## Analysis & PDE

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

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

#### 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
Revised: 13 June 2013
Accepted: 23 July 2013
First available in Project Euclid: 20 December 2017

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

#### 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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