Poisson boundaries of monoidal categories

TL;DR

This paper introduces the Poisson boundary for rigid C*-tensor categories with simple units, establishing a universal functor that characterizes amenability and unifies several existing results in quantum algebra.

Contribution

It defines the Poisson boundary for C*-tensor categories and proves its universality in characterizing amenability, linking classical random walks to quantum category properties.

Findings

Poisson boundary constructed for C*-tensor categories

Universal property of the boundary functor when the boundary has simple unit

Unification of results on amenability in quantum groups and subfactors

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.