Operator algebras in rigid C*-tensor categories

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 $\mathbf{A}$ in $\mathrm{ind}$-$\mathcal{C}$ whose category of free modules ${\mathsf{FreeMod}}_{\mathcal{C}}(\mathbf{A})$ is a $\mathcal{C}$-module C*/W*-category respectively. When $\mathcal{C}=\mathsf{Hil}{\mathsf{b}}_{f.d.}$, 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 maps 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 $\mathbf{M}$ in $\mathcal{C}$. Our definitions simultaneously unify the definitions of analytic properties for discrete quantum groups and rigid C*-tensor categories.

65 pages, many figures. Comments welcome! In version 2, we restrict the discussion of analytic properties to connected W*-algebra objects

References in corpus (4)

• Fermionic Modular Categories and the 16-fold Way
• Monoidal supercategories
• Planar Para Algebras, Reflection Positivity
• Around Property (T) for quantum groups

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