Ideal structure of operator and measure algebras

Abstract

LetH denote a finite dimensional Hilbert space with subspaceE. The set$J(E) = \{ T \in B(H):E \subseteq \ker T\}$ is a subalgebra ofB(H). A complete description of the ideal (two-sided, left and right) structure ofJ(E) is given. LetG denote a compact group with dual object ∑(G), and let σ be an element of ∑(G). The results concerningJ(E) are applied to certain convolution subalgebras ofM(G), the algebras having the property that the set of operators, μ(σ), where μ lies in the algebra, is of the formJ(E). In particular, all the minimal two-sided and right ideals are listed. The technique used is an extension of one employed byHewitt andRoss in [1] to study the closed ideals of some convolution subalgebras ofM(G) which containT(G).