Left Ideals of Banach Algebras and Dual Banach Algebras

摘要

We investigate topologically left Noetherian Banach algebras.We showthat if G is a compact group, then L~1(G)is topologically leftNoetherian if and onlyif G is metrisable.We prove that, given a Banach space E′ such that E has BAP, thealgebra of compact operators K(E) is topologically left Noetherian if and only ifE′ is separable; it is topologically right Noetherian if and only if E is separable.Wethen give some examples of dual Banach algebras which are topologically left Noetherianin the weak*-topology. Finally we give a unified approach to classifyingthe weak*-closed left ideals of certain dual Banach algebras that are also multiplieralgebras, with applications to M(G) for G a compact group, and B(E) for E areflexive Banach space with AP.