C^{∗}-algebras are quasi-cyclic. A von Neumann algebra M is quasi-cyclic if every normal state on the commutant M′ is a vector state [Smi91, Lemma 2.3].

(2.2) The bilinear case

∗ Let A ⊂ B(H) be a C^{∗}-algebra and E ⊂ B(H) an A-operator bimodule. A bilinear mapping Φ : E × E → B(H) is called A-multimodular if

Φ(aeb, fc) = aΦ(e, bf)c

holds for all a, b, c ∈ A and e, f ∈ E.

We have a bilinear analogue of the above theorem:

Let A ⊂ B(H) be a quasi-cyclic C^{∗}-algebra and E ⊂ B(H) be an A-operator bimodule. Let further Φ : E × E → B(H) be a bilinear, A-multimodular mapping such that the corresponding linearization Φ̃ : E ⊗ E → B(H) is bounded on E ⊗_{h} E. Then Φ is completely bounded and ∥Φ∥_{cb} = ∥Φ̃∥ [SS95, Thm. 1.6.2].

10 Convexity

Since convex sets are important in the study of ordered or normed vector spaces, it is only natural to ask for a non-commutative version of convexity that is more suited for vector spaces of operators. Thus matrix convex sets were introduced, which play the same role in operator space theory as ordinary convex sets in classical functional analysis.

Some time earlier C^{∗}-convex sets were defined for C^{∗}-algebras. Both notions are different but similar, so a section about C^{∗}-convex sets is also included in this survey.

Some publications about non-commutative convexity are [EW97b], [WW99], [FZ98], [Mor94], [Fuj94].

10.1 Matrix convexity

Let V be a complex vector space. A set K of matrices over V consists of subsets K_{n} ⊂ M_{n}(V) for all n ∈ ℕ. Subsets K_{1} ⊂ V can be considered as sets of matrices over V by putting K_{n} = ∅ for n ≥ 2. More generally, if for some n ∈ N the corresponding set K_{n} is not given a priori, put this K_{n} as the empty set. For sets of matrices, we have the following notions of convexity:^{61}

61With the notation M(V) = ⋃{M_{n}(V) | n ∈ ℕ}, a set K of matrices over V is simply a subset of M(V).

The sets K_{n} can be regained as K_{n} = K ∩ M_{n}(V).

Then the definitions have the following form:

Let K ⊂ M(V). K is called matrix convex if

K ⊕ K ⊂ K and α^{∗}Kα ⊂ K for all matrices α with α^{∗}α = 𝟙.

K is called absolutely matrix convex if

K ⊕ K ⊂ K and αKβ ⊂ K for all matrices α, β with ∥α∥, ∥β∥ ≤ 1.

K is called a matrix cone if

K ⊕ K ⊂ K and α^{∗}Kα ⊂ K for all matrices α.

A set K of matrices over V is called matrix convex or a matrix convex set [Wit84b] if for all x ∈ K_{n} and y ∈ K_{m}

x ⊕ y ∈ K_{n+m},

and for all x ∈ K_{n} and α ∈ M_{n,m} with α^{∗}α = 𝟙_{m}

α^{∗}xα ∈ K_{m}.

A set K of matrices over V is called absolutely matrix convex [EW97a] if for all x ∈ K_{n} and y ∈ K_{m}

x ⊕ y ∈ K_{n+m},

and for all x ∈ K_{n}, α ∈ M_{n,m} and β ∈ M_{m,n} with ∥α∥, ∥β∥ ≤ 1

αxβ ∈ K_{m}.

A set K of matrices over V is called a matrix cone [Pow74] if for all x ∈ K_{n} and y ∈ K_{m}

x ⊕ y ∈ K_{n+m},

and for all x ∈ K_{n} and α ∈ M_{n,m}

α^{∗}xα ∈ K_{m}.

A set K of matrices over V is matrix convex if and only if all matrix convex combinations of elements of K are again in K. K is absolutely matrix convex if and only if all absolutely matrix convex combinations of elements of K are again in K. Here, a matrix convex combination of x_{1}, …, x_{n} (x_{i} ∈ K_{k_{i}}) is a sum of the form ∑_{i=1}^{n} α_{i}^{∗}x_{i}α_{i} with matrices α_{i} ∈ M_{k_{i},j} such that ∑_{i=1}^{n} α_{i}^{∗}α_{i} = 𝟙_{j}. An absolutely matrix convex combination of x_{1}, …, x_{n} is a sum of the form ∑_{i=1}^{n} α_{i}x_{i}β_{i} with matrices α_{i} ∈ M_{j,k_{i}} and β_{i} ∈ M_{k_{i},j} such that ∑_{i=1}^{n} α_{i}α_{i}^{∗} ≤ 𝟙_{j} and ∑_{i=1}^{n} β_{i}^{∗}β_{i} ≤ 𝟙_{j}.

If V is a topological vector space, topological terminology is to be considered at all matrix levels: For instance, a set K of matrices over V is called closed if all K_{n} are closed.

Examples

matrix convexity

1. The unit ball of an operator space X, given by the family Ball(X)_{n} = {x ∈ M_{n}(X) | ∥x∥ ≤ 1} for all n ∈ ℕ,^{62} is absolutely matrix convex and closed.

2. The set of matrix states of a unital C^{∗}-algebra A, given by the family CS(A)_{n} = {φ : A → M_{n} | φ completely positive and unital} for all n ∈ ℕ, is matrix convex and weak-∗-compact.

3. If A is a C^{∗}-algebra, the positive cones M_{n}(A)^{+} for all n ∈ ℕ form a closed matrix cone.

62 Also Ball = {x ∈ M(X) | ∥x∥ ≤ 1}.

10.1.1 Separation theorems

An important tool in the theory are the following separation theorems.

Let ⟨V, W ⟩ be a (non degenerate) duality of complex vector spaces. Thus V and W have weak topologies, and the matrix levels have the corresponding product topology. For v = [v_{i,j}] ∈ M_{n}(V) and w = [w_{k,l}] ∈ M_{m}(W), ⟨v, w⟩ is defined by the joint amplifications of the duality:

⟨v, w⟩ = [⟨v_{i,j}, w_{k,l}⟩]_{(i,k),(j,l)}.

Note that the matrices, ordered by the cone of the positive semidefinite matrices, are not totally ordered; ≰ does not imply ≥.

Theorem:^{63} Let ⟨V, W ⟩ be a duality of complex vector spaces, K a closed set of matrices over V and v_{0} ∈ M_{n}(V) \ K_{n} for some n.

a) [WW99, Thm. 1.6] If K is matrix convex, then there are w ∈ M_{n}(W) and α ∈ (M_{n})_{sa} such that for all m ∈ ℕ and v ∈ K_{m}

Re⟨v, w⟩ ≤ 𝟙_{m} ⊗ α, but Re⟨v_{0}, w⟩ ≰ 𝟙_{n} ⊗ α.

b) [EW97b, Thm. 5.4] If K is matrix convex and 0 ∈ K_{1}, then there is w ∈ M_{n}(W) such that for all m ∈ ℕ and v ∈ K_{m}

Re⟨v, w⟩ ≤ 𝟙_{nm}, but Re⟨v_{0}, w⟩ ≰ 𝟙_{n^{2}}.

c) [Bet97, p. 57] If K is a matrix cone, then there is w ∈ M_{n}(W) sucht that for all m ∈ ℕ and v ∈ K_{m}

Re⟨v, w⟩ ≤ 0, but Re⟨v_{0}, w⟩ ≰ 0.

d) [EW97a, Thm. 4.1] If K is absolutely matrix convex, then tere is w ∈ M_{n}(W) such that for all m ∈ ℕ and v ∈ K_{m}

∥⟨v, w⟩∥ ≤ 1, but ∥⟨v_{0}, w⟩∥ > 1.

63 From this theorem one can easily get the following sharper version of the parts a), b) and d):

a) If K is matrix convex, then there are w ∈ M_{n}(W), α ∈ (M_{n})_{sa} and ε > 0 such that for all m ∈ ℕ

and v ∈ K_{m}

Re⟨v, w⟩ ≤ 𝟙_{m} ⊗ (α − ε𝟙_{n}), but Re⟨v_{0}, w⟩ ≰ 𝟙_{n} ⊗ α.

b) If K is matrix convex and 0 ∈ K_{1}, then there are w ∈ M_{n}(W) and ε > 0 such that for all m ∈ ℕ

and v ∈ K_{m}

Re⟨v, w⟩ ≤ (1 − ε)𝟙_{nm}, but Re⟨v_{0}, w⟩ ≰ 𝟙_{n^{2}}.

d) If K is absolutely matrix convex, then there are w ∈ M_{n}(W) and ε > 0 such that for all m ∈ ℕ and

v ∈ K_{m}

∥⟨v, w⟩∥ ≤ 1 − ε, but ∥⟨v_{0}, w⟩∥ > 1.

One can prove Ruan’s theorem using the separation theorem for absolutely matrix convex sets, applied to the unit ball of a matricially normed space [].

If V is a complex involutive vector space, one can find selfadjoint separating functionals: Theorem: ^{64} Let ⟨V, W ⟩ be a duality of complex involutive vector spaces, K a closed

set of selfadjoint matrices over V and v_{0} ∈ M_{n}(V) \ K_{n} for some n.

b) If K is matrix convex and 0 ∈ K_{1}, then there is a w ∈ M_{n}(W)_{sa} such that for all m ∈ ℕ and v ∈ K_{m}

⟨v, w⟩ ≤ 𝟙_{nm}, but Re⟨v_{0}, w⟩ ≰ 𝟙_{n^{2}}.

10.1.2 Bipolar theorems

Let ⟨V, W ⟩ be a duality of complex vector spaces and K a set of matrices over V. The matrix polar of K is a set D of matrices over W, given by^{65}

D_{n} = {w ∈ M_{n}(W) | Re⟨v, w⟩ ≤ 𝟙_{nm} for all m ∈ ℕ, v ∈ K_{m} }.

The absolute matrix polar of K is a set D of matrices over W, given by^{66}

D_{n} = {w ∈ M_{n}(W) | ∥⟨v, w⟩∥ ≤ 1 for all m ∈ ℕ, v ∈ K_{m} }.

Polars of sets of matrices over W are defined analogously.

We have the bipolar theorems: Let ⟨V, W ⟩ be a duality of complex vector spaces and K a set of matrices over V.

a) [EW97b] K equals its matrix bipolar if and only if K is closed and matrix convex and 0 ∈ K_{1}.

b) [EW97a] K equals its absolute matrix bipolar if and only if K is closed and absolutely matrix convex.

The matrix bipolar of a set K of matrices over V is therefore the smallest closed and matrix convex set which contains K and 0.

The absolute matrix bipolar of a set K of matrices over V is therefore the smallest closed and absolutely matrix convex set which contains K.

So we get a characterization of the unit balls of MIN (E) and MAX (E) for a normed space E.

64 This theorem can be obtained from the above separation theorem, part a) and b). Note that for selfadjoint v, the mapping w ↦ ⟨v, w⟩ is selfadjoint.

65 D = {w ∈ M(W) | Re⟨v, w⟩ ≤ 𝟙 for all v ∈ K}.

66 D = {w ∈ M(W) | ∥⟨v, w⟩∥ ≤ 1 for all v ∈ K}.

10.2 Matrix extreme points

A part of convexity theory studies the possibility of representing all points of a convex set as convex combinations of special points, the so called extreme points. The well-known results about this are the Krein-Milman theorem and its sharpenings, the Choquet representation theorems. The question arises whether there are analogous results for non-commutative convexity. The sections Matrix extreme points and C^{∗}-extreme points give partial answers to this question.

Let V be a vector space. A matrix convex set of matrices over V is called matrix convex set in V for short. Let A be a set of matrices over V. The matrix convex hull of A is the smallest matrix convex set in V containing A. Its closure is the smallest closed matrix convex set containing A because the closure of matrix convex sets is matrix convex. Two elements x, y ∈ M_{n}(V) are unitarily equivalent, if there is a unitary u ∈ M_{n} such that x = u^{∗}yu. Let U(S) be the set of all elements, that are unitarily equivalent to elements of S ⊂ M_{n}(V). x ∈ M_{n}(V) is called reducible, if it is unitarily equivalent to some block matrix \binom{y 0}{0 z} ∈ M_{n}(V). A matrix convex combination ∑_{i=1}^{k} α_{i}^{∗}x_{i}α_{i} is called proper, if all α_{i} are square matrices different from 0.

Let K be a matrix convex set in V. Then x ∈ K_{n} is a structural element^{67} of K_{n}, if whenever x = ∑_{i=1}^{k} α_{i}^{∗}x_{i}α_{i} is a proper matrix convex combination of x_{i} ∈ K_{n}, then every x_{i} is unitarily equivalent to x. The set of all structural elements of K_{n} is denoted by str(K_{n}). The set of structural elements of K is the set of matrices over V consisting of str(K_{n}) for all n ∈ ℕ.

Example: Let L be an operator system. The generalized state space of L is the matrix convex set CS(L) in the dual L^{∗}, which consists of the matrix states

CS(L)_{n} = {ψ | ψ : L → M_{n} completely positive and unital}.

The generalized state space is weak*-compact. It follows from [CE77, Lemma 2.2] that the structural elements of CS(L)_{n} are exactly those completely positive and unital mappings which are pure. To every compact and matrix convex set K there is an operator system which has K as its generalized state space [WW99, Prop. 3.5].

Let V be a locally convex space and induce the product topology on M_{n}(V). The matrix convex Krein-Milman Theorem is: Let K be a compact matrix convex set in V. Then K is equal to the closed matrix convex hull of the structural elements of K. If V has finite dimension, then K is the matrix convex hull of its structural elements.

The converse result is: Let K be a compact matrix convex set in V. Let S be a closed set of matrices, such that S_{n} ⊂ K_{n} and v^{∗}S_{l}v ⊂ S_{m} for all partial isometries v ∈ M_{lm} and for all n, m, l ∈ ℕ, l ≥ m. If the closed matrix convex hull of S equals K, then all structural elements of K are in S.([WW99], [Fis96]).

67The term structural element is defined by Morenz [Mor94]. Another equivalent definition is given in [WW99]. There the structural elements are called matrix extreme points.

It is possible to sharpen these results for more special matrix convex sets. A matrix convex set K is called simple, if there are n ∈ ℕ and A ⊂ M_{n}(V), such that K is equal to the matrix convex hull of A. K is a simple matrix convex set, if and only if there is n ∈ ℕ such that str(K_{m}) = ∅ for all m > n.

Suppose that K is a matrix convex set in V. Then x ∈ K_{m} is a matrix extreme point, if x ∈ str(K_{m}) and

x ∉ ∪_{m<l}𝟙_{lm}^{∗}str(K_{l})𝟙_{lm}.

Let mext(K) be the set of matrices consisting of all matrix extreme points of K.

Suppose that K is a simple compact matrix convex set in V. Then K is equal to the closed matrix convex hull of mext(K). If V has finite dimension, then the closure is not needed, that means K is the matrix convex hull of mext(K). In this case the following result also holds: Let S be a set of matrices over V not containing reducible elements such that the matrix convex hull of S equals K, then mext(K)_{m} ⊂ U(S_{m}) for all m ∈ ℕ ([Mor94], [Fis96]).

If K is compact and not simple, mext(K) may be empty. As an example take the generalized state space CS(A) of a C^{∗}-algebra A. Its matrix extreme points are exactly the irreducible finite dimensional representations of A. These need not exist in general.

10.3 C^{∗}-convexity

Let A be a unital C^{∗}-algebra. A subset K ⊂ A is a C^{∗}-convex set, if

∑_{i=1}^{n} a_{i}^{∗}x_{i}a_{i} ∈ K

for all x_{i} ∈ K and a_{i} ∈ A such that ∑_{i=1}^{n} a_{i}^{∗}a_{i} = 𝟙. The sum ∑_{i=1}^{n} a_{i}^{∗}x_{i}a_{i} is called a C^{∗}-convex combination.

A subset K ⊂ A is a C^{∗}-absolutely convex set, if

∑_{i=1}^{n} a_{i}^{∗}x_{i}b_{i} ∈ K

for all x_{i} ∈ K and a_{i}, b_{i} ∈ A such that ∑_{i=1}^{n} a_{i}^{∗}a_{i}, ∑_{i=1}^{n} b_{i}^{∗}b_{i} ≤ 𝟙.

In particular C^{∗}-convex sets are convex and C^{∗}-absolutely convex sets are absolutely convex.

Example: Let H be a Hilbert space and T ∈ B(H). The n-th matrix range of T is the set

W (T)_{n} := {φ(T)|φ : B(H) → M_{n} completely positive and unital}.

W (T)_{n} is a compact and C^{∗}-convex subset of M_{n} and to every compact and C^{∗}-convex subset K ⊂ M_{n} there exist a separable Hilbert space H and S ∈ B(H) such that K = W (S)_{n} ([LP81, Prop. 31]).

The sets

Ball(B(H)) = {x ∈ B(H)|∥x∥ ≤ 1} and P = {x ∈ B(H)|0 ≤ x ≤ 𝟙}

are C^{∗}-convex und wot-compact subsets of B(H). Ball(B(H)) is also C^{∗}-absolutely convex.

Loebl and Paulsen introduced C^{∗}-convex sets in [LP81]. At the beginning of the nineties Farenick and Morenz studied C^{∗}-convex subsets of M_{n} ([Far92],[FM93]). Eventually Morenz succeeded in proving an analogue of the Krein-Milman theorem for a compact C^{∗}-convex subset of M_{n} ([Mor94]). At the end of the nineties Magajna generalized the notion of C^{∗}-convex sets to the setting of operator modules and proved some separation theorems [Mag00, Th. 1.1] and also an analogue of the Krein-Milman theorem [Mag98, Th. 1.1].

10.3.1 Separation theorems

Suppose that A, B ⊂ B(H) are unital C^{∗}-algebras and Y ⊂ B(H) is a (A, B)-bimodul. Then K ⊂ Y is (A, B)-absolutely convex, if

∑_{i=1}^{n} a_{i}^{∗}x_{i}b_{i} ∈ K

for all x_{i} ∈ K and a_{i} ∈ A, b_{i} ∈ B such that ∑_{i=1}^{n} a_{i}^{∗}a_{i}, ∑_{i=1}^{n} b_{i}^{∗}b_{i} ≤ 𝟙. Let Y be a A-bimodul. Then K is A-convex, if

∑_{i=1}^{n} a_{i}^{∗}x_{i}a_{i} ∈ K

for all x_{i} ∈ K and a_{i} ∈ A such that ∑_{i=1}^{n} a_{i}^{∗}a_{i} = 𝟙. In the case Y = A this definition is equivalent to the definition of C^{∗}-convex sets.

There are following separation theorems: Let A, B ⊂ B(H) be unital C^{∗}-algebras and Y ⊂ B(H) a A, B-bimodul. Let K ⊂ Y be norm closed and y_{0} ∈ Y \ K.

1) If A = B, 0 ∈ K and K is A-convex, then there is a Hilbert space H_{π}, a cyclic representation π : A → B(H_{π}) and a completely bounded A-bimodul-homomorphism, such that for all y ∈ K

Re ϕ(y) ≤ 𝟙, but Re ϕ(y_{0}) ≰ 𝟙.

2)If K is (A, B)-absolutely convex, then there is a Hilbert space H_{π}, representations π : A → B(H_{π}) and σ : B → B(H_{π}) and a completely bounded (A, B)-bimodul-homomorphism ϕ : Y → B(H_{π}), such that for all y ∈ K

∥ϕ(y)∥ ≤ 1, but ∥ϕ(y_{0})∥ > 1.

10.4 C^{∗}-extreme points

Let A be an unital C^{∗}-algebra and Y ⊂ A. The C^{∗}-convex hull of Y is the smallest C^{∗}-convex set that contains Y.

Suppose K ⊂ A is C^{∗}-convex. Then x ∈ K is a C^{∗}-extreme point, if whenever x = ∑_{i=1}^{n} a_{i}^{∗}x_{i}a_{i} is a C^{∗}-convex combination of x_{i} ∈ K with invertible a_{i} ∈ A, then there are unitaries u_{i} ∈ A such that x = u_{i}^{∗}x_{i}u_{i} for i = 1, …, n.

Suppose now A = M_{n} and let K ⊂ M_{n} be compact and C^{∗}-convex. Let K̃ be the matrix convex hull of K. Then K̃ is a simple compact and matrix convex set in ℂ, such that K̃_{n} = K ([Fis96]). Thus it is possible to conceive a C^{∗}-convex subset of M_{n} as a matrix convex set in ℂ. Now the matrix convex Krein-Milman theorem can be used. Moreover, it follows from the work of Farenick and Morenz that the structural elements of K̃_{n} are exactly the not reducible C^{∗}-extreme points of K. So following theorem holds: Let K ⊂ M_{n} be compact and C^{∗}-convex, then K is equal to the C^{∗}-convex hull of its C^{∗}-extreme points.

In order to get a somewhat more general result, the definition of the extreme points can be changed. Suppose that R is a hyperfinite factor and that K ⊂ R is C^{∗}-convex. Then x ∈ K is a R-extreme point, if whenever x = ∑_{i=1}^{n} a_{i}x_{i}a_{i} is a C^{∗}-convex combination of x_{i} ∈ K such that all a_{i} ∈ A are positive and invertible, then it follows that x = x_{i} and a_{i}x = xa_{i} forr i = 1, …, n.^{68}

With this definition following theorem hold: Let K ⊂ R be C^{∗}-convex and weak* compact. Then K is equal to the weak* closure of the C^{∗}-convex hull of its R-extreme points.

11 Mapping Spaces

Let E, F be Banach spaces. We consider a linear subspace A(E, F) of the space B(E, F) of the continuous operators between E and F which contains all finite rank maps and is a Banach space with respect to a given norm. Furthermore, it is usually required that A(E, F) be defined for all pairs of Banach spaces E and F. Such a space is called – according to A. Grothendieck – a mapping space .

Analogously, we call an operator space A(X, Y) which is a linear subspace of CB(X, Y) a CB-mapping space. Note that generally the algebraic identification of M_{n}(A(X, Y)) with A(X, M_{n}(Y)) fails to be isometric and that the norms on A(X, M_{n}(Y)) do not generate an operator space structure for A(X, Y).

There is a close relationship between mapping spaces and tensor products: The space F (X, Y) of all finite rank maps between X and Y and the algebraic tensor product of X^{∗} with Y are isomorphic:

68If R = M_{n} the R-extreme points are exactly the C^{∗}-extreme points. In general every R-extreme point is also C^{∗}-extreme, but not vice versa.

X^{∗} ⊗_{alg} Y ≅ F (X, Y).

This identification enables us to transfer norms from one space to the other one. To this end, we consider the extension of the mapping X^{∗} ⊗ Y → F (X, Y) to the completion with respect to an operator space tensor norm X^{∗} ˜⊗Y :

Φ : X^{∗} ˜⊗Y → CB(X, Y).

Φ is in general neither injective nor surjective. As a CB-mapping space one obtains

Im(Φ) ⊂ CB(X, Y)

with the operator space norm of

(X^{∗} ˜⊗Y)/Ker(Φ).

We consider now assignments that assign a mapping space A(·, ·) with operator space norm α(·) to every pair of operator spaces. In the Banach space theory A. Pietsch intensified the notion of mapping spaces to that of the operator ideals [Pie78]. Analogously, we consider operator ideals which are mapping spaces with the CB-ideal property [ER94], i.e , the composition

CB(X_{1}, X_{2}) × A(X_{2}, Y_{2}) × CB(Y_{2}, Y_{1}) → A(X_{1}, Y_{1})
(Ψ_{1}, Φ, Ψ_{2}) ↦ Ψ_{2} ∘ Φ ∘ Ψ_{1}

is for all operator spaces X_{1}, X_{2}, Y_{1}, Y_{2} defined and jointly completely contractive. A CB-ideal is called local [EJR98], if its norm satisfies:

α(φ) = sup{α(φ|_{L}) : L ⊂ X, dim L < ∞}.

11.1 Completely nuclear mappings

Let X and Y be operator spaces. The completely nuclear mappings from X to Y [ER94, §2], [EJR98, §3] are defined by the projective operator space tensor norm. We consider the extension of the canonical identity X^{∗} ⊗ Y = F (X, Y):

Φ : X^{∗} ⊗^{∧} Y → X^{∗} ⊗^{∨} Y ⊂ CB(X, Y).

⊗^{∨} is the injective tensor product and ⊗^{∧} the projective tensor product.

A mapping in the range of Φ is called completely nuclear. One denotes by

CN (X, Y) := (X^{∗} ⊗^{∧} Y)/Ker(Φ)

the space of the completely nuclear mappings and endows it with the quotient operator space structure. The operator space norm is denoted by ν(·). M_{n}(CN (X, Y)) and CN (X, M_{n}(Y)) are in general not isometric.

Nuclear ^{69} mappings are completely nuclear. [ER94, 3.10]

In general, the projective tensor norm does not respect complete isometries. Hence,even for subspaces Y_{0} ⊂ Y the canonical embedding CN (X, Y_{0}) → CN (X, Y) is generally only completely contractive and not isometric. Since the projective tensor norm respects quotient mappings, every nuclear map φ : X_{0} → Y on a subspace X_{0} ⊂ X with ν(φ) < 1 has an extension φ̃ to the whole of X satisfying ν(φ̃) < 1.

The completely nuclear mappings enjoy the CB-ideal property. Furthermore, the adjoint φ^{∗} is completely nuclear if φ is, and the inequality: ν(φ^{∗}) ≤ ν(φ) [EJR98, Lemma 3.2] holds.

A mapping φ is completely nuclear, if and only if there is a factorization of the form

B(ℓ_{2}) →^{M(a,b)} T (ℓ_{2})
↑ r  ↓ s
X  →^{φ} Y

Here a, b are Hilbert-Schmidt operators defining the mapping M(a, b) : x ↦ axb. For the completely nuclear norm we have: ν(φ) = 1 precisely if for all ϵ > 0 there exists a factorization with ∥r∥_{cb}∥a∥_{2}∥b∥_{2}∥s∥_{cb} ≤ 1 + ϵ ^{70} [ER94, Thm. 2.1].

The completely nuclear mappings are not local.

11.2 Completely integral mappings

Completely integral mappings are defined by the aid of completely nuclear mappings. A mapping φ : X → Y is said to be completely integral, if there are a constant c > 0 and a net of finite rank maps φ_{α} ∈ CN (X, Y) with ν(φ_{α}) ≤ c converging to φ in the point norm topology ^{71}.

The set of all these mappings forms the space CI (X, Y) of the completely integral mappings.

The infimum of all those constants c satisfying the above condition is actually attained and denoted by ι(φ). ι(·) is a norm turning CI (X, Y) into a Banach space. The unit sphere of CI (X, Y) is merely the point norm closure of the unit sphere of CN (X, Y).

ℓ_{∞} →^{d} ℓ_{1}
r ↑  ↓ s
E →^{φ} F,

69The completely nuclear mappings owe their definition to the one of the nuclear mappings of the Banach space theory. There, one considers a corresponding mapping Φ_{B} : E^{∗} ⊗_{γ} F → B(E, F) for two Banach spaces E and F.

70In the Banach space theory one has an analogous statement: A mapping φ is nuclear, if and only if there’s a diagram where d is a diagonal operator, i.e. , there is a (d_{i}) ∈ ℓ_{1}, such that d((a_{i})) = (d_{i} · a_{i}) for all (a_{i}) ∈ ℓ_{∞}.

The nuclear norm is then computed as: ν_{B}(φ) = inf ∥r∥∥d∥_{ℓ_{1}}∥s∥, where the infimum runs over all possible factorizations.

71 φ_{α} → φ in the point norm topology, if ∥φ_{α}(x) − φ(x)∥ → 0 for all x ∈ X.

One obtains the canonical operator space structure by defining the unit sphere of M_{n}(CI (X, Y)) as the point norm closure of unit sphere of M_{n}(CN (X, Y)).

By definition we have ι(φ) ≤ ν(φ); for finite dimensional X we have moreover [EJR98, Lemma 4.1]

CI (X, Y) =^{cb} CN (X, Y).

Integral ^{72} mappings are completely integral [ER94, 3.10].

The canonical embedding

CI (X, Y) ↪ (X ⊗^{∨} Y^{∗})^{∗}

is a complete isometry [EJR98, Cor. 4.3]. One has moreover that [EJR98, Cor. 4.6] φ is completely integral, if and only if there is a factorization of the form:

B(H) →^{M(ω)} B(K)^{∗}
↑ r  ↓ s
X →^{φ} Y ↪ Y^{∗∗}

with weak^{∗}-continuous s . The mapping M(ω) : B(H) → B(K)^{∗} is for two elements a ∈ B(H), b ∈ B(K) given by (M(ω)(a))(b) = ω(a ⊗ b). We have for the norm ι(φ) = 1, if there is a factorization with ∥r∥_{cb}∥ω∥∥s∥_{cb} = 1 (note that generally ∥M(ω)∥_{cb} ≠ ∥ω∥).

The completely integral mappings enjoy also the CB-ideal property. Contrasting the situation of completely nuclear mappings they are local. In general one only has ι(φ) ≤ ι(φ^{∗}) [EJR98].

12 Appendix

12.1 Tensor products

12.1.1 Tensor products of operator matrices

As usual we define the algebraic tensor product of operator matrices x = [x_{ij}] ∈ M_{p}(X), y = [y_{kl}] ∈ M_{q}(Y) by setting

x ⊗ y := [x_{ij} ⊗ [y_{kl}]_{k,l}]_{i,j} ∈ M_{p}(X ⊗ M_{q}(Y)).

Here we have used the definition M_{p}(X) = M_{p} ⊗ X and the associative law

M_{p}(X) ⊗ M_{q}(Y) = M_{p} ⊗ (X ⊗ M_{q}(Y)) = M_{p}(X ⊗ M_{q}(Y)).  (5)

72 The unit ball of the integral mappings of the Banach space theory is just the point norm closure of the unit ball of the nuclear mappings. One should note that the formulas ι_{B}(φ) = ι_{B}(φ^{∗}) and I_{B}(E, F^{∗}) = (E ⊗_{λ} F)^{∗} have no counterparts for completely integral mappings.

In view of the next identification one should note that the shuffle-map is an algebraic isomorphism:

X ⊗ (M_{q} ⊗ Y) → M_{q} ⊗ (X ⊗ Y),  (6)
x ⊗ (β ⊗ y) ↦ β ⊗ (x ⊗ y),

for β ∈ M_{q}, x ∈ X, y ∈ Y. The shuffle-isomorphism at hand we obtain the identification:

x ⊗ y = [x_{ij} ⊗ [y_{kl}]_{k,l}]_{i,j} = [[x_{ij} ⊗ y_{kl}]_{k,l}]_{i,j} ∈ M_{p}(M_{q}(X ⊗ Y)).

Finally we use the usual^{73} identification M_{p}(M_{q}) = M_{pq}

[[x_{ij} ⊗ y_{kl}]_{k,l}]_{i,j} = [x_{ij} ⊗ y_{kl}]_{(i,k),(j,l)}

to obtain

M_{p}(X) ⊗ M_{q}(Y) = M_{pq}(X ⊗ Y).  (7)

We call this algebraic isomorphism the shuffle-isomorphism.

One should note that for operator space tensor products the algebraic identifications (5) and (6) are only complete contractions:

𝕄_{p}(X) ⊗_{α} Y → 𝕄_{p}(X ⊗_{α} Y),  (8)
X ⊗_{α} 𝕄_{q}(Y) → 𝕄_{q}(X ⊗_{α} Y).  (9)

In general these are not isometries even for p = 1 resp. q = 1.

For an operator space tensor product the shuffle-map

𝕄_{p}(X) ⊗_{α} 𝕄_{q}(Y) → 𝕄_{pq}(X ⊗_{α} Y)  (10)

in general is only completely contractive.

In the case of the injective operator space tensor product this is of course a complete isometry.

More generally, one considers the shuffle–map for rectangular matrices^{74}:

𝕄_{m,n}(X) ⊗_{α} 𝕄_{p,q}(Y) → 𝕄_{mp,nq}(X ⊗_{α} Y).

Another example is provided by the Blecher-Paulsen equation.

73In the matrix so obtained (i, k) are the row indices and (j, l) are the column indices, where i, j = 1, …, p and k, l = 1, …, q. The indices (i, k) resp. (j, l) are ordered lexicographically.

74The shuffle-map

(U ⊗ X) ⊗ (V ⊗ Y) → (U ⊗ V) ⊗ (X ⊗ Y),

(u ⊗ x) ⊗ (v ⊗ y) ↦ (u ⊗ v) ⊗ (x ⊗ y), U,V,X,Y operator spaces, has been studied for various combinations of operator space tensor products [EKR93, Chap. 4].

12.1.2 Joint amplification of a duality

The matrix dualitywhich is fundamental in the duality theory of operator spaces, is a special case of the joint amplification of a bilinear mapping.

The joint amplification of a duality ⟨X, X^{∗}⟩ of vector spaces is defined by

⟨x, φ⟩^{p×q} = ⟨[x_{ij}], [φ_{κλ}]⟩^{p×q} := [⟨x_{ij}, φ_{κλ}⟩] ∈ M_{p}(M_{q}) = M_{pq}

for x = [x_{ij}] ∈ M_{p}(X), φ = [φ_{κλ}] ∈ M_{q}(X^{∗}).

Interpreting φ as a mapping φ : X → M_{q} we have

φ^{(p)}(x) = ⟨x, φ⟩^{p×q}.

Associated to the duality of tensor products^{75} ⟨X ⊗ Y, X^{∗} ⊗ Y^{∗}⟩ is the joint amplification

⟨M_{p}(X ⊗ Y), M_{q}(X^{∗} ⊗ Y^{∗})⟩.

Especially, the equation

⟨x ⊗ y, φ ⊗ ψ⟩ = ⟨[x_{ij}] ⊗ [y_{kl}], [φ_{κλ}] ⊗ [ψ_{μν}]⟩
:= [⟨x_{ij}, φ_{κλ}⟩⟨y_{kl}, ψ_{μν}⟩]_{(ikκμ),(jlλν)} ∈ M_{p_{1}}(M_{p_{2}}(M_{q_{1}}(M_{q_{2}}))) = M_{p_{1}p_{2}q_{1}q_{2}}

obtains, where x = [x_{ij}] ∈ M_{p_{1}}(X), y = [y_{kl}] ∈ M_{p_{2}}(Y), φ = [φ_{κλ}] ∈ M_{q_{1}}(X^{∗}), ψ = [ψ_{μν}] ∈ M_{q_{2}}(Y^{∗}).

12.1.3 Tensor matrix multiplication

The definition of the completely bounded bilinear maps as well as the Haagerup tensor product relies on the tensor matrix multiplication [Eff87]

x ⊙ y = [x_{ij}] ⊙ [y_{jk}] := [∑_{j=1}^{l} x_{ij} ⊗ y_{jk}] ∈ M_{n}(X ⊗ Y)

of operator matrices x = [x_{ij}] ∈ M_{n,l}(X), y = [y_{jk}] ∈ M_{l,n}(Y).

The amplification of the bilinear mapping ⊗ : X × Y → X ⊗ Y is given by

⊗^{(n,l)} = ⊙ : M_{n,l}(X) × M_{l,n}(Y) → M_{n}(X ⊗ Y).

For scalar matrices α, γ ∈ M_{n}, β ∈ M_{l} we have

(αxβ) ⊙ (yγ) = α(x ⊙ (βy))γ.

We use the short hand notation αxβ ⊙ yγ.

75The duality ⟨X ⊗ Y, X^{∗} ⊗ Y^{∗}⟩ is defined by ⟨x ⊗ y, φ ⊗ ψ⟩ := ⟨x, φ⟩⟨y, ψ⟩ for x ∈ X, y ∈ Y, φ ∈ X^{∗}, ψ ∈ Y^{∗}.

For linear maps

Φ = [Φ_{ij}] : x → M_{n,l}(V),
Ψ = [Φ_{jk}] : x → M_{l,n}(W)

we denote by Φ ⊙ Ψ the mapping

Φ ⊙ Ψ = [∑_{j=1}^{l} Φ_{ij} ⊗ Ψ_{jk}] : X ⊗ Y → M_{n}(V ⊗ W),

Φ ⊙ Ψ : x ⊗ y ↦ [∑_{j=1}^{l} Φ_{ij}(x) ⊗ Ψ_{jk}(y)] .

We then have

(Φ ⊙ Ψ)^{(p)}(x ⊙ y) = (Φ^{(p,q)}(x)) ⊙ (Ψ^{(q,p)}(y))

for x ∈ M_{p,q}(X), y ∈ M_{q,p}(Y).

Let ⊗_{α} be an operator space tensor product. We define the tensor matrix multiplication ⊙_{α} of operator matrices S = [S_{i,j}] ∈ M_{n,l}(CB(X_{1}, X_{2})), T = [T_{k,l}] ∈ M_{l,n}(CB(Y_{1}, Y_{2})) of completely bounded maps by setting

S ⊙_{α} T = [∑_{j=1}^{l} S_{ij} ⊗_{α} T_{jk}] ∈ M_{n}(CB(X_{1} ⊗_{α} Y_{1}, X_{2} ⊗_{α} Y_{2})).

12.2 Interpolation

Intersection and sum

Let X and Y be operator spaces such that M_{1}(X) and M_{1}(Y) are embedded in a Hausdorff topological vector space. M_{n}(X ∩ Y) is given a norm via M_{n}(X ∩ Y) := M_{n}(X) ∩ M_{n}(Y). So we have:

∥[x_{ij}]∥_{M_{n}(X∩Y)} = max {∥[x_{ij}]∥_{M_{n}(X)}, ∥[x_{ij}]∥_{M_{n}(Y)}}.

The operator space X ∩ Y is called the intersection of X and Y. For operator spaces^{76} X and Y, by embedding X ⊕Y in (X^{∗}⊕_{∞}Y^{∗})^{∗} we obtain an operator

∥x∥_{E+F} = inf_{x=x_{E}+x_{F}} (∥x_{E}∥_{E} + ∥x_{F} ∥_{F} ) .

76 Let E, F be Banach spaces. Then we have their 1-direct sum E ⊕_{1} F with the norm

∥(x_{E}, x_{F})∥ = ∥x_{E}∥ + ∥x_{F} ∥ and their sum E + F with the quotient norm

space structure X ⊕_{1} Y. We write ♢ := {(x, −x)} ⊂ X ⊕_{1} Y. The quotient operator space (X ⊕_{1} Y) /♢ is called the sum of X and Y and is denoted by X + Y. We have

∥[x_{ij}]∥_{M_{n}(X+Y)} = inf_{[x_{ij}]=[x_{ij}]_{X}+[x_{ij}]_{Y}} ∥[(x_{ij_{X}} , x_{ij_{Y}})]∥_{M_{n}(X⊕_{1}Y)}.

Interpolation

Let E_{0}, E_{1} be Banach spaces continuously embedded in a Hausdorff topological vector space. The pair (E_{0}, E_{1}) is called a compatiple couple in the sense of interpolation theory [BL76]. Then we can define the interpolation space E_{θ} := (E_{0}, E_{1})_{θ} for 0 < θ < 1.

Pisier introduced the analogous construction for operator spaces [Pis96, §2]: Let X_{i} (i = 0, 1) be operator spaces continuously embedded in a Hausdorff topological vector space V. Then we have specific norms on M_{n}(X_{i}) and continuous linear inclusions M_{n}(X_{i}) ↪ M_{n}(V) for all n ∈ ℕ.^{77} The interpolated operator space X_{θ} is defined via M_{n}(X_{θ}) := (M_{n}(X_{0}), M_{n}(X_{1}))_{θ}. Let X be an operator space, H a Hilbert space and V : H → X a bounded linear and injective mapping with dense range such that the mapping^{78} V V^{∗} : \overline{X^{∗}} → X also is bounded, linear and injective with dense range. Then we have completely isometrically [Pis96, Cor. 2.4]:

(\overline{X^{∗}}, X)_{\frac{1}{2}} =^{cb} OH_{H}.

Examples

1. (R_{H}, C_{H})_{\frac{1}{2}} =^{cb} OH_{H} =^{cb} (MIN_{H}, MAX_{H})_{\frac{1}{2}}

2. (C_{H} ⊗_{h} R_{H}, R_{H} ⊗_{h} C_{H})_{\frac{1}{2}} =^{cb} OH_{H} ⊗_{h} OH_{H} =^{cb} OH_{H⊗H}

In this manner one also obtains operator space structures on the Schatten ideals S_{p} = (S_{∞}, S_{1})_{\frac{1}{p}} for 1 ≤ p ≤ ∞.

13 Symbols

Sets

ℕ the set of natural numbers
ℤ the set of integers
ℚ the set of rational numbers
ℝ the set of real numbers
ℂ the set of complex numbers

77We identify M_{n}(V) with V^{n^{2}}.

78As usual we identify H with its dual.

What are operator spaces ? January 6, 2001 http://www.math.uni-sb.de/∼wittck/ 62

Banach spaces

E, F Banach spaces
H, K Hilbert spaces
⨁2 Hilbert space direct sum
H^{n}  ⨁_{i=1}^{n} H
ℓ_{2}^{n}  the n-dimensional Hilbert space

Algebras

B(H) the algebra of bounded linear operators on H
A, B C^{∗}-algebras
A^{op}  the opposite algebra of an algebra A
𝟙_{A}  the identity in A
M, N von Neumann algebras

Ideals

S_{p}  the Schatten-p-classes
K(·, ·) the ideal of compact operators
K(·, ·) the closed linear span of the elementary operators; ideal in B^{ad}(·, ·)
N(·, ·) the ideal of nuclear operators
HS(·, ·) the ideal of Hilbert-Schmidt operators (S_{2})
Π_{2}(·, ·) the ideal of absolutely 2-summing operators
π_{2}(·)  the absolutely 2-summing norm

Mapping spaces

Γ_{2}(·, ·) operators factorizing through the column Hilbert space
γ_{2}(·)  the norm on Γ_{2}
CN (·, ·) completely nuclear operators
ν(·)  the norm on CN
CI (·, ·) completely integral operators
ι(·)  the norm on CI

What are operator spaces ? January 6, 2001 http://www.math.uni-sb.de/∼wittck/ 63

Operator spaces

X, Y  operator spaces
B(H)  aqlgebra of bounded linear operators on H
M_{n}(X)  M_{n} ⊗ X matrices with entries from X (algebraically)
M_{1}(X)  first level of the operator space X
CB(X, Y)  the operator space of completely bounded mappings

CB(X, Y)_{A} the operator space of completely bounded right A-module homomorphisms

CB(X × Y; Z) the operator space of completely bounded bilinear mappings
JCB(X × Y; Z) the operator space of jointly completely bounded bilinear mappings
∥ · ∥_{jcb}  norm of a jointly completely bounded bilinear mapping
X_{0}, Y_{0}  operator subspace of the corresponding operator spaces
X^{∗}  dual of the operator space X

Special operator spaces

MIN (E) the minimal operator space on E
MAX (E) the maximal operator space on E
MIN_{H}  the minimal operator space on H
MAX_{H}  the maximal operator space on H
R_{H}  the row Hilbert space
R_{n}  R_{ℓ_{2}^{n}}
C_{H}  the column Hilbert space
C_{n}  C_{ℓ_{2}^{n}}
OH_{H}  the operator Hilbert space

Norms

∥ · ∥_{cb} completely bounded norm
∥ · ∥_{row} row norm
∥ · ∥_{col} column norm
∥ · ∥_{n}  norm on M_{n}(X)
∥ · ∥_{m,n} norm on M_{m,n}(X)

What are operator spaces ? January 6, 2001 http://www.math.uni-sb.de/∼wittck/ 64

Matrices

M_{n,m}(X) n × m-matrices over X
M_{n}(X) M_{n,n}(X)
M_{n,m}  M_{n,m}(ℂ)
M_{n}  M_{n,n}
𝕄_{n,m}(X) the operator space of n × m-matrices over X
𝕄_{n}(X)  𝕄_{n,n}(X)
𝕄_{n,m}  𝕄_{n,m}(ℂ)
𝕄_{n}  𝕄_{n,n}
C_{n}(X)  𝕄_{n,1}(X), the columns of an operator space
R_{n}(X)  𝕄_{1,n}(X), the rows of an operator space

Tensor products

⊙  tensor matrix product
⊗  algebraic tensor product
⊗_{A}  algebraic module tensor product
˜⊗  completion of the algebraic tensor product
⊗_{α}  operator space tensor product
⊗_{α^{∗}}  dual operator space tensor product
⊗_{h}  the Haagerup tensor product
⊗_{hA}  the module Haagerup tensor product
⊗^{∨}  the injective tensor product
⊗^{∧}  the projective tensor product
⊗_{λ}  the injective Banach space tensor product
⊗_{γ}  the projective Banach space tensor product
⊗_{ext}  the external tensor product
⊗_{Θ}  the internal tensor product
S ⊗_{α} T α-tensor product of the completely bounded operators S, T

Tensor norms

∥ · ∥_{α} α-operator space tensor norm
∥ · ∥_{α,n} α-operator space tensor norm on the nth level
∥ · ∥_{α^{∗}} dual operator space tensor norm of ∥ · ∥_{α}
∥ · ∥_{∨} injective operator space tensor norm
∥ · ∥_{∧} projective operator space tensor norm
∥ · ∥_{h} Haagerup operator space tensor norm
∥ · ∥_{λ} injective Banach space tensor norm
∥ · ∥_{γ} projective Banach space tensor norm

Operator modules

A_{1}, A_{2}  C^{∗}-algebras
M, N  von Neumann algebras
Aut(M)  the set of ^{∗}-automorphisms of M

CB_{(A_{1},A_{2})}(X, Y) the operator space of completely bounded left A_{1}- right A_{2}-module homomorphisms between X and Y

CB_{(A_{1},A_{2})}^{σ}(X, Y) the operator space of normal completely bounded left A_{1}- right A_{2}- module homomorphisms between X and Y

CB_{(A_{1},A_{2})}^{s}(X, Y) the operator space of singular completely bounded left A_{1}- right A_{2}- module homomorphisms between X and Y

Hilbert-C^{∗}-modules

\overline{X}  conjugate Hilbert-C^{∗}-module
B(·, ·)_{A}  the space of bounded right A-module homomorphisms
⟨·, ·⟩_{A}  inner product of a right Hilbert-A-module
A⟨·, ·⟩  inner product of a left Hilbert-A-module
⟨·, ·⟩  inner product of a Hilbert space

B^{ad}(X, Y) the operator space of adjointable A-module homomorphisms between Hilbert-C^{∗}-modules X and Y

Mappings

𝟙_{B(H)} the identity of B(H)
𝟙_{n}  the identity of M_{n}
π  representations
Φ^{(n)}  nth amplification of a linear mapping Φ
Φ^{(n,l)} (n, l)th amplification of a bilinear mapping Φ
Φ^{(n)}  Φ^{(n,n)}, nth amplification of a bilinear mapping Φ
Φ^{(n)}  nth amplification of a multilinear mapping Φ
Φ^{(p×q)} joint amplification of a bilinear mapping Φ
Φ̃  linearization of a bilinear mapping Φ

Φ^{σ} the normal (=w^{∗}-w^{∗}-continuous) part of a mapping Φ between dual spaces

Φ^{s}  the singular part of a mapping Φ between dual spaces
Θ  faithful non-degenerate ∗-representation of a C^{∗}-algebra

Isomorphisms

=^{cb} completely isometrically isomorphic
≃^{cb} completely isomorphic
≃ isomorphic

Miscellanea

Ball  unit ball
⨁_{i=1}^{n} X_{i} nth direct sum
⨁_{n∈ℕ} X_{n} countable direct sum
T^{∗}  the adjoint of the operator T

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Index

Subindex see amplification see axioms see bilinear mapping see C^{∗}-algebra see completelely bounded mapping see examples see Haagerup tensor product see Hilbertian operator space see injective op. space tensor product see lemma of see mapping see mapping space see module homomorphism see multilinear mapping see operator algebra see operator module see operator space see operator space tensor product see projective op. space tensor product see special operator spaces see theorem of

absolutely matrix convex, 48 absolutely matrix convex combination, 48 abstract operator space, see

operator space algebra homomorphism, 24 amplification, 4,

bilinear mapping, see

bilinear mapping joint, see

bilinear mapping automatic complete boundedness, 45 axioms,

of Ruan operator module, 20 operator space, 3

bilinear mapping,

amplification, 23, 38 to quadratic matrices Φ^{(n)}, 38

to rectangular matrices Φ^{(n,l)}, 38 completely bounded, 23, 37, see Haagerup tensor product, 39

norm ∥·∥_{cb}, 39 representation, 40 space CB(X × Y; Z), 40 transposition, 40 joint amplification, 38

Φ^{(p×q)}, 38

matrix duality, 38 jointly completely bounded, 37, 37,

see

projective operator space tensor product, 39

norm ∥·∥_{jcb}, 39

space JCB(X × Y; Z), 39

transposition, 39 linearization Φ̃, 38 transposed mapping Φ^{t}, 39, 40

bipolar theorem, 50

Calkin algebra, 22 CB-ideal property, 55 CB-mapping space, 54 characterization

completely locally reflexive operator

spaces, 42 Christensen

theorem of, 43 column Hilbert space, see

hilbertian operator space column norm, 17 column space C_{p}, 7 complete quotient mapping, see

completely bounded mapping completely bounded mapping, 4,

complete quotient mapping, 4 completely contractive, 4 completely isometric, 4 completely contractive, see

completely bounded mapping completely isometric, see

completely bounded mapping, 9 completely locally reflexive, 42 concrete operator space, see

operator space convex combination

absolutely matrix convex combina-

tion, 48 matrix convex combination, 48

cross norm, 28 C^{∗}-algebra, 5, 19, 23,

commutative, 5, 11, 43 injective, 22 nuclear, 43 quasi-cyclic, 46

decomposition theorem

for completely bounded symmetric

multilinear mappings, 45 for self-adjoint completely bounded

bimodule homomorphisms, 22 of Tomiyama-Takesaki, 22

derivations completely bounded, 43 inner, 43

dominate, 4, 24 dual

of an operator space, see

operator space operator space tensor norm, see

operator space tensor norm operator space tensor product, see

operator space tensor product

embedding

canonical, 8, 57 examples, 48,

Haagerup tensor product, 34 hilbertian operator spaces, 15 injective op. space tensor product, 30 module Haagerup tensor product, 42 of operator modules, 20 operator algebra, 23, 24 operator space, 5, 7 operator spaces

completely locally reflexive, 43 projective op. space tensor product,

32 extension theorem

for completely bounded module ho-

momorphisms, 21 for completely bounded multilinear

mappings, 44

factorization

through a column Hilbert space, see hilbertian operator space

finite dimensional mappings, 54

Grothendieck H-tensor product, 34

H-tensor product, 34 Haagerup tensor, norm, 33

infimum formula, 33 minimum formula, 34 sum formula, 34 supremum formula, 35

product, 33 H-tensor product, 34 interpolation, 34 involving column Hilbert space, 36,

37 involving row Hilbert space, 36, 37 lemma of Blecher and Paulsen, 36 shuffle map, 35 tensor matrix product ⊙_{h}, 36

hilbertian operator space, 15

column Hilbert space C, 15, 17 as an operator algebra, 24 characterizations, 18 factorization through a, 18

homogeneous, 15

completely self dual, 15 maximal, 15 minimal, 15 morphisms, 16 operator Hilbert space OH , 15, 61

as an operator algebra, 24 row Hilbert space R, 15

as an operator algebra, 24 Hochschild cohomology, 43

ICM 1986, 3 ideal

of an operator algebra, see

operator algebra Schatten class, 16 symmetrically normed, 16

ideal property, 55 injective operator space,

tensor norm, 29 tensor product, 29

and λ-tensor product, 30 interpolation of operator spaces, 61 interpolation space, 61 intersection of operator spaces, 60

joint

amplification, see bilinear mapping

lemma of, see

theorem of Blecher and Paulsen, 36 Smith, 5, 46

locally reflexive, 42 completely, 42

M-ideal, 42 mapping,

complete quotient mapping, 9 completely bounded, 3, see

completely bounded mapping completely integral, 56

characterization, 57 completely nuclear, 55

characterization, 56 completely positive, 3 Hilbert-Schmidt, 16 integral, 57 nuclear, 16, 56 shuffle, see

operator space tensor product, 27 mapping space, 54, see

Abbildungsraum, completely integral mappings, 56 completely nuclear mappings, 55

matricially normed space, 4, see

space matrix

set of matrices, 47 matrix cone, 48 matrix convex, 48 absolutely, 48

matrix convex combination, 48 matrix duality, see

bilinear mapping matrix multiplication, 23 matrix polar, 50 absolute, 50

module homomorphism, 22 completely bounded

decomposition theorem, 22 extension theorem, 21 module Haagerup tensor product,

23, 42 representation (K; π; v^{∗}; w), 21 representation theorem for, 20

normal, 22 self-adjoint, 21 singular, 22 multilinear mapping, amplification, 44 completely bounded, 44 extension theorem, 44 representation theorem, 44 completely bounded symmetric decomposition theorem, 45

linearization, 44

Haagerup tensor product, 44 multimodular, 47

normal

module homomorphism, see module homomorphism operator

2-summing, 16 adjoint, 9

operator algebra, 23 abstract, 23 concrete, 23 ideal in an, 24 Ruan’s theorem, 23

operator ideal, 55 operator module, see

module homomorphisms, (A_{1}, A_{2})-, 20 abstract, 20 concrete, 20 examples, 20 operator bimodule

A-, 20 representation theorem, 20

operator space, 3, 4, abstract, 5 completely locally reflexive, 42 concrete, 5 dual, 8 finite dimensional, 42 hilbertian, 15, see

hilbertian operator space homogeneous, 4

completely self dual, see

hilbertian operator space injective, 17 interpolated, 61 matrix level, 4, 6 maximal, 10, 50 minimal, 10, 50 operator space norm, 3 quotient of, 6 reflexive, 8 subspace of, 6

operator space tensor,

norm, 26

Haagerup, see

Haagerup tensor norm injective, see

injective maximal, see projective minimal, see injective projective, see projective spatial, see injective product, 25

associativity of, 28

axioms, 26 cross norm, 28 Haagerup, see

Haagerup tensor product injective, see injective injectivity of, 28 maximal, see projective minimal, see injective projective, see projective projectivity of, 28 self-duality of, 28 shuffle mapping, 27 spatial, see injective symmetry of, 28 operator space tensor norm

projective, 55 operator systems, 3

Paulsen

conjecture of, 16 projective operator space, tensor norm, 31 tensor product, 31

and γ-tensor product, 32 column Hilbert space, 32 Fourier algebra, 32 predual of a von Neumann algebra,

32 row Hilbert space, 32 quasi-cyclic, 46 quotient

of an operator space, see operator space

quotient mapping

complete, see mapping

representation theorem

for completely bounded module ho-

momorphisms, 20 for completely bounded multilinear

mappings, 44 operator modules, 20

row Hilbert space, see

hilbertian operator space row norm, 17 row space R_{p}, 7 Ruan

axioms operator modules, 20 operator space, 3

theorem of, 5

Schatten ideals S_{p}, 16, 24

operator space structure, 61 Schur product, 24 self-adjoint

algebra in B(H), 23 separation theorem, 49 set of matrices, 47 shuffle mapping, see

operator space tensor product, 27 similarity problem, 43 singular

module homomorphism, see module homomorphism

singular values, 16 special operator spaces,

CB(X, Y), 7

intersection of operator spaces X ∩Y,

60 operator Hilbert space OH , 61 sum of op. spaces X + Y, 61 standard dual, 8 ∗-representation, 20 state space

generalized, 51 structural element, 51 subspace

of an operator space, see operator space sum of operator spaces, 60, 61

tensor norm

of operator spaces, see

operator space tensor norm tensor product

of operator spaces, see

operator space tensor product cross norm, 28 Haagerup, see Haagerup

theorem bipolar theorem, 50 separation theorem, 49

theorem of, see

lemma of Christensen, 43 Ruan, 5

for operator algebras, 23 topology

non commutative, 3