Advances in Operator Theory

Non-isomorphic $C^{*}$-algebras with isomorphic unitary groups

Ahmed Al-Rawashdeh

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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.

Article information

Adv. Oper. Theory, Volume 1, Number 2 (2016), 160-163.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L05: General theory of $C^*$-algebras
Secondary: 46L35: Classifications of $C^*$-algebras

$C^{*}$-algebra unitary group *-isomorphism


Al-Rawashdeh, Ahmed. Non-isomorphic $C^{*}$-algebras with isomorphic unitary groups. Adv. Oper. Theory 1 (2016), no. 2, 160--163. doi:10.22034/aot.1609.1004.

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