Open Access
Autumn 2016 Non-isomorphic $C^{*}$-algebras with isomorphic unitary groups
Ahmed Al-Rawashdeh
Adv. Oper. Theory 1(2): 160-163 (Autumn 2016). DOI: 10.22034/aot.1609.1004


H. Dye proved that the discrete unitary group in a factor determines the algebraic type of the factor. Afterwards, for a large class of simple unital $C^{*}$-algebras, Al-Rawashdeh, Booth and Giordano proved that the algebras are $*$-isomorphic if and only if their unitary groups are isomomorphic as abstract groups. In this paper, we give a counter example in the non-simple case. Indeed, we give two $C^{*}$-algebras with isomorphic unitary groups but the algebras themselves are not $*$-isomorphic.


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Ahmed Al-Rawashdeh. "Non-isomorphic $C^{*}$-algebras with isomorphic unitary groups." Adv. Oper. Theory 1 (2) 160 - 163, Autumn 2016.


Received: 20 September 2016; Accepted: 4 December 2016; Published: Autumn 2016
First available in Project Euclid: 4 December 2017

zbMATH: 1367.46042
MathSciNet: MR3723617
Digital Object Identifier: 10.22034/aot.1609.1004

Primary: 46L05
Secondary: 46L35

Keywords: $C^{*}$-algebra , *-isomorphism , unitary group

Rights: Copyright © 2016 Tusi Mathematical Research Group

Vol.1 • No. 2 • Autumn 2016
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