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.
"Isometries on Positive Definite Operators with Unit Fuglede-Kadison Determinant." Taiwanese J. Math. 23 (6) 1423 - 1433, December, 2019. https://doi.org/10.11650/tjm/190205