Classification of regular subalgebras of injective type III factors

We provide a complete classification for regular subalgebras $B\subset M$ of injective factors satisfying a natural relative commutant condition. We show that such subalgebras are classified by their associated amenable discrete measured groupoid $\mathcal{G}={\mathcal{G}}_{B\subset M}$ and the action $\text{mod}(\alpha )$ of $\mathcal{G}$ on the flow of weights induced by the cocycle action $(\alpha ,u)$ of $\mathcal{G}$ on $B$. We obtain a similar result for triple inclusions $A\subset B\subset M$ where $M$ is an injective factor, $A$ is a Cartan subalgebra of $M$, and $B\subset M$ is regular, showing that such inclusions are also classified by their associated groupoid $\mathcal{G}={\mathcal{G}}_{B\subset M}$ and the induced action on the flow of weights. Given such a discrete measured amenable groupoid $\mathcal{G}$, we also construct a model action of $\mathcal{G}$ on a field of Cartan inclusions with prescribed action on the associated field of flows.