Abstract
Let \({\mathcal{A}}\) be an Abelian unital C *-algebra and let \(\hat {\mathcal{A}}\) denote its Gelfand spectrum. We give some necessary and sufficient conditions for a nondegenerate representation of \({\mathcal{A}}\) to be unitarily equivalent to a representation in which the elements of \({\mathcal{A}}\) act multiplicatively, by their Gelfand transforms, on a space L 2(\(\hat {\mathcal{A}}\),μ), where μ is a positive measure on the Baire sets of \(\hat {\mathcal{A}}\). We also compare these conditions with the multiplicity-free property of a representation.
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Cavallaro, S. Spectral Properties of Abelian C*-Algebras. Algebras and Representation Theory 3, 175–186 (2000). https://doi.org/10.1023/A:1009941513419
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DOI: https://doi.org/10.1023/A:1009941513419