Operator algebras in rigid C*-tensor categories

TL;DR

This paper develops a framework for operator algebras within rigid C*-tensor categories, generalizing classical results and properties of operator algebras to a categorical setting, unifying quantum group and tensor category theories.

Contribution

It introduces the concept of C*-algebra objects in rigid C*-tensor categories, extending classical operator algebra results and properties to this new categorical context.

Findings

Generalized Gelfand-Naimark and bicommutant theorems

Established Stinespring dilation theorem in the categorical setting

Unified analytic properties for quantum groups and 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 $\operatorname{ind}$-$\mathcal{C}$ whose category of free modules ${\sf FreeMod}_{\mathcal{C}}(\mathbf{A})$ is a $\mathcal{C}$-module C*/W*-category respectively. When $\mathcal{C}={\sf Hilb_{f.d.}}$, 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.