Open Access
July, 2002 Free relative entropy for measures and a corresponding perturbation theory
Fumio HIAI, Masaru MIZUO, Dénes PETZ
J. Math. Soc. Japan 54(3): 679-718 (July, 2002). DOI: 10.2969/jmsj/1191593914


Voiculescu's single variable free entropy is generalized in two different ways to the free relative entropy for compactly supported probability measures on the real line. The one is introduced by the integral expression and the other is based on matricial (or microstates) approximation; their equivalence is shown based on a large deviation result for the empirical eigenvalue distribution of a relevant random matrix. Next, the perturbation theory for compactly supported probability measures via free relative entropy is developed on the analogy of the perturbation theory via relative entropy. When the perturbed measure via relative entropy is suitably arranged on the space of selfadjoint matrices and the matrix size goes to infinity, it is proven that the perturbation via relative entropy on the matrix space approaches asymptotically to that via free relative entropy. The whole theory can be adapted to probability measures on the unit circle.


Download Citation

Fumio HIAI. Masaru MIZUO. Dénes PETZ. "Free relative entropy for measures and a corresponding perturbation theory." J. Math. Soc. Japan 54 (3) 679 - 718, July, 2002.


Published: July, 2002
First available in Project Euclid: 5 October 2007

zbMATH: 1033.46051
MathSciNet: MR1900962
Digital Object Identifier: 10.2969/jmsj/1191593914

Primary: 46L54
Secondary: 94A17

Keywords: Free entropy , free relative entropy , large deviation , Random matrix

Rights: Copyright © 2002 Mathematical Society of Japan

Vol.54 • No. 3 • July, 2002
Back to Top