Ideals and hereditary subalgebras in operator algebras

TL;DR

This paper explores the structure of operator algebras with contractive approximate identities, focusing on ideals, hereditary subalgebras, and their properties in biduals, providing new insights and examples in the Banach algebra context.

Contribution

It advances the understanding of ideal structures and hereditary subalgebras in operator algebras from a Banach algebra perspective, including properties related to biduals and weak compactness.

Findings

Characterization of hereditary subalgebras in operator algebras

Analysis of ideal structures in algebras with contractive approximate identities

Examples illustrating properties of weakly compact operator algebras

Abstract

This paper may be viewed as having two aims. First, we continue our study of algebras of operators on a Hilbert space which have a contractive approximate identity, this time from a more Banach algebraic point of view. Namely, we mainly investigate topics concerned with the ideal structure, and hereditary subalgebras (HSA's), which are in some sense generalization of ideals. Second, we study properties of operator algebras which are hereditary subalgebras in their bidual, or equivalently which are `weakly compact'. We also give several examples answering natural questions that arise in such an investigation.