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Communications in Mathematical Physics

, Volume 355, Issue 3, pp 1121–1188 | Cite as

Operator Algebras in Rigid C*-Tensor Categories

  • Corey Jones
  • David Penneys
Article
  • 117 Downloads

Abstract

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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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Australian National UniversityCanberraAustralia
  2. 2.The Ohio State UniversityColumbusUSA

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