The Cuntz semigroup and stability of close C*-algebras

TL;DR

This paper demonstrates that close C*-algebras have isomorphic Cuntz semigroups and preserves several properties like stability and strict comparison, with implications for algebras related to Kadison's Similarity Problem.

Contribution

It establishes the invariance of the Cuntz semigroup under closeness and explores stability and comparison properties in this context.

Findings

Isomorphic Cuntz semigroups for close C*-algebras

Stability is preserved under closeness if one algebra has stable rank one

Strict comparison is maintained when C*-algebras are sufficiently close

Abstract

We prove that separable C*-algebras which are completely close in a natural uniform sense have isomorphic Cuntz semigroups, continuing a line of research developed by Kadison - Kastler, Christensen, and Khoshkam. This result has several applications: we are able to prove that the property of stability is preserved by close C*-algebras provided that one algebra has stable rank one; close C*-algebras must have affinely homeomorphic spaces of lower-semicontinuous quasitraces; strict comparison is preserved by sufficient closeness of C*-algebras. We also examine C*-algebras which have a positive answer to Kadison's Similarity Problem, as these algebras are completely close whenever they are close. A sample consequence is that sufficiently close C*-algebras have isomorphic Cuntz semigroups when one algebra absorbs the Jiang-Su algebra tensorially.