Standard invariants for crossed products inclusions of factors

Abstract

When N ⊂ M is an inclusion of factors with finite index and a group G acts on N ⊂ M, we compare the standard invariants of N ⊂ M and the crossed product inclusion N G ⊂ M G. The cases when G is a discrete group and when G is a locally compact abelian group are separately considered. Applying to a common crossed product decomposition, we obtain comparison results between the type II and type III standard invariants of an inclusion of type III factors. Introduction.Since Jones [21] initiated the index theory of type II 1 subfactors, a great progress has been made particularly on classification of hyperfinite type II 1 subfactors (see [35,[37][38][39] and also [20,23] In the course of classifying hyperfinite type II 1 subfactors with small indices, it has been observed that symmetries on principal graphs (or paragroup symmetries) play a vital role typically in orbifold constructions (see [4,8,10,20,23]). A paragroup symmetry is an action of a (finite) group G on the principal graph and Ocneanu's connection made from an inclusion N ⊂ M. When this symmetry can extend to the subfactor level, the crossed product inclusion N G ⊂ M G arises and its standard invariant 237

Amenability and Strong Amenability for Fusion Algebras With Applications to Subfactor Theory

“…The first part of the following is already known [33,73]. More examples of fusion algebra homomorphisms can be given related with free product fusion algebras (see Sec.…”

“…This notion gives typical examples of fusion algebra homomorphisms in connection with the type II and type III standard invariants of type III subfactors. Our systematic analysis of fusion algebra homomorphisms covers some results in [33], where the behavior of the standard invariants under simultaneous crossed product was investigated. By using the Kesten invariant and the entropic densities, we also show that amenability and strong amenability behave quite well in non-degenerate commuting squares.…”