Lieb type convexity for positive operator monotone decreasing functions

Abstract

We prove Lieb type convexity and concavity results for trace functionals associated with positive operator monotone (decreasing) functions and certain monotone concave functions, together with strictly positive linear maps of matrices. This gives a partial generalization of Hiai's recent work on trace functionals associated with power functions, by allowing positive operator monotone decreasing functions instead of power maps.Our proof is based on variational formula for trace functionals based on the Legendre transform, and a strengthened convexity of positive operator monotone decreasing functions in a previous work of Kirihata and the second named author. We also provide the generalization to the framework of unital tracial C -algebras based on Petz's work.