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Operator Systems from Discrete Groups

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Abstract

We express various sets of quantum correlations studied in the theoretical physics literature in terms of different tensor products of operator systems of discrete groups. We thus recover earlier results of Tsirelson and formulate a new approach for the study of quantum correlations. To do this we formulate a general framework for the study of operator systems arising from discrete groups. We study in detail the operator system of the free group \({\mathbb{F}_n}\) on n generators, as well as the operator systems of the free products of finitely many copies of the two-element group \({\mathbb{Z}_2}\). We examine various tensor products of group operator systems, including the minimal, the maximal, and the commuting tensor products. We introduce a new tensor product in the category of operator systems and formulate necessary and sufficient conditions for its equality to the commuting tensor product in the case of group operator systems.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Regina, Regina, SK, S4S 0A2, Canada

    Douglas Farenick

  2. Department of Mathematics, University of Illinois, Urbana, IL, 61801, USA

    Ali S. Kavruk

  3. Department of Mathematics, University of Houston, Houston, TX, 77204-3476, USA

    Vern I. Paulsen

  4. Pure Mathematics Research Centre, Queen’s University Belfast, Belfast, BT7 1NN, UK

    Ivan G. Todorov

Authors
  1. Douglas Farenick
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  2. Ali S. Kavruk
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  3. Vern I. Paulsen
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  4. Ivan G. Todorov
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Corresponding author

Correspondence to Douglas Farenick.

Additional information

Communicated by A. Winter

This work supported in part by NSERC (Canada), NSF (USA), and the Royal Society (UK).

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Farenick, D., Kavruk, A.S., Paulsen, V.I. et al. Operator Systems from Discrete Groups. Commun. Math. Phys. 329, 207–238 (2014). https://doi.org/10.1007/s00220-014-2037-6

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  • DOI: https://doi.org/10.1007/s00220-014-2037-6

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