Partial and Total Ideals of Von Neumann Algebras

TL;DR

This paper introduces the concept of partial ideals in operator algebras, showing that in von Neumann algebras, these correspond to ultraweakly closed ideals, linking algebraic and spectral properties.

Contribution

It establishes a correspondence between total ideals and unitarily invariant partial ideals in von Neumann algebras, providing a new perspective on their structure.

Findings

Total ideals correspond to unitarily invariant partial ideals

The result can be expressed via central projections

Connects algebraic ideals with spectral notions in noncommutative algebras

Abstract

A notion of partial ideal for an operator algebra is a weakening the notion of ideal where the defining algebraic conditions are enforced only in the commutative subalgebras. We show that, in a von Neumann algebra, the ultraweakly closed two-sided ideals, which we call total ideals, correspond to the unitarily invariant partial ideals. The result also admits an equivalent formulation in terms of central projections. We place this result in the context of an investigation into notions of spectrum of noncommutative $C^*$-algebras.