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

## TL;DR

This paper studies invariant subspaces of generalized Hardy algebras linked to compact abelian group actions on von Neumann algebras, establishing conditions for their hereditary reflexivity.

## Contribution

It introduces conditions under which the Hardy algebra associated with such group actions is hereditarily reflexive, extending understanding of operator algebra structures.

## Key findings

- Hardy algebra is hereditarily reflexive if each spectral subspace contains a unitary operator.
- Hereditary reflexivity holds for dual actions on crossed products and ergodic actions.
- Fixed point algebras being factors also ensure the Hardy algebra's hereditary reflexivity.

## 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.