Measure algebras on idempotent semigroups

Measure algebras of semilattices with finite breadth

“…3* P-algebras* In [9], Newman defines a P-algebra to be a semisimple convolution measure algebra M such that whenever μ is positive element of M and h e ΔM, the h(μ) ^ 0. He shows (Theorem 1 of [9]) that these are precisely the semisimple convolution measure algebras whose structure semigroups are in fact semilattices.…”

“…He shows (Theorem 1 of [9]) that these are precisely the semisimple convolution measure algebras whose structure semigroups are in fact semilattices. It is easily checked that the equivalence (1) <==> (2) <=> (3) of Theorem 1 of [9] is true without assuming semisimplicity, so we shall define a Palgebra to be a convolution measure algebra M such that h{μ) ^ 0 for all h e ΔM and all μeM such that μ^O.…”

“…3.15], and consequently, the structure semigroup of M(S) in the sense of Taylor [10] is itself a semilattice [9,Th. 3].…”

“…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.…”

Measure Algebras and Functions of Bounded Variation on Idempotent Semigroups

“…We show that the point evaluation maps are complex homomorphisms of BV(T), and, as such, form an idempotent subsemigroup of the maximal ideal space S of BV(T). We use results obtained in [5] to exhibit an idempotent semigroup T such that the structure semigroup S of the algebra BV(T) is not idempotent. We conclude that S is not idempotent, even though T is embedded in § as an idempotent subsemigroup which separates points in BV(T).…”