Joint Convexity And Concavity Of N-tuples τ-Measurable Positive Operator Trace Functions

In this thesis, we consider the noncommutative Lp(M;τ) spaces associated with asemifinite von Neumann algebra M.Firstly, We give the prove that n- tuplesτ-measurable trace functionF(x,y) =τ(x^p +y^p)^1/pis jointly convex and jointly concave in (x,y). Secondly, We obtain the necessary andsuffcient condition of theτ-measurable trace functionis jointly convex and jointly concave in (x1,x2,···,xn).