Unitary orbits of normal operators in von Neumann algebras

TL;DR

This paper characterizes the closures of the unitary orbit of normal operators in von Neumann algebras using spectral data, clarifying when different closures coincide and exploring implications for infinite algebras.

Contribution

It provides spectral criteria for the norm and strong* closures of unitary orbits of normal operators in von Neumann algebras, extending known results and analyzing their relationships.

Findings

Characterization of when norm and strong* closures of unitary orbits coincide.

Conditions under which different orbit closures are equal.

In infinite algebras, the strong-closed orbit intersects the center in the essential spectrum.

Abstract

The starting points for this paper are simple descriptions of the norm and strong* closures of the unitary orbit of a normal operator in a von Neumann algebra. The statements are in terms of spectral data and do not depend on the type or cardinality of the algebra. We relate this to several known results and derive some consequences, of which we list a few here. Exactly when the ambient von Neumann algebra is a direct sum of sigma-finite algebras, any two normal operators have the same norm-closed unitary orbit if and only if they have the same strong*-closed unitary orbit if and only if they have the same strong-closed unitary orbit. But these three closures generally differ from each other and from the unclosed unitary orbit, and we characterize when equality holds between any two of these four sets. We also show that in a properly infinite von Neumann algebra, the strong-closed unitary orbit of any operator, not necessarily normal, meets the center in the (non-void) left essential central spectrum of Halpern. One corollary is a "type III Weyl-von Neumann-Berg theorem" involving containment of essential central spectra.