Metric freedom and projectivity for classical and quantum normed modules

TL;DR

This paper develops a categorical framework for projective modules in functional analysis, characterizes metric projectivity for classical and quantum modules, and identifies metrically projective normed spaces as finitely supported $l_1$-subspaces.

Contribution

It introduces a general categorical approach to projectivity, characterizes metric free modules in classical and quantum contexts, and solves the case for normed spaces.

Findings

Metrically projective normed spaces are exactly finitely supported $l_1$-subspaces.

Extreme projectivity is a form of asymptotic metric projectivity.

All projective modules in the framework are retracts of free modules.

Abstract

In functional analysis there are several reasonable approaches to the notion of a projective module. We show that a certain general-categorical framework contains, as particular cases, all known versions. In this scheme, the notion of a free object comes to the forefront, and in the best of categories, called freedom-loving, all projective objects are exactly retracts of free objects. We concentrate on the so-called metric version of projectivity and characterize metrically free `classical', as well as quantum (= operator) normed modules. Hitherto known the so-called extreme projectivity turns out to be, speaking informally, a kind of `asymptotically metric projectivity'. Besides, we answer the following concrete question: what can be said about metrically projective modules in the simplest case of normed spaces? We prove that metrically projective normed spaces are exactly $l_1^0(M)$, the subspaces of $l_1(M)$, where $M$ is a set, consisting of finitely supported functions. Thus in this case the projectivity coincides with the freedom.