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 $X \otimes_h Y$ of two operator spaces $X$ and $Y$ is characterized by the complete isometry

$$(X\otimes_h Y)^* \stackrel{\mathrm{cb}}{=}\mathit{CB}(X \times Y;{\mathbb{C}}),$$

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

$$\mathit{CB}(X \otimes_h Y; Z) \stackrel{\mathrm{cb}}{=}\mathit{CB}(X \times Y; Z)$$

completely isometrically.

Here, $\mathit{CB}(X \times Y; Z)$ denotes the operator space of completely bounded bilinear mappings.

For operator spaces $X$ and $Y$ the Haagerup operator space tensor norm of $u \in M_n(X\otimes Y)$ is explicitly given by (cf. [ER91, Formel (2.11)], [BP91, Lemma 3.2])

$$\Vert u\Vert _h = \inf \Vert x\Vert _{n,l}\ \Vert y\Vert _{l,n},$$

where $l \in {\mathbb{N}}$, $x \in M_{n,l}(X)$, $y \in M_{l,n}(Y)$ and $u$ is the tensor matrix product $u = x \odot y$. The Haagerup tensor product $X \otimes_h Y$ then of course is the completion of the algebraic tensor product $X \otimes Y$ with respect to this operator space tensor norm. There are several other useful formulae , 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

$$X^* \otimes_h Y^* \hookrightarrow (X \otimes_h Y)^*$$

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

$$X \otimes_h Y \subset X \stackrel{\scriptscriptstyle \vee}{\otimes} Y.$$

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 $(X_0 \otimes _h Y_0, X_1 \otimes_h Y_1)$ is a compatible pair of operator spaces and we have completely isometrically

$$(X_0 \otimes _h Y_0, X_1 \otimes_h Y_1)_\vartheta \stackrel{\mathrm{cb}}{=} (X_0,X_1)_\vartheta \otimes_h (Y_0,Y_1)_\vartheta$$

for $0 \leq \vartheta \leq 1$.

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 Grothendieck³⁹ for normed spaces $E$ and $F$ [BP91, pp. 277-279, Prop. 4.1]. In fact, on the first matrix level we have:

$$\mathit{MIN}(E)\otimes_h \mathit{MIN}(F) = E \otimes_H F,$$

$$\mathit{MAX}(E)\otimes_h \mathit{MAX}(F) = E \otimes_{H^*} F$$

isometrically. The non-associativity of the $\H$-tensor product is reflected by the fact that in general $\mathit{MIN}(E)\otimes_h \mathit{MIN}(F)$ and $\mathit{MIN}(E \otimes_H F)$ are not completely isometric.

Footnotes

... Grothendieck³⁹

This tensor norm also is known as $\gamma_2$ [Pis86].