Poisson boundaries of monoidal categories

Abstract

Given a rigid C*-tensor category C with simple unit and a probability measure $\mu$ on the set of isomorphism classes of its simple objects, we define the Poisson boundary of $(C,\mu )$. This is a new C*-tensor category P, generally with nonsimple unit, together with a unitary tensor functor $\Pi :C\to P$. Our main result is that if P has simple unit (which is a condition on some classical random walk), then $\Pi$ is a universal unitary tensor functor defining the amenable dimension function on C. Corollaries of this theorem unify various results in the literature on amenability of C*-tensor categories, quantum groups, and subfactors.

v2: 37 pages, minor changes, to appear in Ann. Sci. Ecole Norm. Sup.; v1: 37 pages

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Cited by in corpus (7)

• Graded twisting of comodule algebras and module categories
• Classification of non-Kac compact quantum groups of SU(n) type
• Noncommutative Poisson boundaries and Furstenberg-Hamana boundaries of Drinfeld doubles
• Martin boundaries of the duals of free unitary quantum groups
• Probabilistic boundaries of finite extensions of quantum groups
• Riesz transforms on compact quantum groups and strong solidity
• Towards a classification of compact quantum groups of Lie type