Projective operator space tensor product

The projective operator space tensor product $X \stackrel{\scriptscriptstyle \wedge}{\otimes}Y$ of two operator spaces $X$ and $Y$ is characterized by the following complete isometry:

$$(X \stackrel{\scriptscriptstyle \wedge}{\otimes}Y)^* %%\cbgl JCB... ...\mathrm{cb}}{=}\mathit{CB}(X,Y^*) \stackrel{\mathrm{cb}}{=}\mathit{CB}(Y,X^*).$$

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]

$$\mathit{CB}(X \stackrel{\scriptscriptstyle \wedge}{\otimes}Y,Z) \stackrel{\mathrm{cb}}{=}\mathit{JCB}(X \times Y; Z),$$

where $Z$ is an operator space.

Here, $\mathit{JCB}(X \times Y; Z)$ 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 $u \in M_n(X\otimes Y)$: (cf. [ER91, Formel (2.10)])

$$\Vert u\Vert _\wedge = \inf\left\{ \Vert\alpha\Vert \Vert x\Vert _p \Vert y\Vert _q \Vert\beta\Vert \ :\ u = \alpha(x \otimes y)\beta \right\},$$

where $p,q\in{\mathbb{N}}$, $x \in M_p(X)$, $y\in M_q(Y)$ and $\alpha \in M_{n,pq}$, $\beta \in M_{pq,n}$.

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].