On Maps Preserving Unitarily Invariant Norms of the Spectral Geometric Mean

Hongjie Chen,Liguang Wang,Zheng Shi,Lei Li

doi:10.4236/jamp.2021.94041

Abstract

We consider maps on positive definite cones of von Neumann algebras preserving unitarily invariant norms of the spectral geometric means. The main results concern Jordan *-isomorphisms between C*-algebras, and show that they are characterized by the preservation of unitarily invariant norms of those operations.

Highlights

In order to study the geometry on the cones n of the positive definite n × n matrices, people consider different means of positive definite matrices
For the log-Euclidean mean, the Gaál and Nagy ([2], Theorem 2) obtained a general results concerning the p-norm in a C*-algebra A equipped with a faithful tracial state, where p ∈[1, ∞)
All the results showed that any correspondence preserver is the restriction of a Jordan *-isomorphism of A multiplied by a central positive invertible element [3] [4]

Summary

Introduction

Open AccessIn order to study the geometry on the cones n of the positive definite n × n matrices, people consider different means of positive definite matrices. Spectral Geometric Mean, Positive Cone, Jordan *-Isomorphisms, Unitarily The authors in considered the maps preserving γ a A, B Gaál and Nagy ([2], Theorem 1) obtained the same results as in [1] concerning the bijective transforms of n which preserve any unitary invariant norm of some quasi-arithmetric means of elements (it includes the weighted arithmetic mean and the weighted log-Euclidean mean) for all t ∈[0,1] .

Objectives

Results

Conclusion