𝐿-subalgebras of 𝑀(𝐺)

Abstract

Let G be a locally compact abelian group with dual group F. Let M(G) be the algebra of bounded regular Borel measures on G under convolution multiplication, and let L(G) be the subalgebra of M(G) consisting of all absolutely continuous measures in M(G). By an L-subalgebra of M(G) we mean a closed subalgebra N, with the property that if k E N and v E M(G), with v absolutely continuous with respect to pu, then v E N. Clearly L(G) is an L-subalgebra of M(G). Let N be an L-subalgebra of M(G). If y E r and h,(1k) = f S d[k for Zt E N, then hy is a multiplicative linear functional on N. Hence, if A is the space of all multiplicative linear functionals on N, then y -h, maps r into A. If this map is one to one and onto, then we shall say that the maximal ideal space of N is F. Note that the maximal ideal space of L(G) is r (cf. [5, Chapter 1]), but L(G) is not unique in this respect. If we define (L(G))112 to be the intersection of all maximal ideals of M(G) containing L(G), then (L(G))112 is also an L-subalgebra of M(G) with maximal ideal space F (cf. [7, Lemma 1 and Theorem 1]). If e E M(G) and Aun E L(G) for some n, then tu E (L(G))112. Hewitt and Zuckerman have recently shown in [3], that for every nondiscrete l.c.a. group G, there is a singular measure Z E M(G) such that 2 E L(G). This shows that L(G) = (L(G))112 if G is nondiscrete. Our main theorem (Theorem 1) characterizes completely those L-subalgebras of M(G) that have maximal ideal space r. This result was conjectured in [7] and proved there for a special case. The missing ingredient for a proof in the general case is supplied by one of the results of [8].