no. 1
A log-Sobolev type inequality for free entropy of two projections
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 1, pp. 239-249.

Nous prouvons un genre d'inégalité de Sobolev logarithmique qui montre que l'information de Fisher libre domine l'entropie de micro-états libre adaptée aux projections dans le cas de deux projections.

We prove a kind of logarithmic Sobolev inequality claiming that the mutual free Fisher information dominates the microstate free entropy adapted to projections in the case of two projections.

DOI : 10.1214/08-AIHP164
Classification : 46L54, 94A17, 60E15
Mots clés : logarithmic Sobolev inequality, free entropy, mutual free Fisher information 
@article{AIHPB_2009__45_1_239_0,
     author = {Hiai, Fumio and Ueda, Yoshimichi},
     title = {A {log-Sobolev} type inequality for free entropy of two projections},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {239--249},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {1},
     year = {2009},
     doi = {10.1214/08-AIHP164},
     mrnumber = {2500237},
     zbl = {1178.46066},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/08-AIHP164/}
}
TY  - JOUR
AU  - Hiai, Fumio
AU  - Ueda, Yoshimichi
TI  - A log-Sobolev type inequality for free entropy of two projections
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2009
SP  - 239
EP  - 249
VL  - 45
IS  - 1
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/08-AIHP164/
DO  - 10.1214/08-AIHP164
LA  - en
ID  - AIHPB_2009__45_1_239_0
ER  - 
%0 Journal Article
%A Hiai, Fumio
%A Ueda, Yoshimichi
%T A log-Sobolev type inequality for free entropy of two projections
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2009
%P 239-249
%V 45
%N 1
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/08-AIHP164/
%R 10.1214/08-AIHP164
%G en
%F AIHPB_2009__45_1_239_0
Hiai, Fumio; Ueda, Yoshimichi. A log-Sobolev type inequality for free entropy of two projections. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 1, pp. 239-249. doi : 10.1214/08-AIHP164. http://www.numdam.org/articles/10.1214/08-AIHP164/

[1] D. Bakry and M. Emery. Diffusion hypercontractives. Séminaire Probabilités XIX 177-206. Lecture Notes in Math. 1123. Springer, Berlin, 1985. | Numdam | MR | Zbl

[2] P. Biane. Free Brownian motion, free stochastic calculus and random matrices. In Free Probability Theory 1-19. D. V. Voiculescu (Ed.). Fields Inst. Commun. 12. Amer. Math. Soc. Providence, RI, 1997. | MR | Zbl

[3] P. Biane. Logarithmic Sobolev inequalities, matrix models and free entropy. Acta Math. Sinica 19 (2003) 497-506. | MR | Zbl

[4] P. Biane, M. Capitaine and A. Guionnet. Large deviation bounds for matrix Brownian motion. Invent. Math. 152 (2003) 433-459. | MR | Zbl

[5] B. Collins. Product of random projections, Jacobi ensembles and universality problems arising from free probability. Probab. Theory Related Fields 133 (2005) 315-344. | MR | Zbl

[6] S. Gallot, D. Hulin and J. Lafontaine. Riemannian Geometry, 2nd edition. Universitext, Springer, Berlin, 1990. | MR | Zbl

[7] F. Hiai and D. Petz. The Semicircle Law, Free Random Variables and Entropy. Amer. Math. Soc., Providence, RI, 2000. | MR | Zbl

[8] F. Hiai and D. Petz. Large deviations for functions of two random projection matrices. Acta Sci. Math. (Szeged) 72 (2006) 581-609. | MR | Zbl

[9] F. Hiai, D. Petz and Y. Ueda. Free logarithmic Sobolev inequality on the unit circle. Canad. Math. Bull. 49 (2006) 389-406. | MR | Zbl

[10] F. Hiai and Y. Ueda. Notes on microstate free entropy of projections. Publ. Res. Inst. Math. Sci. 44 (2008), 49-89. | MR | Zbl

[11] R. Hunt, B. Muckenhoupt and R. Wheeden. Weighted norm inequalities for the conjugate function and Hilbert transform. Trans. Amer. Math. Soc. 176 (1973) 227-251. | MR | Zbl

[12] M. Ledoux. A (one-dimensional) free Brunn-Minkowski inequality. C. R. Math. Acad. Sci. Paris 340 (2005) 301-304. | MR | Zbl

[13] G. I. Ol'Shanskij. Unitary representations of infinite dimensional pairs (g, k) and the formalism of R. Howe. In Representation of Lie Groups and Related Topics 269-463. A. M. Vershik and D. P. Zhelobenko (Eds). Adv. Stud. Contemp. Math. 7. Gordon and Breach, New York, 1990. | MR | Zbl

[14] E. B. Saff and V. Totik. Logarithmic Potentials with External Fields. Springer, Berlin, 1997. | MR | Zbl

[15] C. Villani. Topics in Optimal Transportation. Amer. Math. Soc., Providence, RI, 2003. | MR | Zbl

[16] D. Voiculescu. The analogues of entropy and of Fisher's information measure in free probability theory, I. Comm. Math. Phys. 155 (1993) 71-92. | MR | Zbl

[17] D. Voiculescu. The analogues of entropy and of Fisher's information measure in free probability theory, II. Invent. Math. 118 (1994) 411-440. | MR | Zbl

[18] D. Voiculescu. The analogues of entropy and of Fisher's information measure in free probability theory, IV: Maximum entropy and freeness. In Free Probability Theory 293-302. D. V. Voiculescu (Ed.). Fields Inst. Commun. 12. Amer. Math. Soc., Providence, RI, 1997. | MR | Zbl

[19] D. Voiculescu. The analogues of entropy and of Fisher's information measure in free probability theory, V: Noncommutative Hilbert transforms. Invent. Math. 132 (1998) 189-227. | MR | Zbl

[20] D. Voiculescu. The analogue of entropy and of Fisher's information measure in free probability theory VI: Liberation and mutual free information. Adv. Math. 146 (1999) 101-166. | MR | Zbl

[21] D. V. Voiculescu, K. J. Dykema and A. Nica. Free Random Variables. Amer. Math. Soc., Providence, RI, 1992. | MR | Zbl

Cité par Sources :