A Comment on Jones Inclusions with infinite Index

Given an irreducible inclusion of infinite von-Neumann-algebras \cn \subset \cm together with a conditional expectation E : \cm \rightarrow \cm such that the inclusion has depth 2, we show quite explicitely how \cn can be viewed as the fixed point algebra of \cm w.r.t. an outer action of a compact Kac-algebra acting on \cm. This gives an alternative proof, under this special setting of a more general result of M. Enock and R. Nest, [E-N], see also S. Yamagami, [Ya2].Comment: latex, 40 page

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

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 (M\cros\A). Here M is an A-module algebra, N is the fixed point algebra and \M\cros\A 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 l\in\A 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_l\in\A obeying the Jones relations as an identity in M\cros\A. Finally, we prove that N\subset 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 M\cros\A, where under appropriate conditions M_1 = M\cros\A. 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.Comment: Latex, 63 page