Measure algebras and functions of bounded variation on idempotent semigroups
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- by Stephen E. Newman
- Trans. Amer. Math. Soc. 163 (1972), 189-205
- DOI: https://doi.org/10.1090/S0002-9947-1972-0308686-4
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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.References
- Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053
- Richard B. Darst, A decomposition of finitely additive set functions, J. Reine Angew. Math. 210 (1962), 31–37. MR 137808, DOI 10.1515/crll.1962.210.31
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- Shizuo Kakutani, Concrete representation of abstract $(L)$-spaces and the mean ergodic theorem, Ann. of Math. (2) 42 (1941), 523–537. MR 4095, DOI 10.2307/1968915
- Stephen E. Newman, Measure algebras on idempotent semigroups, Pacific J. Math. 31 (1969), 161–169. MR 275188
- Marc A. Rieffel, A characterization of commutative group algebras and measure algebras, Trans. Amer. Math. Soc. 116 (1965), 32–65. MR 198141, DOI 10.1090/S0002-9947-1965-0198141-3
- Gian-Carlo Rota, On the foundations of combinatorial theory. I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340–368 (1964). MR 174487, DOI 10.1007/BF00531932
- Joseph L. Taylor, The structure of convolution measure algebras, Trans. Amer. Math. Soc. 119 (1965), 150–166. MR 185465, DOI 10.1090/S0002-9947-1965-0185465-9
Bibliographic Information
- © Copyright 1972 American Mathematical Society
- 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