Operator Algebras in Rigid C*-Tensor Categories

In this article, we define operator algebras internal to a rigid C*-tensor category C. A C*/W*-algebra object in C is an algebra object A in ind-C whose category of free modules FreeMod(C) (A) is a C-module C*/W*-category respectively. When 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 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 C. Our definitions simultaneously unify the definitions of analytic properties for discrete quantum groups and rigid C*-tensor categories.The authors would like to thank Marcel Bischoff, Shamindra Ghosh, André Henriques, Zhengwei Liu, Thomas Sinclair, James Tener and Makoto Yamashita for helpful conversations. Corey Jones was supported by Discovery Projects ‘Subfactors and symmetries’ DP140100732 and ‘Low dimensional categories’ DP160103479 from the Australian Research Council. David Penneys was supported by DMS NSF grants 1500387 and 1655912