The Serre-Swan theorem for normed modules

TL;DR

This paper investigates the structure of $L^0$-normed $L^0$-modules over metric measure spaces, establishing conditions for their representation as sections of measurable Banach bundles and exploring categorical equivalences.

Contribution

It provides a detailed analysis of the conditions under which $L^0$-modules correspond to measurable Banach bundles, extending the Serre-Swan theorem to this setting.

Findings

Identifies conditions for $L^0$-modules to be sections of Banach bundles

Establishes an equivalence of categories between modules and bundles

Advances the differential calculus on metric measure spaces

Abstract

The aim of this note is to analyse the structure of the $L^0$-normed $L^0$-modules over a metric measure space. These are a tool that has been introduced by N. Gigli to develop a differential calculus on spaces verifying the Riemannian Curvature Dimension condition. More precisely, we discuss under which conditions an $L^0$-normed $L^0$-module can be viewed as the space of sections of a suitable measurable Banach bundle and in which sense such correspondence can be actually made into an equivalence of categories.