The measure algebra of a locally compact semigroup

Measure algebras on idempotent semigroups

“…Our goal in this section is to show that the measure algebra on a finite product of totally ordered, locally compact, idempotent semigroups is a P-algebra. This result follows trivially from a theorem of Baartz (Theorem 3.5, [1]); however, we shall give an independent development using Theorem 2. We will need the three lemmas that follow.…”

“…Examples of P-algebras are the measure algebra M(T), under convolution, of the compact semigroup T - [a,b] with multiplication x-y = max {x, y} [3], and more generally, the measure algebra M(T) of a finite product T of locally compact, totally ordered spaces with co-ordinate wise maximum multiplication [1]. In both examples, each complex homomorphism of M(T) has the form…”

“…for some subsemigroup A of T whose complement T\A is a (prime) ideal of T (Definition 1.5, [1]). Consequently, M(T) is a P-algebra.…”

Measure algebras of semilattices with finite breadth

“…In earlier work Baartz and Newman have shown that if S is the finite product of totally ordered locally compact semilattices, then every complex homomorphism is given by integration over a Borel subsemilattice whose complement is an ideal [1,Th. 3.15], and consequently, the structure semigroup of M(S) in the sense of Taylor [10] is itself a semilattice [9,Th.…”

“…lower bound. The reader should note that this convention differs from that in [1,9], in which the product of two elements in a semilattice is viewed as their least upper bound. A semilattice on a Hausdorff space S is a topological (semitopological) semilattice if the multiplication function which sends (x, y) to xy from S x S to S is jointly (separately) continuous.…”