Isometries of the unitary groups and Thompson isometries of the spaces of invertible positive elements in C*-algebras

Hatori, Osamu and Molnár, Lajos (2014) Isometries of the unitary groups and Thompson isometries of the spaces of invertible positive elements in C*-algebras. Journal of Mathematical Analysis and Applications, 409 (1). pp. 158-167. ISSN 0022-247X

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Abstract

We show that the existence of a surjective isometry (which is merely a distance preserving map) between the unitary groups of unital C*-algebras implies the existence of a Jordan *-isomorphism between the algebras. In the case of von Neumann algebras we describe the structure of those isometries showing that any of them is extendible to a real linear Jordan *-isomorphism between the underlying algebras multiplied by a fixed unitary element. We present a result of similar spirit for the surjective Thompson isometries between the spaces of all invertible positive elements in unital C*-algebras.

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