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

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.

C*-algebras, perturbation, Cuntz semigroup, stability, quasitraces, traces
Mathematical Subject Classification 2010
Primary: 46L05, 46L35, 46L85
Received: 4 March 2013
Revised: 13 June 2013
Accepted: 23 July 2013
Published: 27 August 2014
Francesc Perera
Department of Mathematics
Universitat Autònoma de Barcelona
08193 Bellaterra
Andrew Toms
Department of Mathematics
Purdue University
Room 720, Mathematical Sciences Building
150 N. University St.
West Lafayette, IN 47907-2067
United States
Stuart White
School of Mathematics and Statistics
University of Glasgow
University Gardens
Q12 8QW
Wilhelm Winter
Mathematisches Institut der WWU Münster
Einsteinstraße 62