Non-isomorphic C*-algebras with isomorphic unitary groups

Dye, [Ann. of Math. (2) 61 (1955), 73-89] 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 [J. Funct. Anal. 262 (2012), 4711-4730] proved that the algebras are *-isomorphic if and only if their unitary groups are isomorphic as abstract groups. In this paper, we give a counterexample in the non-simple case. Indeed, we give two C*-algebras with isomorphic unitary groups but the algebras themselves are not *-isomorphic.