Operator algebras with contractive approximate identities: Weak compactness and the spectrum

TL;DR

This paper explores operator algebras with contractive approximate identities, providing examples that link weak compactness properties to spectral characteristics, thus addressing open questions in the field.

Contribution

It introduces new examples of operator algebras with cais that demonstrate the relationship between weak compactness and spectral properties, answering prior open questions.

Findings

Examples of operator algebras with cais illustrating spectral properties.

Connection established between weak compactness and spectra.

Clarification of previous open questions in operator algebra theory.

Abstract

We continue our study of operator algebras with contractive approximate identities (cais) by presenting a couple of interesting examples of operator algebras with cais, which in particular answer questions raised in previous papers in this series, for example about whether, roughly speaking, `weak compactness' of an operator algebra, or the lack of it, can be seen in the spectra of its elements.