Operator Algebras in Rigid C*-Tensor Categories

Corey Jones, David Penneys*

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    21 Citations (Scopus)

    Abstract

    In this article, we define operator algebras internal to a rigid C*-tensor category C. A C*/W*-algebra object in C is an algebra object A in ind-C whose category of free modules FreeModC(A) is a C-module C*/W*-category respectively. When C= Hilbfd, 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 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 C. Our definitions simultaneously unify the definitions of analytic properties for discrete quantum groups and rigid C*-tensor categories.

    Original languageEnglish
    Pages (from-to)1121-1188
    Number of pages68
    JournalCommunications in Mathematical Physics
    Volume355
    Issue number3
    DOIs
    Publication statusPublished - 1 Nov 2017

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