On multiplicative conditionally free convolution
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- by Mihai Popa and Jiun-Chau Wang PDF
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Abstract:
Using the combinatorics of non-crossing partitions, we construct a conditionally free analogue of Voiculescu’s $S$-transform. The result is applied to the analytical description of conditionally free multiplicative convolution and the characterization of infinite divisibility.References
- S. T. Belinschi and H. Bercovici, Partially defined semigroups relative to multiplicative free convolution, Int. Math. Res. Not. 2 (2005), 65–101. MR 2128863, DOI 10.1155/IMRN.2005.65
- S. T. Belinschi and H. Bercovici, Hinčin’s theorem for multiplicative free convolution, Canad. Math. Bull. 51 (2008), no. 1, 26–31. MR 2384735, DOI 10.4153/CMB-2008-004-3
- Hari Bercovici and Vittorino Pata, Stable laws and domains of attraction in free probability theory, Ann. of Math. (2) 149 (1999), no. 3, 1023–1060. With an appendix by Philippe Biane. MR 1709310, DOI 10.2307/121080
- Hari Bercovici and Dan Voiculescu, Lévy-Hinčin type theorems for multiplicative and additive free convolution, Pacific J. Math. 153 (1992), no. 2, 217–248. MR 1151559
- Hari Bercovici and Jiun-Chau Wang, Limit theorems for free multiplicative convolutions, Trans. Amer. Math. Soc. 360 (2008), no. 11, 6089–6102. MR 2425704, DOI 10.1090/S0002-9947-08-04507-8
- Florin Boca, Free products of completely positive maps and spectral sets, J. Funct. Anal. 97 (1991), no. 2, 251–263. MR 1111181, DOI 10.1016/0022-1236(91)90001-L
- Marek Bożejko and Roland Speicher, $\psi$-independent and symmetrized white noises, Quantum probability & related topics, QP-PQ, VI, World Sci. Publ., River Edge, NJ, 1991, pp. 219–236. MR 1149828
- Marek Bożejko, Michael Leinert, and Roland Speicher, Convolution and limit theorems for conditionally free random variables, Pacific J. Math. 175 (1996), no. 2, 357–388. MR 1432836
- Kenneth J. Dykema, Multilinear function series and transforms in free probability theory, Adv. Math. 208 (2007), no. 1, 351–407. MR 2304321, DOI 10.1016/j.aim.2006.02.011
- Kenneth J. Dykema, On the $S$-transform over a Banach algebra, J. Funct. Anal. 231 (2006), no. 1, 90–110. MR 2190164, DOI 10.1016/j.jfa.2005.04.008
- U. Franz. Boolean convolution of probability measures on the unit circle, arXiv:math/0403243.
- Uffe Haagerup, On Voiculescu’s $R$- and $S$-transforms for free non-commuting random variables, Free probability theory (Waterloo, ON, 1995) Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 127–148. MR 1426838, DOI 10.1215/s0012-7094-97-09004-9
- A. Hinčin. Zur Theorie der unbeschränkt teilbaren Verteilungsgesetze, Mat. Sb. 2 (1937), 79–119.
- Anna Dorota Krystek, Infinite divisibility for the conditionally free convolution, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10 (2007), no. 4, 499–522. MR 2376439, DOI 10.1142/S0219025707002919
- Franz Lehner, Cumulants in noncommutative probability theory. I. Noncommutative exchangeability systems, Math. Z. 248 (2004), no. 1, 67–100. MR 2092722, DOI 10.1007/s00209-004-0653-0
- Alexandru Nica and Roland Speicher, Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, vol. 335, Cambridge University Press, Cambridge, 2006. MR 2266879, DOI 10.1017/CBO9780511735127
- Mihai Popa, A combinatorial approach to monotonic independence over a $C^*$-algebra, Pacific J. Math. 237 (2008), no. 2, 299–325. MR 2421124, DOI 10.2140/pjm.2008.237.299
- Mihai Popa, A new proof for the multiplicative property of the Boolean cumulants with applications to the operator-valued case, Colloq. Math. 117 (2009), no. 1, 81–93. MR 2539549, DOI 10.4064/cm117-1-5
- T. P. Speed, Cumulants and partition lattices, Austral. J. Statist. 25 (1983), no. 2, 378–388. MR 725217
- Roland Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory, Mem. Amer. Math. Soc. 132 (1998), no. 627, x+88. MR 1407898, DOI 10.1090/memo/0627
- Roland Speicher and Reza Woroudi, Boolean convolution, Free probability theory (Waterloo, ON, 1995) Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 267–279. MR 1426845
- D. V. Voiculescu, K. J. Dykema, and A. Nica, Free random variables, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. MR 1217253, DOI 10.1090/crmm/001
- Jiun-Chau Wang, Limit laws for Boolean convolutions, Pacific J. Math. 237 (2008), no. 2, 349–371. MR 2421126, DOI 10.2140/pjm.2008.237.349
- J.C. Wang. Limit theorems for additive c-free convolution, (2008), preprint.
Additional Information
- Mihai Popa
- Affiliation: Department of Mathematics, Indiana University at Bloomington, Rawles Hall, 831 E 3rd Street, Bloomington, Indiana 47405
- Email: mipopa@indiana.edu
- Jiun-Chau Wang
- Affiliation: Department of Mathematics and Statistics, Queen’s University, Jeffery Hall, Kingston, Ontario, Canada K7M 7H7
- Email: jiuwang@mast.queensu.ca
- Received by editor(s): May 29, 2008
- Received by editor(s) in revised form: June 23, 2009, and October 20, 2009
- Published electronically: June 27, 2011
- Additional Notes: The first author was partially supported by the grant 2-CEx06-11-34 of the Romanian Government
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 6309-6335
- MSC (2000): Primary 46L53; Secondary 05A18, 60E07
- DOI: https://doi.org/10.1090/S0002-9947-2011-05242-6
- MathSciNet review: 2833556