Vol. 69, No. 1, 1977

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Measure algebras of semilattices with finite breadth

Jimmie Don Lawson, John Robie Liukkonen and Michael William Mislove

Vol. 69 (1977), No. 1, 125–139

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.

Mathematical Subject Classification 2000
Primary: 43A10
Secondary: 22A20
Received: 24 June 1976
Published: 1 March 1977
Jimmie Don Lawson
John Robie Liukkonen
Michael William Mislove