Isometries of the unitary groups in C*-algebras

@article{OsamuHatori2014,
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.},
author = {Osamu Hatori},
journal = {Studia Mathematica},
keywords = {isometry; Jordan *-isomorphism; unitary group; -algebra; von Neumann algebra},
language = {eng},
number = {1},
pages = {61-86},
title = {Isometries of the unitary groups in C*-algebras},
url = {http://eudml.org/doc/285618},
volume = {221},
year = {2014},
}

TY - JOUR
AU - Osamu Hatori
TI - Isometries of the unitary groups in C*-algebras
JO - Studia Mathematica
PY - 2014
VL - 221
IS - 1
SP - 61
EP - 86
AB - 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.
LA - eng
KW - isometry; Jordan *-isomorphism; unitary group; -algebra; von Neumann algebra
UR - http://eudml.org/doc/285618
ER -