Analysis & PDE

  • Anal. PDE
  • Volume 7, Number 4 (2014), 929-952.

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

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

Full-text: Open access

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.

Article information

Source
Anal. PDE, Volume 7, Number 4 (2014), 929-952.

Dates
Received: 4 March 2013
Revised: 13 June 2013
Accepted: 23 July 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731543

Digital Object Identifier
doi:10.2140/apde.2014.7.929

Mathematical Reviews number (MathSciNet)
MR3254349

Zentralblatt MATH identifier
1310.46055

Subjects
Primary: 46L05: General theory of $C^*$-algebras 46L35: Classifications of $C^*$-algebras 46L85: Noncommutative topology [See also 58B32, 58B34, 58J22]

Keywords
C*-algebras perturbation Cuntz semigroup stability quasitraces traces

Citation

Perera, Francesc; Toms, Andrew; White, Stuart; Winter, Wilhelm. The Cuntz semigroup and stability of close $C^*$-algebras. Anal. PDE 7 (2014), no. 4, 929--952. doi:10.2140/apde.2014.7.929. https://projecteuclid.org/euclid.apde/1513731543


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