Convolution Measure Algebras with Group Maximal Ideal Spaceso

Let G denote a locally compact abelian topological group (an l.c.a. group) with dual group C\ We will denote by A7(G) the Banach algebra of bounded regular Borel measures on G under convolution multiplication and by L(G) the algebra of bounded measures absolutely continuous with respect to Haar measure on G (for discussions of these Banach algebras cf. [1], [2], and [5]). In this paper we shall be concerned with closed subalgebras 501 of M(G) with the following two properties: (1) if p e 50c and v is absolutely continuous with respect to p, then vsïïî; (2) the maximal ideal space of 501 is G~, where the Gelfand transform p~ of P e 501 is given by p~(x)=J x dp for ^êGa; i.e., the Gelfand transform coincides with the Fourier-Stieltjes transform on 50Î. Any closed subalgebra of M(G) satisfying (1) will be called an L-subalgebra of M(G). It is well known thatL(G) satisfies (1) and (2) (cf. [5, Chapter 1]). In Theorem 1 we show that any L-subalgebra 50c of M(G), with L(G)<=50cc(L(G))1,2) also satisfies (2), where (L(G))112 is the intersection of all maximal ideals of M(G) containing L(G). We conjecture that the converse is also true; i.e., if 501 satisfies (1) and (2) then L(G)c 50c c(L(G))1/2. In Theorem 2 we prove that this is true provided G contains no copy of 7?" for «> 1. The problem arises in the following way : in [6] we define the concept of abstract convolution measure algebra and prove that any such algebra 501, provided it is commutative and semisimple, may be represented as an L-subalgebra of M(S), where S is a compact topological semigroup called the structure semigroup of 5DÎ. M(G) and L(G) are convolution measure algebras as is any L-subalgebra of the measure algebra on a semigroup. The map p-> ps which embeds 50c in M(S) is an isometry as well as an algebraic isomorphism and it preserves the order theoretic properties of 50Î as a space of measures. The maximal ideal space of 50c may be identified as S~, the set of all semicharacters of S, where the Gelfand transform of p e 50c is given by p~(f) = $fdp.s for fe S~ (a semicharacter of S is a continuous homomorphism of S into the unit disc which is not identically zero). Under pointwise multiplication S~ is a semigroup provided 50c has an approximate identity.