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 -> 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.
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