Aspects of free probability

XIVth International Congress on Mathematical Physics: Lisbon, 28 July - 2 August 2003

Volumn , Issue , 2006, Pages 145-157

  Voiculescu, Dan ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

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EID: 84967647351     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1142/9789812704016_0012     Document Type: Chapter

References (39)

1. P. Biane, "Processes with free increments", Math. Z no. 1, 143-174 (1998).
2. P. Biane, M. Capitaine, A. Guionnet, "Large deviation bounds for matrix Brownian motion", Invent. Math. 152, 433-459 (2003).
3. P. Biane, R. Speicher, "Free diffusions, free entropy and free Fisher information", Ann. Inst. H. Poincaré Probab. Statist. 37, 581-606 (2001).
4. E. Brezin, C. Itzykson, G. Parisi, J. B. Zuber, "Planar diagrams", Comm. Math. Phys. 59, 35-51 (1978).
5. T. Cabanal-Duvillard, A. Guionnet, "Large deviations upper bounds for the laws of matrixvalued processes and noncommutative entropies", Ann. Probab. 29, 1205-1261 (2001).
6. M. R. Douglas, "Stochastic master fields", Phys. Lett. B 344, 117-126 (1995).
7. K. J. Dykema, "Free products of hyperfinite von Neumann algebras and free dimension", Duke Math. J. 69, 97-119 (1993).
8. L. Ge, "Applications of free entropy to finite von Neumann algebras II", Ann. of Math. 147, 143-157 (1998).
9. R. Gopakumar, D. J. Gross, "Mastering the master field", Nucl. Phys. B 451, 379-415 (1995).
10. U. Haagerup, S. Thorbjoernsen, "A new application of random matrices: Ext(C£(F2)) is not a group", preprint (2002).
11. A. S. Joni, G.-C. Rota, "Coalgebras and bialgebras in combinatorics", in Studies in Applied Mathematics 61, Elsevier North Holland, 1979, pp. 93-139.
12. A. Nica, D. Shlyakhtenko, R. Speicher, "Some minimization problems for the free analogue of the Fisher information", Adv. Math. 121, 282-347 (1999).
13. ∞)", in Subfactor Theory, preprint (2002).
14. ∞) ⊗ B(H) scaling the trace", C. R. Acad. Sci. Paris t. 314, Serie I, 1027-1032 (1992).
15. F. Radulescu, "Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group of noninteger index", Invent. Math. 115, 347-389 (1994).
16. D. Shlyakhtenko, "Random Gaussian band matrices and freeness with amalgamation", International Math. Res. Notices 20, 1013-1025 (1996).
17. D. Shlyakhtenko, "Free quasi-free states", Pacific J. Math. 177, 329-368 (1997).
18. D. Shlyakhtenko, "On prime factors of type III", Proc. Nat. Acad. Sci. 97, 12439-12441 (2000).
19. I. M. Singer, "On the master field in two dimensions", Functional Analysis on the Eve of the 21st Century, Essays in Honor of the 80th Birthday of I. M. Gelfand, Progress in Math. 131, 263-283 (1995).
20. R. Speicher, "Combinatorial theory of the free product with amalgamation and operator-valued free probability theory", Mem. Amer. Math. Soc. 627 (1998).
21. M. B. Stefan, "The indecomposability of free group factors over nonprime-subfactors and abelian subalgebras", preprint.
22. G. t’Hooft, "A two-dimensional model for mesons", Nucl. Phys. B75, 461-470 (1974).
23. Y. Ueda, D. Shlyakhtenko, "Irreducible subfactors of index λ > 4", preprint (2000).
24. D. Voiculescu, "Symmetries of some reduced free product C*-algebras", in Operator Algebras and Their Connections with Topology and Ergodic Theory, Lecture Notes in Math 1132, Springer, 1985, pp. 556-588.
25. D. Voiculescu, "Circular and semicircular systems and free product factors", in Operator Algebras, Unitary Representations, Enveloping Algebras and Invariant Theory, Progr. Math. 92, Birkhàuser, Boston, MA, 1990, pp. 45-60.
26. D. Voiculescu, "Limit laws for random matrices and free products", Invent. Math. 104, 201-220 (1991).
27. D. Voiculescu, "Operations on certain noncommuting operator-valued random variables", Asterisque no. 232, 243-275 (1995).
28. D. Voiculescu, "The analogues of entropy and of Fisher’s information measure in free probability theory", Comm. Math. Phys. 155, 71-92 (1993).
29. D. Voiculescu, "The analogues of entropy and of Fisher’s information measure in free probability theory II", Invent. Math. 118, 411-440 (1994).
30. D. Voiculescu, "The analogues of entropy and of Fisher’s information measure in free probability theory III: The absence of Cartan subalgebras", Geom. Fund. Anal. 6, 172-199 (1996).
31. D. Voiculescu, "The analogues of entropy and of Fisher’s information measure in free probability theory V: Noncommutative Hilbert transforms", Invent. Math. 132, 182-227 (1998).
32. D. Voiculescu, "The analogues of entropy and of Fisher’s information measure in free probability theory VI: Liberation and mutual free information", Adv. Math. 146, 101-166 (1999).
33. D. Voiculescu, "A strengthened asymptotic freeness result for random matrices with applications to free entropy", International Math. Res. Notices № 1, 41-63 (1998).
34. D. Voiculescu, "Lectures on free probability theory", Lectures on probability theory and statistics, Ecole d’été des Probabilités de Saint-Flour XXVIII, Lecture Notes in Math. 1738, Springer, 1998, pp. 280-349.
35. D. Voiculescu, "The coalgebra of the free difference quotient in free probability theory", International Math. Res. Notices 2, 79-106 (2000).
36. D. Voiculescu, "Free entropy", Bull. London Math. Soc. 34, 257-278 (2002).
37. X:B", to appear in International Math Research Notices (2004).
38. D. Voiculescu, K. J. Dykema, A. Nica, Free random variables, CRM Monograph Series 1, Amer. Math. Soc., Providence, RI, 1992.
39. E. P. Wigner, "Characteristic vectors of bordered matrices with infinite dimensions", Ann. Math. 62, 54 (1955).