For any densely defined, lower semi-continuous trace τ on a C*-algebra A with mutually commuting C*-subalgebras A1, A2,...An, and a convex function f of n variables, we give a short proof of the fact that the function (x1, x2,...,xn) → τ(f(x1, x2,...,xn)) is convex on the space ⊕i=1n(Ai)sa. If furthermore the function f is log-convex or root-convex, so is the corresponding trace function. We also introduce a generalization of log-convexity and root-convexity called l-convexity, show how it applies to traces, and give some examples. In particular we show that the Kadison-Fuglede determinant is concave and that the trace of an operator mean is always dominated by the corresponding mean of the trace values.
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics
- Operator algebras
- Trace functions
- Trace inequalities