Isometries of the unitary groups in $C^{*}$-algebras
Volume 221 / 2014
Abstract
We give a complete description of the structure of surjective isometries between the unitary groups of unital $C^*$-algebras. While any surjective isometry between the unitary groups of von Neumann algebras can be extended to a real-linear Jordan $^{*}$-isomorphism between the relevant von Neumann algebras, this is not the case for general unital $C^*$-algebras. We show that the unitary groups of two $C^*$-algebras are isomorphic as metric groups if and only if the $C^*$-algebras are isomorphic in the sense that each of them can be decomposed as the direct sum of two $C^*$-algebras with the first parts being linear $^{*}$-algebra isomorphic and the second parts being conjugate-linear $^{*}$-algebra isomorphic. We emphasize that in this paper by an isometry we merely mean a distance preserving transformation; we do not assume that it respects any algebraic operation.
