Maximal One-Sided Ideals in Operator Algebras

Abstract

Maximal right ideals in a Banach algebra are closed provided the algebra has a unit. Maximal modular right ideals are closed in any Banach algebra. It seems natural to ask whether maximal right ideals are always closed. If d is a Banach algebra and d2 (={xy: x,y^d}) is contained in a proper subspace, then that subspace is a right ideal and so, if d2 is contained in a dense maximal subspace of d then this subspace is a non-closed maximal right ideal. This does occur, for example if d is the algebra of Hilbert Schmidt operators, CL2 consists of all operators of finite trace. It has been conjectured that if Q2 = {2 then every maximal right ideal in d is closed. We shall consider only algebras CL such that d2 = d and for brevity say that d satisfies (*) provided every maximal right ideal and every maximal left ideal is closed. If d is commutative then a maximal ideal is necessarily modular and hence closed. (7*-algebras and group algebras satisfy d2 = d. We have been able to prove that some of these satisfy (*). For example if d is the algebra of compact operators on a Hilbert space,