A Galois Correspondence for Finite Depth II₁ Factors
by
Dmitri Nikshych
University of California at Los Angeles
Coauthors: Leonid Vainerman

We introduce a Galois-type correspondence between finite depth II₁ factors and weak Hopf C^*-algebras. The latter objects generalize both groupoid algebras and usual Hopf C^*-algebras (quantum groups) and were recently employed to characterize (not necessarily irreducible) depth 2 subfactors of finite index : any such an inclusion has a weak Hopf C^*-algebra as a complete invariant.

We will show that to any finite depth inclusion N subset M of II₁ factors ([M:N] < \infty) one can canonically associate a weak Hopf C^*-algebra B acting outerly on N and a left coideal C^*-subalgebra I subset B such that M=<N, I> subset N×B (in the case when depth <= 2 one has M = N×B). Thus, all finite depth inclusions can be described in terms of weak Hopf C^*-algebras.

Moreover, we will show that given an outer action of B on N there is a natural isomorphism between the lattices of intermediate von Neumann subalgebras of N subset N×B and left coideal C^*-subalgebras of B, which provides a version of a Galois correspondence for subfactors. This extends the previous result of Izumi, Longo, and Popa for usual Hopf C^*-algebras (Kac algebras).

Date received: April 28, 1999