Operator algebras in rigid C*-tensor categories

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 ${\mathsf{F}\mathsf{r}\mathsf{e}\mathsf{e}\mathsf{M}\mathsf{o}\mathsf{d}}_{\mathcal{C}}(\mathbf{A})$ 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.