Vol. 31, No. 1, 1969

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Measure algebras on idempotent semigroups

Stephen E. Newman

Vol. 31 (1969), No. 1, 161–169

Taylor has shown that for every commutative convolution measure algebra M there is a compact topological semigroup S, called the structure semigroup of M, and an embedding μ μS of M into M(S) such that every complex homomorphism of M has the form hf(μ) = sfdμs for some semicharacler f on S.

This paper deals with commutative convolution measure algebras whose structure semigroups are idempotent. The measure algebra on the interval [0,1], where the interval is given the semigroup operation of maximum multiplication, is an algebra of this type. These algebras are studied in this general setting in the hope of shedding new light on the known theory of measure algebras on locally compact idempotent semigroups and in the hope of extending attempts to classify a convolution measure algebra in terms of the algebraic nature of its structure semigroup.

An example is given of a measure algebra on a compact idempotent semigroup whose structure semigroup is not idempotent.

Mathematical Subject Classification
Primary: 46.80
Received: 30 May 1968
Published: 1 October 1969
Stephen E. Newman