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Operator Algebras in Rigid C*-Tensor Categories

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In this article, we define operator algebras internal to a rigid C*-tensor category \({\mathcal{C}}\). A C*/W*-algebra object in \({\mathcal{C}}\) is an algebra object A in ind-\({\mathcal{C}}\) whose category of free modules \({\mathsf{FreeMod}_\mathcal{C}(\mathbf{A})}\) is a \({\mathcal{C}}\)-module C*/W*-category respectively. When \({\mathcal{C}= \mathsf{Hilb}_\mathsf{fd}}\), the category of finite dimensional Hilbert spaces, we recover the usual notions of operator algebras. We generalize basic representation theoretic results, such as the Gelfand-Naimark and von Neumann bicommutant theorems, along with the GNS construction. We define the notion of completely positive morphisms between C*-algebra objects in \({\mathcal{C}}\) and prove the analog of the Stinespring dilation theorem. As an application, we discuss approximation and rigidity properties, including amenability, the Haagerup property, and property (T) for a connected W*-algebra M in \({\mathcal{C}}\). Our definitions simultaneously unify the definitions of analytic properties for discrete quantum groups and rigid C*-tensor categories.

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Authors and Affiliations

  1. Australian National University, Canberra, Australia

    Corey Jones

  2. The Ohio State University, Columbus, OH, USA

    David Penneys

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  1. Corey Jones
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  2. David Penneys
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Communicated by Y. Kawahigashi

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Jones, C., Penneys, D. Operator Algebras in Rigid C*-Tensor Categories. Commun. Math. Phys. 355, 1121–1188 (2017). https://doi.org/10.1007/s00220-017-2964-0

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