On jointly concavity of some trace functions

Abstract

Let M be a finite von Neumann algebra with a normal faithful finite trace τ, L0(M) be the set of all measurable operators with respect to (M,τ) and μt(x) be the generalized singular number of x∈L0(M). Set L0(M)+={x:x∈L0(M),x≥0} and M++={x:x∈M,x≥0andxis invertible}. We prove that if f:[0,∞)→[0,∞) is an operator concave function, 0<p≤1, 0<s≤1p and Φj is a continuous positive linear map from L0(Mj) to L0(M) with Φj(Mj)⊂M, where Mj is finite von Neumann algebra, j=1,2,⋯,n, then for 0≤t<τ(1)∫tτ(1)μv((∑j=1nΦj(f(xjp)))s)dvand∫tτ(1)μv((∑j=1nΦj(f(xj)p))s)dv are jointly concave in (x1,x2,⋯,xn)∈L0(M1)+×L0(M2)+×⋯×L0(Mn)+. We also prove that if f:(0,∞)→(0,∞) is an operator concave function, Φj is a strictly positive linear map from finite von Neumann algebra Mj to M, j=1,2,⋯,n, 0<p≤1 and 0<s≤1p, then for 0≤t<τ(1),∫tτ(1)μv((∑j=1nΦj(f(xj−p)))−s)dvand∫tτ(1)μv((∑j=1nΦj(f(xj)−p))−s)dv are jointly concave in (x1,x2,⋯,xn)∈M1++×M2++×⋯×Mn++.