Algebra Of Compact Operators Research Articles

Maximal One-Sided Ideals in Operator Algebras

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,

Nonfactorization in subsets of the measure algebra

In this note we unify and simplify some recent results showing the impossibility of factoring in certain convolution subalgebras of the group algebra of a nondiscrete LCAG. A new result is a direct proof of nonfactorization of the classical Hardy spaces, regarded as convolution algebras, on the circle. By considering the ideal of Hilbert-Schmidt operators in the algebra of compact operators on a Hilbert space we illustrate that nonfactorization is not peculiar to convolution.