Matrices over an operator space

The vector space $M_p(X)$ of matrices over a matricially normed space $X$ itself is matricially normed in a natural manner: The norm on the $n$th level $M_n(M_p(X))$ is given by the identification

$$M_n(M_p(X)) = M_{np}(X)$$

[BP91, p. 265]. We⁸write

$${\mathbb{M}}_p(X)$$

for $M_p(X)$ with this operator space structure. In particular,

$${M_1}({\mathbb{M}}_p(X)) = M_p(X)$$

holds isometrically. Analogously $M_{p,q}(X)$ becomes a matricially normed space ${\mathbb{M}}_{p,q}(X)$ by the identification

$$M_n(M_{p,q}(X))=M_{np,nq}(X)$$   .

By adding zeros it is a subspace of ${\mathbb{M}}_r(X)$ for $r\geqslant p$, $q$.

Examples: For a $C^*$-algebra $A$, ${\mathbb{M}}_p(A)$ is the $C^*$-Algebra of $p\times p$-matrices over $A$ with its natural operator space structure.

The Banach space $M_p(A)$ is the first matrix level of the operator space ${\mathbb{M}}_p(A)$.

The complex numbers have a unique operator space structure which on the first matrix level is isometric to ${\mathbb{C}}$, and for this $M_p({\mathbb{C}}) = M_p$ holds isometrically. We write

$${\mathbb{M}}_p := {\mathbb{M}}_p({\mathbb{C}})$$   .

Then ${\mathbb{M}}_p$ always stands for the $C^*$-algebra of $p\times p$-matrices with its operator space structure. The Banach space $M_p$ is the first matrix level of the operator space ${\mathbb{M}}_p$.

Footnotes

... We⁸

In the literature, the symbol $M_p(X)$ stands for both the operator space with first matrix level $M_p(X)$ and for the $p$th level of the operator space $X$. We found that the distinction between ${\mathbb{M}}_p(X)$ and $M_p(X)$ clarifies for instance the definition of the operator space structure of $CB(X,Y)$.