Isometries of the unitary groups in $C^{*}$-algebras

Osamu Hatori ¹

¹ Department of Mathematics Faculty of Science Niigata University 950-2181 Niigata, Japan

Studia Mathematica, Tome 221 (2014) no. 1, pp. 61-86

Résumé

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.

Détail

DOI : 10.4064/sm221-1-4

Keywords: complete description structure surjective isometries between unitary groups unital * algebras while surjective isometry between unitary groups von neumann algebras extended real linear jordan * isomorphism between relevant von neumann algebras general unital * algebras unitary groups * algebras isomorphic metric groups only * algebras isomorphic sense each decomposed direct sum * algebras first parts being linear * algebra isomorphic second parts being conjugate linear * algebra isomorphic emphasize paper isometry merely mean distance preserving transformation assume respects algebraic operation