On a preserver problem concerning Kubo-Ando means of positive operators

Abstract

We study bijective transformations ϕ of the set of positive operators on a Hilbert space such that given a Kubo-Ando mean σ with certain properties, ‖AσB‖=‖ϕ(A)σϕ(B)‖ holds for every pair A,B of positive operators. It is shown that whenever the Hilbert space is finite dimensional, then ϕ is necessarily conjugation by a fixed unitary or antiunitary operator. Coupled with earlier results, this completely resolves the above preserver problem in the finite dimensional case. Most ideas of the proof carry over to the infinite dimensional setting, thus some partial results are obtained in that case as well.