Convex multivariable trace functions

Elliott H. Lieb, Gert K. Pedersen

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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.

Original languageEnglish (US)
Pages (from-to)631-648
Number of pages18
JournalReviews in Mathematical Physics
Issue number7-8
StatePublished - 2002

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


  • Operator algebras
  • Trace functions
  • Trace inequalities


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