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 $\Psi : X \times Y \rightarrow W$ is called balanced, if the equation

$$\Psi (x \cdot a, y) = \Psi(x,a \cdot y)$$

obtains for all $x \in X, y \in Y, a \in A$. The module Haagerup tensor product is defined to be the operator space $X \otimes_{hA} Y$ (which is unique up to complete isometry) together with a bilinear, completely contractive, balanced mapping

$$\otimes_{hA} : X \times Y \rightarrow X \otimes_{hA} Y,$$

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

$$\Psi : X \times Y \rightarrow W$$

there is a unique linear completely bounded map

$$\widetilde{\Psi} : X \otimes_{hA} Y \rightarrow W$$

satisfying $\widetilde{\Psi} \circ \otimes_{hA} = \Psi$ and $\Vert \widetilde{\Psi} \Vert _{\mathrm{cb}} = \Vert \Psi \Vert _{\mathrm{cb}}$. The module Haagerup tensor product can be realized in different ways:

1. Let
   $$N := \mathrm{lin} \{ (x \cdot a) \otimes y - x \otimes (a \cdot y) \mid a \in A, x \in X, y \in Y\}$$.
   The quotient space $(X \otimes_h Y) \big/ \overline{N}$ with its canonical matrix norms is an operator space which satisfies the defintion of $X \otimes_{hA} Y$ [BMP].
2. Let us denote by $X \otimes_A Y$ the algebraic module tensor product, i.e. the quotient space $(X \otimes_{\mathrm{alg}} Y) \big{/}N$. For $n\in{\mathbb{N}}$ and $u \in M_n(X \otimes_A Y)$, by
   $$p_n(u) := \inf \left \{ \Vert S\Vert \Vert T\Vert \left\vert u =... ...ght ], l \in {\mathbb{N}}, S \in M_{nl}(X), T \in M_{ln}(Y) \right. \right\}$$
   we define a semi-norm on $M_n( X \otimes_A Y)$. We obtain
   $$\mathrm{Kern}(p_n) = M_n(\mathrm{Kern}(p_1)),$$
   and the semi-norms $p_n$ give an operator space norm on $(X \otimes_A Y) / \mathrm{Kern}(p_1)$ [Rua89]. The completion of this space satisfies the definition of $X \otimes_{hA} Y$ [BMP].