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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm221-1-4,
author = {Osamu Hatori},
title = {Isometries of the unitary groups in C*-algebras},
journal = {Studia Mathematica},
volume = {223},
year = {2014},
pages = {61-86},
zbl = {1306.47048},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm221-1-4}
}
Osamu Hatori. Isometries of the unitary groups in C*-algebras. Studia Mathematica, Tome 223 (2014) pp. 61-86. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm221-1-4/