Left ideals of Banach algebras and dual Banach algebras

TL;DR

This paper characterizes when certain Banach algebras, including group algebras and operator algebras, are topologically left Noetherian, revealing connections between algebraic properties and topological or geometric features of underlying spaces.

Contribution

It provides new characterizations of topologically left Noetherian Banach algebras, especially for group and operator algebras, linking algebraic properties to topological and geometric conditions.

Findings

L^{1}(G) is topologically left Noetherian iff G is metrisable for compact G

The algebra of compact operators al K(E) is topologically left Noetherian iff E' is separable

Examples of dual Banach algebras that are topologically left Noetherian in the weak*-topology

Abstract

We investigate topologically left Noetherian Banach algebras. We show that if $G$ is a compact group, then $L^{\, 1}(G)$ is topologically left Noetherian if and only if $G$ is metrisable. We prove that, given a Banach space $E$ such that $E'$ has BAP, the algebra of compact operators $\mathcal{K}(E)$ is topologically left Noetherian if and only if $E'$ is separable; it is topologically right Noetherian if and only if $E$ is separable. We then give some examples of dual Banach algebras which are topologically left Noetherian in the weak*-topology. Finally we give a unified approach to classifying the weak*-closed left ideals of certain dual Banach algebras that are also multiplier algebras, with applications to $M(G)$ for $G$ a compact group, and $\mathcal{B}(E)$ for $E$ a reflexive Banach space with AP.