In this article, we define operator algebras internal to a rigid C*-tensor category $${\mathcal{C}}$$ C . A C*/W*-algebra object in $${\mathcal{C}}$$ C is an algebra object A in ind- $${\mathcal{C}}$$ C whose category of free modules $${\mathsf{FreeMod}_\mathcal{C}(\mathbf{A})}$$ FreeMod C ( A ) is a $${\mathcal{C}}$$ C -module C*/W*-category respectively. When $${\mathcal{C}= \mathsf{Hilb}_\mathsf{fd}}$$ C = Hilb 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 $${\mathcal{C}}$$ 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 $${\mathcal{C}}$$ C . Our definitions simultaneously unify the definitions of analytic properties for discrete quantum groups and rigid C*-tensor categories.
Communications in Mathematical Physics – Springer Journals
Published: Aug 7, 2017
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