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The Cuntz semigroup and
stability of close $C^*$-algebras
Francesc Perera, Andrew Toms, Stuart White and Wilhelm Winter |
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Vol. 7 (2014), No. 4, 929–952
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Abstract |
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We prove that separable -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 -algebras provided that one algebra has stable rank one; close -algebras must have affinely homeomorphic spaces of lower-semicontinuous quasitraces; strict comparison is preserved by sufficient closeness of -algebras. We also examine -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 -algebras have isomorphic Cuntz semigroups when one algebra absorbs the Jiang–Su algebra tensorially. |
Keywords
C*-algebras, perturbation, Cuntz semigroup, stability,
quasitraces, traces
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Mathematical Subject Classification 2010
Primary: 46L05, 46L35, 46L85
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Milestones
Received: 4 March 2013
Revised: 13 June 2013
Accepted: 23 July 2013
Published: 27 August 2014
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Authors |
| Francesc Perera | |
| Department of Mathematics Universitat Autònoma de Barcelona 08193 Bellaterra Barcelona Spain |
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| Andrew Toms | |
| Department of Mathematics Purdue University Room 720, Mathematical Sciences Building 150 N. University St. West Lafayette, IN 47907-2067 United States |
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| Stuart White | |
| School of Mathematics and
Statistics University of Glasgow University Gardens Glasgow Q12 8QW Scotland |
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| Wilhelm Winter | |
| Mathematisches Institut der WWU
Münster Einsteinstraße 62 48149 Münster Germany |
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