Subspaces and quotients

Let $X$ be a matricially normed space and $X_0\subset X$ a linear subspace. Then $M_n(X_0)\subset M_n(X)$, and $X_0$ together with the restriction of the operator space norm again is a matricially normed space. The embedding $X_0 \hookrightarrow X$ is completely isometric. If $X$ is an operator space and $X_0\subset X$ is a closed subspace, then $M_n(X_0)\subset M_n(X)$ is closed and $X_0$ is an operator space.

Algebraically we have $M_n(X/X_0)=M_n(X)/M_n(X_0)$. If $X_0$ is closed, then $X/X_0$ together with the quotient norm on each matrix level is matricially normed (an operator space if $X$ is one). The quotient mapping $X \twoheadrightarrow X/X_0$ is a complete quotient mapping.

More generally, a subspace of a matricially normed space (operator space) $X$ is a matricially normed space (operator space) $Y$ together with a completely isometric operator $Y\to X$. A quotient of $X$ is a matricially normed space (operator space) $Y$ together with a complete quotient mapping $X\to Y$.