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

Abstract

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.