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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
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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  • [1] G. Birkhoff, Lattice theory, 3rd ed., Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, R. I., 1967. MR 37 #2638. MR 0227053 (37:2638)
  • [2] R. B. Darst, A decomposition of finitely additive set functions, J. Reine Angew. Math. 210 (1962), 31-37. MR 25 #1257. MR 0137808 (25:1257)
  • [3] N. Dunford and J. T. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 22 #8302. MR 0117523 (22:8302)
  • [4] S. Kakutani, Concrete representation of abstract $ (L)$-spaces and the mean ergodic theorem, Ann. of Math. (2) 42 (1941), 523-537. MR 2, 318. MR 0004095 (2:318d)
  • [5] S. E. Newman, Measure algebras on idempotent semigroups, Pacific J. Math. 31 (1969), 161-169. MR 0275188 (43:945)
  • [6] M. A. Rieffel, A characterization of commutative group algebras and measure algebras, Trans. Amer. Math. Soc. 116 (1965), 32-65. MR 33 #6300. MR 0198141 (33:6300)
  • [7] G.-C. Rota, On the foundations of combinatorial theory. I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340-368. MR 30 #4688. MR 0174487 (30:4688)
  • [8] J. L. Taylor, The structure of convolution measure algebras, Trans. Amer. Math. Soc. 119 (1965), 150-166. MR 32 #2932. MR 0185465 (32:2932)

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Keywords: Bounded variation, convolution measure algebra
Article copyright: © Copyright 1972 American Mathematical Society

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