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Measure algebras and functions of bounded variation on idempotent semigroups


Author: Stephen E. Newman
Journal: Trans. Amer. Math. Soc. 163 (1972), 189-205
MSC: Primary 43A10
DOI: https://doi.org/10.1090/S0002-9947-1972-0308686-4
MathSciNet review: 0308686
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Abstract: Our main result establishes an isomorphism between all functions on an idempotent semigroup $ S$ with identity, under the usual addition and multiplication, and all finitely additive measures on a certain Boolean algebra of subsets of $ S$, under the usual addition and a convolution type multiplication. Notions of a function of bounded variation on $ S$ and its variation norm are defined in such a way that the above isomorphism, restricted to the functions of bounded variation, is an isometry onto the set of all bounded measures. Our notion of a function of bounded variation is equivalent to the classical notion in case $ S$ is the unit interval and the ``product'' of two numbers in $ S$ is their maximum.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0308686-4
Keywords: Bounded variation, convolution measure algebra
Article copyright: © Copyright 1972 American Mathematical Society

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