The group algebra of a locally compact group

Abstract

Introduction. This paper concerns the theory of ideals in the algebra (called the group algebra) of all complex-valued functions on a locally compact (abbreviated to LC) group which are integrable with respect to Haar measure, multiplication being defined as convolution. It is proved that the group algebra of a group which is either LC abelian or compact is semisimple, an algebra being called semi-simple in case the intersection of all regular maximal ideals is the null ideal (a regular ideal is defined as one modulo which the algebra has an identity). This extends the theorem that the group algebra of a finite group is semi-simple. A weaker kind of semi-simplicity is proved in the case of a general LC group, an algebra being called weakly semi-simple when both the intersection of all regular maximal right ideals is the null ideal and the same for left ideals. (In the case of a finite-dimensional algebra these concepts of semi-simplicity are equivalent to that of Wedderburn, and in the case of a commutative Banach algebra with an identity-which Gelfand designates as a normed ring-are equivalent to Gelfand's concept of the vanishing of the radical of the algebra.) It is shown that an ideal in a (strongly) semi-simple algebra can be resolved into regular ideals in an approximate fashion, the approximation being in terms of a topology which is (algebraically) introduced into the familv of all regular maximal ideals. This topologized family (which is called the spectrum of the algebra, this term being suggested by the case of the algebra generated by a linear operator) is determined in explicit fashion for the group algebra of a group which is either LC abelian or compact, and partially determined when the group is discrete. Specifically, in the case of a group which is respectively either LC abelian, compact, or discrete, the spectrum of the group algebra is homeomorphic with the group of continuous characters, discrete or compact. A closed ideal in a group algebra is the kernel of a kind of extension from the group to the group algebra of a bounded strongly continuous representation of the group by linear operators on a Banach space. If the