Classification of regular subalgebras of injective type III factors

Abstract

We provide a complete classification for regular subalgebras [Formula: see text] of injective factors satisfying a natural relative commutant condition. We show that such subalgebras are classified by their associated amenable discrete measured groupoid [Formula: see text] and the action modnew([Formula: see text]) of [Formula: see text] on the flow of weights induced by the cocycle action [Formula: see text] of [Formula: see text] on [Formula: see text]. We obtain a similar result for triple inclusions [Formula: see text], where [Formula: see text] is an injective factor, [Formula: see text] is a Cartan subalgebra of [Formula: see text] and [Formula: see text] is regular, showing that such inclusions are also classified by their associated groupoid [Formula: see text] and the induced action on the flow of weights. Given such a discrete measured amenable groupoid [Formula: see text], we also construct a model action of [Formula: see text] on a field of Cartan inclusions with prescribed action on the associated field of flows.