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

The relationship between index theory and random walks on fusion algeb ras is discussed. Popa's notion of amenability is reformulated as a pr operty of fusion algebras, and several equivalent conditions of amenab ility are obtained. A ratio limit theorem is proved as a characterizat ion of amenability. A number of conditions, all equivalent to Popa's n otion of strong amenability in the case of subfactors, of a pair of a fusion algebra and a probability measure are proposed, and their relat ionship is studied from the viewpoint of random walks and entropic den sities. Fusion algebra homomorphisms and free products of fusion algeb ras are also discussed.