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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.
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