TY - JOUR

T1 - Operator Algebras in Rigid C*-Tensor Categories

AU - Jones, Corey

AU - Penneys, David

N1 - Publisher Copyright:
© 2017, Springer-Verlag GmbH Germany.

PY - 2017/11/1

Y1 - 2017/11/1

N2 - 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 FreeModC(A) is a C-module C*/W*-category respectively. When C= Hilbfd, 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.

AB - 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 FreeModC(A) is a C-module C*/W*-category respectively. When C= Hilbfd, 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.

UR - http://www.scopus.com/inward/record.url?scp=85026923598&partnerID=8YFLogxK

U2 - 10.1007/s00220-017-2964-0

DO - 10.1007/s00220-017-2964-0

M3 - Article

SN - 0010-3616

VL - 355

SP - 1121

EP - 1188

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 3

ER -