Isometries on Positive Definite Operators with Unit Fuglede-Kadison Determinant

Abstract

In this paper we explore the structure of certain surjective generalized isometries (which are transformations that leave any given member of a large class of generalized distance measures invariant) of the set of positive invertible elements in a finite von Neumann factor with unit Fuglede-Kadison determinant. We conclude that any such map originates from either an algebra $^*$-isomorphism or an algebra $^*$-antiisomorphism of the underlying operator algebra.