Weak Hopf Algebras and Reducible Jones Inclusions of Depth 2. I: From Crossed products to Jones towers

Hans-Werner Wiesbrock, Kornél Szlachányi, Florian Nill

Abstract: We apply the theory of finite dimensional weak C^*-Hopf algebras A as developed by G. B"ohm, F. Nill and K. Szlach'anyi to study reducible inclusion triples of von-Neumann algebras N subset M subset (McrosA). Here M is an A-module algebra, N is the fixed point algebra and McrosA is the crossed product extension. ``Weak means that the coproduct Delta on A is non-unital, requiring various modifications of the standard definitions for (co-)actions and crossed products. We show that acting with normalized positive and nondegenerate left integrals linA gives rise to faithful conditional expectations E_l: M-->N, where under certain regularity conditions this correspondence is one-to-one. Associated with such left integrals we construct ``Jones projections e_linA obeying the Jones relations as an identity in McrosA. Finally, we prove that Nsubset M always has finite index and depth 2 and that the basic Jones construction is given by the ideal M_1:=M e_l M subset McrosA, where under appropriate conditions M_1 = McrosA. In a subsequent paper we will show that converseley any reducible finite index and depth-2 Jones tower of von-Neumann factors (with finite dimensional centers) arises in this way.

This page was built for publication: Weak Hopf Algebras and Reducible Jones Inclusions of Depth 2. I: From Crossed products to Jones towers

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6500940)