#### Vol. 7, No. 4, 2014

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

### Francesc Perera, Andrew Toms, Stuart White and Wilhelm Winter

Vol. 7 (2014), No. 4, 929–952
##### 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.

##### Keywords
C*-algebras, perturbation, Cuntz semigroup, stability, quasitraces, traces
##### Mathematical Subject Classification 2010
Primary: 46L05, 46L35, 46L85