AMENABILITY AND STRONG AMENABILITY FOR FUSION ALGEBRAS WITH APPLICATIONS TO SUBFACTOR THEORY

Abstract

The relationship between index theory and random walks on fusion algebras is discussed. Popa's notion of amenability is reformulated as a property of fusion algebras, and several equivalent conditions of amenability are obtained. A ratio limit theorem is proved as a characterization of amenability. A number of conditions, all equivalent to Popa's notion of strong amenability in the case of subfactors, of a pair of a fusion algebra and a probability measure are proposed, and their relationship is studied from the viewpoint of random walks and entropic densities. Fusion algebra homomorphisms and free products of fusion algebras are also discussed.