Left ideals of Banach algebras and dual Banach algebras

Jared T. White

Published 2018-11-06Version 1

We define what it means for a Banach algebra to be topologically left Noetherian. 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 define what it means for a dual Banach algebra to be weak*-topologically left Noetherian, and give some examples for which this condition holds. 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.