Representation theory for subfactors, $\lambda$-lattices and C*-tensor categories

Abstract

We develop a representation theory for $\lambda$-lattices, arising as standard invariants of subfactors, and for rigid C*-tensor categories, including a definition of their universal C*-algebra. We use this to give a systematic account of approximation and rigidity properties for subfactors and tensor categories, like (weak) amenability, the Haagerup property and property (T). We determine all unitary representations of the Temperley-Lieb-Jones $\lambda$-lattices and prove that they have the Haagerup property and the complete metric approximation property. We also present the first subfactors with property (T) standard invariant and that are not constructed from property (T) groups.

v3: minor changes, final version to appear in Communications in Mathematical Physics. v2: improved exposition; permanence of property (T) under quotients added

References in corpus (2)

• On a Class of Type II${}_1$ Factors with Betti Numbers Invariants
• A classification of $SU(d)$-type C${}^{\ast }$-tensor categories

Cited by in corpus (26)

• Operator algebras in rigid C*-tensor categories
• Drinfeld center and representation theory for monoidal categories
• Around Property (T) for quantum groups
• L^2-Betti numbers of rigid C*-tensor categories and discrete quantum groups
• Property (T) discrete quantum groups and subfactors with triangle presentations
• C*-tensor categories and subfactors for totally disconnected groups
• Comparison of unitary duals of Drinfeld doubles and complex semisimple Lie groups
• Compact Hypergroups from Discrete Subfactors
• Admissibility Conjecture and Kazhdan's Property (T) for quantum groups
• A few remarks on the tube algebra of a monoidal category
• The Fourier algebra of a rigid $C^{\ast }$-tensor category
• Categorically Morita equivalent compact quantum groups
• Noncommutative Poisson boundaries and Furstenberg-Hamana boundaries of Drinfeld doubles
• $K$-theory of AF-algebras from braided C*-tensor categories
• Howe-Moore type theorems for quantum groups and rigid C*-tensor categories
• Quantum groups, property (T), and weak mixing
• Pointwise convergence of noncommutative Fourier series
• The approximation property for locally compact quantum groups
• Realizing the braided Temperley-Lieb-Jones C*-tensor categories as Hilbert C*-modules
• Local topological order and boundary algebras
• Some remarks on free products of rigid $C^{\ast }$-2-categories
• Positive definiteness and Fell bundles over discrete groups
• Riesz transforms on compact quantum groups and strong solidity
• Tube algebra of group-type subfactors
• Tube representations and twisting of graded categories
• Discrete Inclusions of C*-algebras