Maximal subalgebras of group-algebras

Abstract

A closed subalgebra of a Banach algebra is called maximal if it is not contained in any larger proper closed subalgebra. Let G be a discrete abelian topological group and L its group-algebra, i.e. L is the Banach algebra of functions f on G with . G If (X) I < oo and multiplication defined as convolution. What are the maximal subalgebras of L? The complete answer is not known even when G is the group of integers. Here we assume that G is ordered. Let G+ be the semi-group of nonnegative elements of G and L+ the subset of L consisting of functions which vanish outside of G+. Then L+ is a proper closed subalgebra of L.