  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.
