Invariant subspaces of generalized Hardy algebras associated with compact abelian group actions on W*-algebras

Abstract

We consider an action of a compact group whose dual is archimedean linearly ordered or a direct product (or sum) of such groups on a von Neumann algebra, M. We define the generalized Hardy subspace of the Hilbert space of a standard representation the algebra, and the Hardy subalgebra of analytic elements of M with respect to the action. We find conditions in order that the Hardy algebra is a hereditarily reflexive algebra of operators. In particular if every non zero spectral subspace, contains a unitary operator, the condition is satisfied and therefore the Hardy algebra is hereditarily reflexive. This is the case if the action is the dual action on a crossed product, or an ergodic action, or, if, in some situations, the fixed point algebra is a factor.