Al-Rawashdeh, A. (2016). Non-isomorphic C*-algebras with isomorphic unitary groups. Advances in Operator Theory, 1(2), 160-163. doi: 10.22034/aot.1609.1004
Ahmed Al-Rawashdeh. "Non-isomorphic C*-algebras with isomorphic unitary groups". Advances in Operator Theory, 1, 2, 2016, 160-163. doi: 10.22034/aot.1609.1004
Al-Rawashdeh, A. (2016). 'Non-isomorphic C*-algebras with isomorphic unitary groups', Advances in Operator Theory, 1(2), pp. 160-163. doi: 10.22034/aot.1609.1004
Al-Rawashdeh, A. Non-isomorphic C*-algebras with isomorphic unitary groups. Advances in Operator Theory, 2016; 1(2): 160-163. doi: 10.22034/aot.1609.1004
Non-isomorphic C*-algebras with isomorphic unitary groups
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