Convex Multivariable Trace Functions

Abstract

For any densely defined, lower semi-continuous trace τon a C*-algebra A with mutually commuting C*-subalgebras A_1, A_2, ... A_n, and a convex function f of n variables, we give a short proof of the fact that the function (x_1, x_2, ..., x_n) --> τ(f(x_1, x_2, ..., x_n)) is convex on the space \bigoplus_{i=1}^n (A_i)_{self-adjoint}. 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 \ell-convexity, show how it applies to traces, and give a few examples. In particular we show that the trace of an operator mean is always dominated by the corresponding mean of the trace values.

13 pages, AMS TeX, Some remarks and results added

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