Projective Hilbert Modules and Sequential Approximation

TL;DR

This paper investigates the projectivity of countably generated Hilbert modules over separable C*-algebras and explores conditions under which the Cuntz semigroup aligns with classes of these modules, linking algebraic properties to module theory.

Contribution

It establishes the projectivity of countably generated Hilbert modules over separable C*-algebras and characterizes when the Cuntz semigroup corresponds to classes of these modules in certain algebraic settings.

Findings

Countably generated Hilbert modules are projective over separable C*-algebras.

The Cuntz semigroup matches the class of Hilbert modules iff the algebra has stable rank one.

Conditions involving strict comparison and affine functions on quasitraces are crucial.

Abstract

We show that, when $A$ is a separable C*-algebra, every countably generated Hilbert $A$-module is projective (with bounded module maps as morphisms). We also study the approximate extensions of bounded module maps. In the case that $A$ is a $\sigma$-unital simple C*-algebra with strict comparison and every strictly positive lower semicontinuous affine function on quasitraces can be realized as the rank of an element in Cuntz semigroup, we show that the Cuntz semigroup is the same as unitarily equivalent class of countably generated Hilbert $A$-modules if and only if $A$ has stable rank one.