Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-20T03:38:04.468Z Has data issue: false hasContentIssue false
  • English
  • Français

Remarks on free mutual information and orbital free entropy

Published online by Cambridge University Press:  11 January 2016

Masaki Izumi  and
Yoshimichi Ueda
Masaki Izumi
Affiliation:
Department of Mathematics, Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japanizumi@math.kyoto-u.ac.jp
Yoshimichi Ueda
Affiliation:
Graduate School of Mathematics, Kyushu University, Fukuoka, 819-0395, Japanueda@math.kyushu-u.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The present notes provide a proof of i* (ℂP + ℂ(I - P); ℂQ + ℂ(I - Q)) = – χorb(P,Q) for any pair of projections P,Q with τ(P) = τ(Q) = 1/2. The proof includes new extra observations, such as a subordination result in terms of Loewner equations. A study of the general case is also given.

Keywords

Primary 46L54Secondary 94A17
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 220 , December 2015 , pp. 45 - 66
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

References

[1] Biane, Ph., “Free Brownian motion, free stochastic calculus and random matrices” in Free Probability Theory (Waterloo, Canada, 1995), Fields Inst. Commun 12, Amer. Math. Soc., Providence, 1997, 1–19. MR 1426833.Google Scholar
[2] Biane, Ph., Segal-Bargmann transform, functional calculus on matrix spaces and the theory of semi-circular and circular systems, J. Funct. Anal. 144 (1997), 232–286. MR 1430721. DOI 10.1006/jfan.1996.2990.Google Scholar
[3] Biane, Ph., Processes with free increments, Math. Z. 227 (1998), 143–174. MR 1605393. DOI 10.1007/PL00004363. Google Scholar
[4] Biane, Ph. and Dabrowski, Y., Concavification of free entropy, Adv. Math. 234 (2013), 667–696. MR 3003941. DOI 10.1016/j.aim.2012.11.003.CrossRefGoogle Scholar
[5] Biane, Ph. and Speicher, R., Stochastic calculus with respect to free Brownian motion and analysis on Wigner space, Probab. Theory Related Fields 112 (1998), 373–409. MR 1660906. DOI 10.1007/s004400050194.CrossRefGoogle Scholar
[6] Collins, B. and Kemp, T., Liberation of projections, J. Funct. Anal. 266 (2014), 1988–2052. MR 3150150. DOI 10.1016/j.jfa.2013.10.034.CrossRefGoogle Scholar
[7] Dabrowski, Y., A free stochastic partial differential equation, Ann. Inst. Henri Poincaré Probab. Stat. 50 (2014), 1404–1455. MR 3270000. DOI 10.1214/13-AIHP548.CrossRefGoogle Scholar
[8] Demni, N., Free Jacobi process, J. Theoret. Probab. 21 (2008), 118–143. MR 2384475. DOI 10.1007/s10959-007-0110-1.CrossRefGoogle Scholar
[9] Demni, N., Hamdi, T., and Hmidi, T., Spectral distribution of the free Jacobi process, Indiana Univ. Math. J. 61 (2012), 1351–1368. MR 3071702. DOI 10.1512/iumj. 2012.61.5034.CrossRefGoogle Scholar
[10] Duren, P. L., Theory of Hp Spaces, Dover, New York, 2000.Google Scholar
[11] Geronimus, Ya. L., Polynomials orthogonal on a circle and their applications, Amer. Math. Soc. Transl. Ser. 2 1954 (1954), 79 pp. MR 0061706.Google Scholar
[12] Halmos, P. R., A Hilbert Space Problem Book, 2nd ed., Grad. Texts in Math. 19, Springer, New York, 1982. MR 0675952.Google Scholar
[13] Hiai, F., Miyamoto, T., and Ueda, Y., Orbital approach to microstate free entropy, Internat. J. Math. 20 (2009), 227–273. MR 2493361. DOI 10.1142/S0129167X09005261.CrossRefGoogle Scholar
[14] Hiai, F. and Petz, D., Large deviations for functions of two random projection matrices, Acta Sci. Math. (Szeged) 72 (2006), 581–609. MR 2289756.Google Scholar
[15] Hiai, F. and Ueda, Y., Notes on microstate free entropy of projections, Publ. Res. Inst. Math. Sci. 44 (2008), 49–89. MR 2405867.Google Scholar
[16] Hiai, F. and Ueda, Y., A log-Sobolev type inequality for free entropy of two projections, Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009), 239–249. MR 2500237. DOI 10.1214/08-AIHP164.Google Scholar
[17] Kargin, V., On free stochastic differential equations, J. Theoret. Probab. 24 (2011), 821–848. MR 2822483. DOI 10.1007/s10959-011-0341-z.CrossRefGoogle Scholar
[18] King, F. W., Hilbert Transforms, Vol. 1, Encyclopedia Math. Appl. 124, Cambridge University Press, Cambridge, 2009. MR 2542214.Google Scholar
[19] Koosis, P., Introduction to Hp Spaces, 2nd ed., with two appendices by Havin, V. P., Cambridge Tracts in Math. 115, Cambridge University Press, Cambridge, 1998. MR 1669574.Google Scholar
[20] Lawler, G. F., Conformally Invariant Processes in the Plane, Math. Surveys Monogr. 114, Amer. Math. Soc., Providence, 2005. MR 2129588.Google Scholar
[21] Ueda, Y., Orbital free entropy, revisited, Indiana Univ. Math. J. 63 (2014), 551–577. MR 3233219. DOI 10.1512/iumj.2014.63.5220.CrossRefGoogle Scholar
[22] Voiculescu, D., The analogues 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 1711843. DOI 10.1006/aima.1998.1819.CrossRefGoogle Scholar
[23] Voiculescu, D., “Lectures on free probability theory” in Lectures on Probability Theory and Statistics (Saint-Flour, 1998), Lecture Notes in Math. 1738, Springer, Berlin, 2000, 279–349. MR 1775641. DOI 10.1007/BFb0106703.Google Scholar
[24] Voiculescu, D., Free entropy, Bull. Lond. Math. Soc. 34 (2002), 257–278. MR 1887698. DOI 10.1112/S0024609301008992.Google Scholar
[25] Zhong, P., On the free convolution with a free multiplicative analogue of the normal distribution, to appear in J. Theor. Probab., preprint, arXiv: 1211.3160v3 [math.PR].Google Scholar