Projective and free matricially normed spaces

Abstract

We study metrically projective and metrically free matricially normed spaces. We describe these spaces in terms of a special space ${\hat{M}}_n$, the space of $n\times n$ matrices, endowed with a special matrix-norm. We show that metrically free matricially normed spaces are matricial ${\ell }_1$-sums of some distinguished families of matricially normed spaces ${\hat{M}}_n$, whereas metrically projective matricially normed spaces are complete direct summands of matricial ${\ell }_1$-sums of arbitrary families of the spaces ${\hat{M}}_n$. At the end we specify the underlying normed space of ${\hat{M}}_n$ and show that the spaces ${\hat{M}}_n$; $n>1$ do not belong to any of the classes $L^p$; $p\in [1,\infty ]$, introduced by Effros and Ruan. However, in a certain sense the behavior of ${\hat{M}}_n$ resembles that of $L^1$-spaces.