Abstract
It is shown that Gelfand transforms of elements \({f\in L^{\infty} (\mu)}\) are almost constant at almost every fiber \({\Pi^{-1}(\{x\})}\) of the spectrum of L ∞(μ) in the following sense: for each \({f\in L^{\infty} (\mu)}\) there is an open dense subset U = U(f) of this spectrum having full measure and such that the Gelfand transform of f is constant on the intersection \({\Pi^{-1}(\{x\})\cap U}\). As an application a new approach to disintegration of measures is presented, allowing one to drop the usually taken separability assumption.
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Bourbaki N.: Élements de mathématique. Livre VI, Intégration. Hermann, Paris (1959)
Dixmier J.: Sur certain espaces considérés par M. H. Stone. Summa Brasil. Math. 2, 151–182 (1951)
Gamelin T.W.: Uniform Algebras. Prentice Hall, Inc.,, Englewood Clifs, N.J (1969)
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The authors wish to thank Professors Christian Berg, Jan Stochel and Edward Tutaj for valuable remarks.
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Marek Kosiek was supported by Ministry of Science and Higher Education Grant NN201 546438.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Kosiek, M., Rudol, K. Fibers of the L ∞ algebra and disintegration of measures. Arch. Math. 97, 559–567 (2011). https://doi.org/10.1007/s00013-011-0332-4
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DOI: https://doi.org/10.1007/s00013-011-0332-4