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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
DOI: https://doi.org/10.1090/tran6661
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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Additional Information

Kenneth J. Dykema
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: kdykema@math.tamu.edu

Claus Köstler
Affiliation: School of Mathematical Sciences, Western Gateway Building, Western Road, University College Cork, Cork, Ireland
Email: claus@ucc.ie

John D. Williams
Affiliation: Department of Mathematics, Fachrichtung Mathematik, Universität des Saarlandes, Campus E24, 66123 Saarbrücken, Germany
Email: williams@math.uni-sb.de

DOI: https://doi.org/10.1090/tran6661
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