The structure of certain measure algebras
Kenneth A. Ross · Pacific Journal of Mathematics · 1961
Introduction* In their paper [3], Hewitt and Zuckerman study the measure algebra ^(G) where G is a topological semigroup of the following type: G is a linearly ordered set topologized with the order topology, is compact in this topology, and multiplication is defined by xy = max (x, y).In this study, we will suppose that G has the above properties except that compactness will be replaced by local compactness.(See § 8.5 [3]).As the reader will readily observe, we are heavily indebted to Hewitt and Zuckerman for their initial study of these measure algebras.For completeness, we have listed, without proof, a few of their results; they are stated in their paper for compact semigroups but the proofs easily carry over to locally compact semigroups.In §2 we study G and G o .The characterization of the Gel'fand topology on G is somewhat simpler than that of Theorem 5.5 [3].The major result of this study is Theorem 3.4, stating that every closed ideal in ^?f(G) is the intersection of maximal ideals; i.e., spectral synthesis holds for ^£(G).Malliavin [7] has recently shown that spectral synthesis fails for ^£ (G) when G is a non-compact locally compact commutative group. 1 Theorem 3.4 shows that this result cannot be generalized to locally compact commutative semigroups.In § 4, a generalization of Theorem 6.7 [3] is indicated; see Theorem 4.5.This is used to obtain additional facts about ^f(G) ( §5).In 5.8 we show that our theory is not a special case of the theory of function algebras.1. Preliminaries-1.1.We will be concerned with linearly ordered sets; i.e. sets ordered by transitive, irreflexive relations y] = {ze X: x ^ z g y}.The half-open intervals [x, y[ and ]x, y] are defined analogously.We also define ] -oo, x[ = {ze X: z < x) and ] -oo, x] = {ze X: z < x) with analogous definitions for [x, oo[,]x, oo[, and ] -oo, oo[.The symbols -oo and oo will never denote actual elements of X.The order topology for X is the topology having the family {] -c ° f x[} xex U