### 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 ⊕_{i=1}^{n}(A_{i})_{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 language | English (US) |
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Pages (from-to) | 631-648 |

Number of pages | 18 |

Journal | Reviews in Mathematical Physics |

Volume | 14 |

Issue number | 7-8 |

DOIs | |

State | Published - Jan 1 2002 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Keywords

- Operator algebras
- Trace functions
- Trace inequalities

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## Cite this

*Reviews in Mathematical Physics*,

*14*(7-8), 631-648. https://doi.org/10.1142/s0129055x02001260