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Transactions of the American Mathematical Society

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Quantum symmetric states on free product $ C^*$-algebras

Authors: Kenneth J. Dykema, Claus Köstler and John D. Williams
Journal: Trans. Amer. Math. Soc. 369 (2017), 645-679
MSC (2010): Primary 46L53; Secondary 46L54, 81S25, 46L10
Published electronically: March 21, 2016
MathSciNet review: 3557789
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Abstract: We introduce symmetric states and quantum symmetric states on universal unital free product $ C^*$-algebras of the form $ {\mathfrak{A}}=\operatornamewithlimits {\ast }_1^\infty A$ for an arbitrary unital $ C^*$-algebra $ A$ as a generalization of the notions of exchangeable and quantum exchangeable random variables. We prove the existence of conditional expectations onto tail algebras in various settings and we define a natural $ C^*$-subalgebra of the tail algebra, called the tail $ C^*$-algebra. Extending and building on the proof of the noncommutative de Finetti theorem of Köstler and Speicher, we prove a de Finetti type theorem that characterizes quantum symmetric states in terms of amalgamated free products over the tail $ C^*$-algebra, and we provide a convenient description of the set of all quantum symmetric states on $ {\mathfrak{A}}$ in terms of $ C^*$-algebras generated by homomorphic images of $ A$ and the tail $ C^*$-algebra. This description allows a characterization of the extreme quantum symmetric states. Similar results are proved for the subset of tracial quantum symmetric states, though in terms of von Neumann algebras and normal conditional expectations. The central quantum symmetric states are those for which the tail algebra is in the center of the von Neumann algebra, and we show that the central quantum symmetric states form a Choquet simplex whose extreme points are the free product states, while the tracial central quantum symmetric states form a Choquet simplex whose extreme points are the free product traces.

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  • [1] Beatriz Abadie and Ken Dykema, Unique ergodicity of free shifts and some other automorphisms of $ C^\ast $-algebras, J. Operator Theory 61 (2009), no. 2, 279-294. MR 2501005 (2010f:46097)
  • [2] Stephen Curran, Quantum exchangeable sequences of algebras, Indiana Univ. Math. J. 58 (2009), no. 3, 1097-1125. MR 2541360 (2010f:46096),
  • [3] Kenneth J. Dykema and Claus Köstler, Tail algebras of quantum exchangeable random variables, Proc. Amer. Math. Soc. 142 (2014), no. 11, 3853-3863. MR 3251725,
  • [4] Rolf Gohm and Claus Köstler, Noncommutative independence from the braid group $ \mathbb{B}_\infty $, Comm. Math. Phys. 289 (2009), no. 2, 435-482. MR 2506759 (2012d:46152),
  • [5] Edwin Hewitt and Leonard J. Savage, Symmetric measures on Cartesian products, Trans. Amer. Math. Soc. 80 (1955), 470-501. MR 0076206 (17,863g)
  • [6] Olav Kallenberg, Probabilistic symmetries and invariance principles, Probability and its Applications (New York), Springer, New York, 2005. MR 2161313 (2006i:60002)
  • [7] Claus Köstler, A noncommutative extended de Finetti theorem, J. Funct. Anal. 258 (2010), no. 4, 1073-1120. MR 2565834 (2011a:46101),
  • [8] 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 (2010h:46101),
  • [9] E. C. Lance, Hilbert $ C^*$-modules, London Mathematical Society Lecture Note Series, vol. 210, Cambridge University Press, Cambridge, 1995. A toolkit for operator algebraists. MR 1325694 (96k:46100)
  • [10] Weihua Liu. ,private communication.
  • [11] Robert R. Phelps, Lectures on Choquet's theorem, 2nd ed., Lecture Notes in Mathematics, vol. 1757, Springer-Verlag, Berlin, 2001. MR 1835574 (2002k:46001)
  • [12] Shôichirô Sakai, $ C^*$-algebras and $ W^*$-algebras, Springer-Verlag, New York-Heidelberg, 1971. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60. MR 0442701 (56 #1082)
  • [13] 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 (98i:46071),
  • [14] Erling Størmer, Symmetric states of infinite tensor products of $ C^{\ast } $-algebras, J. Functional Analysis 3 (1969), 48-68. MR 0241992
  • [15] 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 (87d:46075),
  • [16] 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 (94c:46133)
  • [17] Shuzhou Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998), no. 1, 195-211. MR 1637425,

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Additional Information

Kenneth J. Dykema
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Claus Köstler
Affiliation: School of Mathematical Sciences, Western Gateway Building, Western Road, University College Cork, Cork, Ireland

John D. Williams
Affiliation: Department of Mathematics, Fachrichtung Mathematik, Universität des Saarlandes, Campus E24, 66123 Saarbrücken, Germany

Keywords: Quantum symmetric states, quantum exchangeable, de Finetti theorem, amalgamated free product
Received by editor(s): June 25, 2013
Received by editor(s) in revised form: September 24, 2014, and January 17, 2015
Published electronically: March 21, 2016
Additional Notes: The first author was supported in part by NSF grant DMS-1202660.
The second author was supported in part by EPSRC grant EP/H016708/1.
Article copyright: © Copyright 2016 American Mathematical Society

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