Free relative entropy for measures and a corresponding perturbation theory

Journal of the Mathematical Society of Japan

Volumn 54, Issue 3, 2002, Pages 679-718

  Hiai, Fumio ^a   Mizuo, Masaru ^a   Petz, Dénes ^b  

^a TOHOKU UNIVERSITY   (Japan)

^b BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS   (Hungary)

Author keywords

Free entropy; Free relative entropy; Large deviation; Random matrix

Indexed keywords

EID: 0036629981     PISSN: 00255645     EISSN: 18811167     Source Type: Journal    
DOI: 10.2969/jmsj/1191593914     Document Type: Article

References (21)

1. H. Araki, Relative entropy for states of von Neumann algebras II, Publ. Res. Inst. Math. Sci., 13 (1977), 173-192.
2. G. Ben Arous and A. Guionnet, Large deviation for Wigner’s law and Voiculescu’s non-commutative entropy, Probab. Theory Related Fields, 108 (1997), 517-542.
3. P. Biane and R. Speicher, Free diffusions, free entropy and free Fisher information. Ann. Inst. H. Poincare Probab. Statist., 37 (2001), 581-606.
4. I. Ciszar, Information-type measures of difference of probability distributions and indirect observations, Studia Sci. Math. Hungar., 2 (1967), 299-318.
5. A. Dembo and O. Zeitouni, Large Deviation Techniques and Applications, Second edition, Springer, New York, 1998.
6. J. D. Deuschel and D. W. Stroock, Large Deviations. Academic Press, Boston, 1989.
7. M. J. Donald, Relative hamiltonian which are not bounded from above, J. Funct. Anal., 91 (1990), 143-173.
8. U. Haagerup, unpublished notes.
9. F. Hiai, M. Ohya and M. Tsukada, Sufficiency, KMS condition and relative entropy in von Neumann algebras. Pacific J. Math., 96 (1981), 99-109.
10. F. Hiai and D. Petz, Maximizing free entropy. Acta Math. Hungar., 80 (1998), 335-356.
11. F. Hiai and D. Petz, Properties of free entropy related to polar decomposition. Comm. Math. Phys., 202 (1999), 421-444.
12. F. Hiai and D. Petz. A large deviation theorem for the empirical eigenvalue distribution of random unitary matrices. Ann. Inst. H. Poincare Probab. Statist., 36 (2000), 71-85.
13. F. Hiai and D. Petz, The Semicircle Law, Free Random Variables and Entropy, Mathematical Surveys and Monographs, Vol. 77. Amer. Math. Soc., Providence, 2000.
14. M. Mizuo, Large deviations and microstate free relative entropy, Interdiscip. Inform. Sci., 6 (2000), 111-121.
15. M. Ohya and D. Petz, Quantum Entropy and Its Use, Springer, Berlin-Heidelberg, 1993.
16. D. Petz. A variational expression for the relative entropy. Comm. Math. Phys., 114 (1988), 345-349.
17. D. Ruelle, Thermodynamic Formalism, Encyclopedia Math. Appl., Vol. 5. Addison-Wesley, London, 1978.
18. E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer, Berlin-Heidelberg-New York, 1997.
19. D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory, I. Comm. Math. Phys., 155 (1993), 71-92.
20. D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory, II, Invent. Math., 118 (1994), 411-440.
21. D. V. Voiculescu, K. J. Dykema and A. Nica, Free Random Variables. CRM Monogr. Ser., Vol. 1. Amer. Math. Soc., 1992