On the Primitive Ideals of Nest Algebras

TL;DR

This paper characterizes primitive ideals in nest algebras, providing concrete descriptions of the Jacobson radical and classifying primitive ideals, especially in atomic cases, under the Continuum Hypothesis.

Contribution

It establishes that Ringrose's diagonal ideals are primitive in nest algebras and offers a standard form for all left ideals, advancing understanding of their structure.

Findings

Ringrose's diagonal ideals are primitive in nest algebras

Provides a standard form for all left ideals of nest algebras

Classifies primitive ideals in atomic nest algebras by diagonal behavior

Abstract

We show that Ringrose's diagonal ideals are primitive ideals in a nest algebra (subject to the Continuum Hypothesis). This provides for the first time concerete descriptions of enough primitive ideals to obtain the Jacobson radical as their intersection. Separately, we provide a standard form for all left ideals of a nest algebra, which leads to insights into the maximal left ideals. In the case of atomic nest algebras we show how primitive ideals can be categorized by their behaviour on the diagonal, and provide concrete examples of all types.