The structure of convolution measure algebras
Joseph L. Taylor · Transactions of the American Mathematical Society · 1965
Let G be a locally compact abelian topological group, Ly(G) the algebra of Haar integrable functions on G under convolution multiplication, and M(G) the algebra of regular Borel measures on G under convolution multiplication.Ly(G) has been extensively studied and has been the motivation behind many of the results in Banach algebra theory; its maximal ideal space is G~, the character group of G. M(G) is a far more complex algebra and a workable characterization of its maximal ideal space has been elusive.In this paper we attempt to give such a characterization.In analogy with Ly(G), we represent the maximal ideal space of M(G) as the set S~ of all semicharacters on a compact topological semigroup S. The methods used are applicable to a large class of Banach algebras which we have labeled convolution measure algebras.Ly(G) and M(G) are convolution measure algebras as is the measure algebra on any locally compact topological semigroup.The Banach space structure of a convolution measure algebra is that of a complex L-space.This allows us to use the L-space theory developed by Kakutani, Cunningham, and others.In §1 we identify the adjoint space of a complex L-space as the space of all continuous functions on some compact Hausdorff space.This follows from similar results of Cunningham for real L-spaces.For this paper we have chosen to use the abstract definition of complex L-space.A detailed development of the results of §1 based on a concrete definition of L-space may be found in the author's dissertation [11].In §2, we define the concept of convolution measure algebra and prove, using the results of §1, that the maximal ideal space of a commutative convolution measure algebra SOt may be represented as the set S^ of all semicharacters on some compact topological semigroup S. In §3, we investigate the structures of SfJi, S, and SA.We identify a subset 77 of SA on which the Gelfand transform of each element of 9TJÎ attains its maximum modulus.In §4, we discuss briefly three special