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Extended de Finetti theorems for boolean independence and monotone independence

Author: Weihua Liu
Journal: Trans. Amer. Math. Soc. 370 (2018), 1959-2003
MSC (2010): Primary 46L53; Secondary 46L54, 60G09
Published electronically: October 24, 2017
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Abstract: We construct several new spaces of quantum sequences and their quantum families of maps in the sense of Sołtan. The noncommutative distributional symmetries associated with these quantum maps are noncommutative versions of spreadability and partial exchangeability. Then, we study simple relations between these symmetries. We will focus on studying two kinds of noncommutative distributional symmetries: monotone spreadability and boolean spreadability. We provide an example of a spreadable sequence of random variables for which the usual unilateral shift is an unbounded map. As a result, it is natural to study bilateral sequences of random objects, which are indexed by integers, rather than unilateral sequences. At the end of the paper, we will show Ryll-Nardzewski type theorems for monotone independence and boolean independence: Roughly speaking, an infinite bilateral sequence of random variables is monotonically (boolean) spreadable if and only if the variables are identically distributed and monotone (boolean) with respect to the conditional expectation onto its tail algebra. For an infinite sequence of noncommutative random variables, boolean spreadability is equivalent to boolean exchangeability.

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  • [1] Teodor Banica, Stephen Curran, and Roland Speicher, Classification results for easy quantum groups, Pacific J. Math. 247 (2010), no. 1, 1-26. MR 2718205,
  • [2] Teodor Banica, Stephen Curran, and Roland Speicher, De Finetti theorems for easy quantum groups, Ann. Probab. 40 (2012), no. 1, 401-435. MR 2917777,
  • [3] 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
  • [4] Vitonofrio Crismale and Francesco Fidaleo, Exchangeable stochastic processes and symmetric states in quantum probability, Ann. Mat. Pura Appl. (4) 194 (2015), no. 4, 969-993. MR 3357690,
  • [5] Stephen Curran, A characterization of freeness by invariance under quantum spreading, J. Reine Angew. Math. 659 (2011), 43-65. MR 2837010,
  • [6] Stephen Curran and Roland Speicher, Quantum invariant families of matrices in free probability, J. Funct. Anal. 261 (2011), no. 4, 897-933. MR 2803836,
  • [7] P. Diaconis and D. Freedman, Partial exchangeability and sufficiency, Statistics: applications and new directions (Calcutta, 1981) Indian Statist. Inst., Calcutta, 1984, pp. 205-236. MR 786142
  • [8] Uwe Franz, Multiplicative monotone convolutions, Quantum probability, Banach Center Publ., vol. 73, Polish Acad. Sci. Inst. Math., Warsaw, 2006, pp. 153-166. MR 2423123,
  • [9] Uwe Franz, Monotone and Boolean convolutions for non-compactly supported probability measures, Indiana Univ. Math. J. 58 (2009), no. 3, 1151-1185. MR 2541362,
  • [10] Amaury Freslon and Moritz Weber, On bi-free de Finetti theorems, Ann. Math. Blaise Pascal 23 (2016), no. 1, 21-51. MR 3505568
  • [11] Tomohiro Hayase, De Finetti theorems for a boolean analogue of easy quantum groups, arXiv (2015).
  • [12] Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I: Elementary theory, Graduate Studies in Mathematics, vol. 15, American Mathematical Society, Providence, RI, 1997. Reprint of the 1983 original. MR 1468229
  • [13] Olav Kallenberg, Spreading-invariant sequences and processes on bounded index sets, Probab. Theory Related Fields 118 (2000), no. 2, 211-250. MR 1790082,
  • [14] Olav Kallenberg, Probabilistic symmetries and invariance principles, Probability and its Applications (New York), Springer, New York, 2005. MR 2161313
  • [15] Claus Köstler, A noncommutative extended de Finetti theorem, J. Funct. Anal. 258 (2010), no. 4, 1073-1120. MR 2565834,
  • [16] Claus Köstler and Roland Speicher, A noncommutative de Finetti theorem: invariance under quantum permutations is equivalent to freeness with amalgamation, Comm. Math. Phys. 291 (2009), no. 2, 473-490. MR 2530168,
  • [17] Weihua Liu, A noncommutative de Finetti theorem for boolean independence, J. Funct. Anal. 269 (2015), no. 7, 1950-1994. MR 3378866,
  • [18] Weihua Liu, Noncommutative Distributional Symmetries and Their Related de Finetti Type Theorems, ProQuest LLC, Ann Arbor, MI, 2016. Thesis (Ph.D.)-University of California, Berkeley. MR 3640931
  • [19] Naofumi Muraki, Noncommutative Brownian motion in monotone Fock space, Comm. Math. Phys. 183 (1997), no. 3, 557-570. MR 1462227,
  • [20] Naofumi Muraki, The five independences as natural products, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), no. 3, 337-371. MR 2016316,
  • [21] 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
  • [22] Mihai Popa, A combinatorial approach to monotonic independence over a $ C^*$-algebra, Pacific J. Math. 237 (2008), no. 2, 299-325. MR 2421124,
  • [23] C. Ryll-Nardzewski, On stationary sequences of random variables and the de Finetti's equivalence, Colloq. Math. 4 (1957), 149-156. MR 0088823
  • [24] Piotr M. Sołtan, Quantum families of maps and quantum semigroups on finite quantum spaces, J. Geom. Phys. 59 (2009), no. 3, 354-368. MR 2501746,
  • [25] Piotr Mikołaj Sołtan, On quantum semigroup actions on finite quantum spaces, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 (2009), no. 3, 503-509. MR 2572470,
  • [26] Roland Speicher, On universal products, Free probability theory (Waterloo, ON, 1995) Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 257-266. MR 1426844
  • [27] 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
  • [28] Şerban Strătilă, Modular theory in operator algebras, translated from the Roman by the author, Editura Academiei Republicii Socialiste România, Bucharest; Abacus Press, Tunbridge Wells, 1981. MR 696172 (85g:46072)
  • [29] Dan-Virgil Voiculescu, Free probability for pairs of faces I, Comm. Math. Phys. 332 (2014), no. 3, 955-980. MR 3262618,
  • [30] Dan Voiculescu, Symmetries of some reduced free product $ C^\ast$-algebras, Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983) Lecture Notes in Math., vol. 1132, Springer, Berlin, 1985, pp. 556-588. MR 799593,
  • [31] D. V. Voiculescu, K. J. Dykema, and A. Nica, Free random variables: A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups, CRM Monograph Series, vol. 1, American Mathematical Society, Providence, RI, 1992. MR 1217253
  • [32] Shuzhou Wang, Free products of compact quantum groups, Comm. Math. Phys. 167 (1995), no. 3, 671-692. MR 1316765
  • [33] Shuzhou Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998), no. 1, 195-211. MR 1637425,
  • [34] S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), no. 4, 613-665. MR 901157

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Additional Information

Weihua Liu
Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California 94720
Address at time of publication: Department of Mathematics, Indiana University Bloomington, Bloomington, Indiana, 47401

Received by editor(s): April 15, 2016
Received by editor(s) in revised form: June 29, 2016
Published electronically: October 24, 2017
Additional Notes: The author was partially supported by NSF grants DMS-1301727 and DMS-1302713, and by the Natural Science Foundation of China (grant No. 11171301)
Article copyright: © Copyright 2017 American Mathematical Society

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