CONVEX MULTIVARIABLE TRACE FUNCTIONS

Abstract

For any densely defined, lower semi-continuous trace τ on 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 [Formula: see text]. 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.

Locations

• Reviews in Mathematical Physics - View
• arXiv (Cornell University) - View - PDF
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