The space $\mathit{CB}(X,Y)$

Let $X$ and $Y$ be matricially normed spaces. A matrix $[T_{ij}]\in M_n(\mathit{CB}(X,Y))$ determines a completely bounded operator

$$T:X \to {\mathbb{M}}_n(Y)$$

$$x \mapsto \left[T_{ij}(x)\right]$$

Defining $\left\Vert\left[T_{ij}\right]\right\Vert=\Vert T\Vert _\mathrm{cb}$, $\mathit{CB}(X,Y)$ becomes a matricially normed space . It is an operator space , if $Y$ is one. The equation

$${\mathbb{M}}_p(\mathit{CB}(X,Y)) \stackrel{\mathrm{cb}}{=}\mathit{CB}(X,{\mathbb{M}}_p(Y))$$

holds completely isometrically.