Stable Elements and Property (S)

TL;DR

This paper explores the relationship between stability and Property (S) in simple, stably finite C*-algebras, providing characterizations and equivalences among various regularity properties.

Contribution

It offers new characterizations of stable elements via support and links multiple regularity properties under mild conditions.

Findings

Cancellation at infinity on the Cuntz semigroup is equivalent to isomorphism of Hilbert modules.

Various regularity properties such as Regularity, ω-comparison, and Corona Factorization Property are shown to be equivalent.

Characterizations of stable elements in terms of support and projections in the multiplier algebra.

Abstract

We study the relation (and differences) between stability and Property (S) in the simple and stably finite framework. This leads us to characterize stable elements in terms of its support, and study these concepts from different sides : hereditary subalgebras, projections in the multiplier algebra and order properties in the Cuntz semigroup. We use these approaches to show both that cancellation at infinity on the Cuntz semigroup just holds when its Cuntz equivalence is given by isomorphism at the level of Hilbert right-modules, and that different notions as Regularity, $\omega$-comparison, Corona Factorization Property, property R, etc.. are equivalent under mild assumptions.