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.⁵⁷ 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).

(1)

``Criterion: spaces''

(1.1)

The linear case

• Smith's lemma: If $X$ is a matricially normed space and $\Phi: X \rightarrow M_n$, where $n\in{\mathbb{N}}$, is a bounded linear operator, then we have $\Vert\Phi\Vert _{\mathrm{cb}} = \Vert\Phi^{(n)}\Vert$. In particular, $\Phi$ is completely bounded if and only if $\Phi^{(n)}$ is bounded [Smi83, Thm. 2.10].
• Let $X$ be an operator space, $A$ a commutative $C^*$-algebra and $\Phi: X \rightarrow A$ a bounded linear operator. Then $\Phi$ is completely bounded and $\Vert\Phi\Vert _{\mathrm{cb}}=\Vert\Phi\Vert$ ([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 $M_n \otimes C(\Omega)$ for a compact Hausdorff space $\Omega$ [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 $\Phi: A \times A \rightarrow {\mathbb{C}}$ is automatically completely bounded and
  $$\Vert\Phi\Vert \leq \Vert\Phi\Vert _{\mathrm{cb}} \leq K_G \Vert\Phi\Vert$$,
  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.⁵⁸
• Let $A$ and $B$ be $C^*$-subalgebras of $B(\H)$, where $\H$ is a Hilbert space. Let further $E \subset B(\H)$ be an $(A,B)$-operator module and $\Phi: E \rightarrow B(\H)$ be a bounded $(A,B)$-module homomorphism. Let $A$ and $B$ be quasi-cyclic⁵⁹, i.e., for every finite set of vectors $\xi_i, \eta_j \in \H$, $1 \leq i,j \leq n$, there exist $\xi, \eta \in \H$ such that $\xi_i \in {\overline{A \xi}}$, $\eta_j \in {\overline{B \eta}}$, $1 \leq i,j \leq n$. Then $\Phi$ is completely bounded and $\Vert\Phi\Vert _{\mathrm{cb}}=\Vert\Phi\Vert$ ([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].

(2.2)

The bilinear case

• Let $A \subset B(\H)$ be a $C^*$-algebra and $E \subset B(\H)$ an $A$-operator bimodule. A bilinear mapping $\Phi: E \times E \rightarrow B(\H)$ is called $A$-multimodular if
  $$\Phi(aeb, fc) = a \Phi(e, bf) c$$
  holds for all $a, b, c \in A$ and $e, f \in E$.

  We have a bilinear analogue of the above theorem:

  Let $A \subset B(\H)$ be a quasi-cyclic $C^*$-algebra and $E \subset B(\H)$ be an $A$-operator bimodule. Let further $\Phi: E \times E \rightarrow B(\H)$ be a bilinear, $A$-multimodular mapping such that the corresponding linearization ${\widetilde{\Phi}}: E \otimes E \rightarrow B(\H)$ is bounded on $E \otimes_h E$.

  Then $\Phi$ is completely bounded and $\Vert\Phi\Vert _{\mathrm{cb}}=\Vert{\widetilde{\Phi}}\Vert$ [SS95, Thm. 1.6.2].

Footnotes

... mappings.⁵⁷

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

... contractive.⁵⁸

This is evident because $\pi^{(n)}: M_n(A) \rightarrow M_n(B)$ is a $^*$-homomorphism for every $n\in{\mathbb{N}}$.

...quasi-cyclic⁵⁹

For 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].