The Cuntz semigroup and stability of close $C^*$-algebras

Abstract

We prove that separable $C^{\ast }$-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 $C^{\ast }$-algebras provided that one algebra has stable rank one; close $C^{\ast }$-algebras must have affinely homeomorphic spaces of lower-semicontinuous quasitraces; strict comparison is preserved by sufficient closeness of $C^{\ast }$-algebras. We also examine $C^{\ast }$-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 $C^{\ast }$-algebras have isomorphic Cuntz semigroups when one algebra absorbs the Jiang–Su algebra tensorially.