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Quantum rotatability
Author(s):
Stephen
Curran
Journal:
Trans. Amer. Math. Soc.
362
(2010),
4831-4851.
MSC (2000):
Primary 46L54;
Secondary 46L65, 60G09
Posted:
April 27, 2010
MathSciNet review:
2645052
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Abstract:
Recently, Köstler and Speicher showed that de Finetti's theorem on exchangeable sequences has a free analogue if one replaces exchangeability by the stronger condition of invariance of the joint distribution under quantum permutations. In this paper we study sequences of noncommutative random variables whose joint distribution is invariant under quantum orthogonal transformations. We prove a free analogue of Freedman's characterization of conditionally independent Gaussian families; namely, the joint distribution of an infinite sequence of self-adjoint random variables is invariant under quantum orthogonal transformations if and only if the variables form an operator-valued free centered semicircular family with common variance. Similarly, we show that the joint distribution of an infinite sequence of random variables is invariant under quantum unitary transformations if and only if the variables form an operator-valued free centered circular family with common variance. We provide an example to show that, as in the classical case, these results fail for finite sequences. We then give an approximation for how far the distribution of a finite quantum orthogonally invariant sequence is from the distribution of an operator-valued free centered semicircular family with common variance.
References:
-
- 1.
- T. Banica, J. Bichon, and B. Collins, Quantum permutation groups: A survey, Banach Center Publ., vol. 78, Warsaw, 2007 pp. 13-34. MR 2402345 (2009f:46094)
- 2.
- T. Banica and B. Collins, Integration over compact quantum groups, Publ. Res. Inst. Math. Sci., 43 (2007), pp. 277-302. MR 2341011 (2008m:46137)
- 3.
- B. Blackadar, Operator Algebras: Theory of C
 -algebras and von Neumann Algebras, vol. 122 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, 2006. MR 2188261 (2006k:46082) - 4.
- S. Curran, Quantum exchangeable sequences of algebras, Indiana Univ. Math. J. 58 (2009), pp. 1097-1125. MR 2541360
- 5.
- P. Diaconis and D. Freedman, Finite exchangeable sequences., Ann. Probab., 8 (1980), pp. 745-764. MR 577313 (81m:60032)
- 6.
- P. Diaconis and D. Freedman, A dozen de Finetti-style results in search of a theory., Ann. Inst. H. Poincaré Probab. Statist., 23 (1987), pp. 397-423. MR 898502 (88f:60072)
- 7.
- P. Diaconis and D. Freedman, Conditional limit theorems for exponential families and finite versions of de Finetti's theorem., J. Theor. Probab., 1 (1988), pp. 381-410. MR 958245 (89g:60127)
- 8.
- K. Dykema and G. Tucci, Hyperinvariant subspaces for some B-circular operators, Mathematische Annalen, 333 (2005), pp. 485-523. MR 2198797 (2007c:46061)
- 9.
- D. Freedman, Invariants under mixing which generalize de Finetti's theorem., Ann. Math. Stat., 33 (1962), pp. 916-923. MR 0156369 (27:6292)
- 10.
- O. Kallenberg, Probabilistic Symmetries and Invariance Principles, Probability and Its Applications, Springer, 2005. MR 2161313 (2006i:60002)
- 11.
- C. Köstler and R. Speicher, A noncommutative de Finetti theorem: Invariance under quantum permutations is equivalent to freeness with amalgamation, Comm. Math. Phys. 291 (2009), pp. 473-490. MR 2530168
- 12.
- A. Nica, R-transforms of free joint distributions and noncrossing partitions, J. Funct. Anal., 135 (1996), pp. 271-296. MR 1370605 (96k:46119)
- 13.
- A. Nica and R. Speicher, Lectures on the Combinatorics of Free Probability, no. 335 in London Mathematical Society Lecture Note Series, Cambridge University Press, 2006. MR 2266879 (2008k:46198)
- 14.
- R. Speicher, Combinatorial theory of the free product with amalgamation and operator-valued free probability theory., Mem. Amer. Math. Soc., 132 (1998), no. 627, 88 pp. MR 1407898 (98i:46071)
- 15.
- D. Voiculescu, Operations on certain noncommutative operator-valued random variables, Astérisque, no. 232 (1995), pp. 243-275. MR 1372537 (97b:46081)
- 16.
- D. Voiculescu, K. Dykema, and A. Nica, Free Random Variables, vol. 1 of CRM Monograph Series, American Mathematical Society, Rhode Island, 1992. MR 1217253 (94c:46133)
- 17.
- S. Wang, Free products of compact quantum groups, Comm. Math. Phys., 167 (1995), pp. 671-692. MR 1316765 (95k:46104)
- 18.
- S. Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys., 195 (1998), pp. 195-211. MR 1637425 (99h:58014)
- 19.
- S. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys., 111 (1987), pp. 613-665. MR 901157 (88m:46079)
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Additional Information:
Stephen
Curran
Affiliation:
Department of Mathematics, University of California at Berkeley, Berkeley, California 94720
Email:
curransr@math.berkeley.edu
DOI:
10.1090/S0002-9947-10-05119-6
PII:
S 0002-9947(10)05119-6
Keywords:
Free probability,
quantum rotatability,
quantum invariance,
semicircle law
Received by editor(s):
February 11, 2009
Posted:
April 27, 2010
Copyright of article:
Copyright
2010,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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