Measure algebras of semilattices with finite breadth
Jimmie Lawson, John R. Liukkonen, Michael W. Mislove · Pacific Journal of Mathematics · 1977
The main result of this paper is that if S is a locally compact semilattice of finite breadth, then every complex homomorphism of the measure algebra M(S) is given by integration over a Borel filter (subsemilattice whose complement is an ideal), and that consequently M(S) is a P-algebra in the sense of S. E. Newman.More generally it is shown that if S is a locally compact Lawson semilattice which has the property that every bounded regular Borel measure is concentrated on a Borel set which is the countable union of compact finite breadth subsemilattices, then M(S) is a P-algebra.Furthermore, complete descriptions of the maximal ideal space of M(S) and the structure semigroup of M(S) are given in terms of S, and the idempotent and invertible measures in M(S) are identified.