# Projective and free matricially normed spaces

## TL;DR

This paper characterizes metrically projective and free matricially normed spaces, describing their structure in terms of special matrix-normed spaces hat M_n, and explores their relation to classical L^p spaces.

## Contribution

It provides a detailed description of metrically free and projective matricially normed spaces using hat M_n spaces and clarifies their properties and distinctions from L^p spaces.

## Key findings

- Metrically free spaces are matricial l_1-sums of hat M_n spaces.
- Metrically projective spaces are direct summands of l_1-sums of hat M_n spaces.
- hat M_n spaces do not belong to any L^p class but behave similarly to L^1.

## Abstract

We study metrically projective and metrically free matricially normed spaces. We describe these spaces in terms of a special space $\widehat M_n$, 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 $\widehat M_n$, whereas metrically projective matricially normed spaces are complete direct summands of matricial $\ell_1$-sums of arbitrary families of the spaces $\widehat M_n$. At the end we specify the underlying normed space of $\widehat M_n$ and show that the spaces $\widehat M_n$; $n>1$ 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 $\widehat M_n$ resembles that of $L^1$-spaces.