Invariant subspaces of algebras of analytic elements associated with periodic flows on W*-algebras

TL;DR

This paper investigates the structure of invariant subspaces of algebras of analytic elements in von Neumann algebras under circle group actions, establishing conditions for reflexivity and providing various examples.

Contribution

It characterizes when the algebra of analytic elements is reflexive based on spectral properties and extends known results to new classes of operator algebras.

Findings

The algebra of analytic elements is reflexive if the Arveson spectrum is finite.

If the spectrum is infinite, the spectral subspace with the least positive element contains a unitary operator.

Examples include analytic Toeplitz operators, crossed products, and reflexive nest subalgebras.

Abstract

We consider an action of the circle group, T on a von Neumann algebra, M. Similarly to the case when the algebra of essentially bounded functions on T is acted upon by translations, we define the generalized Hardy subspace of H,where H is the Hilbert space of a standard representation of M and the subalgebra of analytic elements of M with respect to the action. We prove that the subalgebra of analytic elements is a reflexive algebra of operators if the Arveson spectrum is finite or, if the spectrum is infinite, the spectral subspace corresponding to the least positive element contains an unitary operator. We also prove that the analytic algebra is reflexive if M is an abelian W*-algebra. Examples include the algebra of analytic Toeplitz operators, crossed products, reduced semicrossed products and some reflexive nest subalgebras of von Neumann algebras.