The spaces

Let $X$ be a complex vector space. A matrix seminorm [EW97b] is a family of mappings $\Vert\cdot\Vert:M_n(X)\to{\mathbb{R}}$, one on each matrix level¹ $M_n(X)=M_n\otimes X$ for $n\in{\mathbb{N}}$, such that

(R1) $\Vert\alpha x\beta\Vert\leqslant\Vert\alpha\Vert\Vert x\Vert\Vert\beta\Vert$ for all $x\in M_n(X)$, $\alpha\in M_{m,n}$, $\beta\in M_{n,m}$
(R2) $\Vert x\oplus y\Vert=\max\{\Vert x\Vert,\Vert y\Vert\}$ for all $x\in M_n(X)$, $y\in M_m(X)$.²

Then every one of these mappings $\Vert\cdot\Vert:M_n(X)\to{\mathbb{R}}$ is a seminorm. If one (and then every one) of them is definite, the operator space seminorm is called a matrix norm.

A matricially normed space is a complex vector space with a matrix norm. It can be defined equivalently, and is usually defined in the literature, as a complex vector space with a family of norms with (R1) and (R2) on its matrix levels.

If $M_n(X)$ with this norm is complete for one $n$ (and then for all $n$), then $X$ is called an operator space³([Rua88], cf. [Wit84a]).

For a matricially normed space (operator space) $X$ the spaces $M_n(X)$ are normed spaces (Banach spaces).⁴These are called the matrix levels of $X$ (first matrix level, second level...).

The operator space norms on a fixed vector space $X$ are partially ordered by the pointwise order on each matrix level $M_n(X)$. One says that a greater operator space norm dominates a smaller one.

Footnotes

... level¹

The term matrix level is to be found for instance in

....²

It suffices to show one of the following two weaker conditions:

(R1${}^\prime$) $\Vert\alpha x\beta\Vert\leqslant\Vert\alpha\Vert\Vert x\Vert\Vert\beta\Vert$ for all $x\in M_n(X)$, $\alpha\in M_{n}$, $\beta\in M_{n}$,
(R2) $\Vert x\oplus y\Vert=\max\{\Vert x\Vert,\Vert y\Vert\}$ for all $x\in M_n(X)$, $y\in M_m(X)$,

which is often found in the literature, or

(R1) $\Vert\alpha x\beta\Vert\leqslant\Vert\alpha\Vert\Vert x\Vert\Vert\beta\Vert$ for all $x\in M_n(X)$, $\alpha\in M_{m,n}$, $\beta\in M_{n,m}$,
(R2${}^\prime$) $\Vert x\oplus y\Vert\leqslant\max\{\Vert x\Vert,\Vert y\Vert\}$ for all $x\in M_n(X)$, $y\in M_m(X)$,

which seems to be appropriate in convexity theory .

... space³

In the literature, the terminology is not conseqent. We propose this distinction between matricially normed space and operator space in analogy with normed space and Banach space.

... spaces).⁴

In the literature, the normed space $M_1(X)$ usually is denoted also by $X$. We found that a more distinctive notation is sometimes usefull.