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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A characterization of nonatomic Hilbert algebras


Authors: Alessandro Figà-Talamanca and Giancarlo Mauceri
Journal: Proc. Amer. Math. Soc. 72 (1978), 468-472
MSC: Primary 46K15; Secondary 22D25
MathSciNet review: 509236
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Abstract: We say that a Hilbert algebra is atomic if its fulfillment is generated by its minimal projections. We prove that Hilbert algebra $ \mathcal{A}$ is not atomic if and only if there is an infinite group $ \mathcal{G}$ of unitary elements of the von Neumann algebra generated by $ \mathcal{A}$, and an element $ {\xi _0}$ of the fulfillment of $ \mathcal{A}$, which commutes with every element of $ \mathcal{G}$, and such that set $ \{ U{\xi _0}:U \in \mathcal{G}\} $ is orthonormal. This result is then applied to gain information on the Plancherel measure of certain unimodular groups.


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DOI: https://doi.org/10.1090/S0002-9939-1978-0509236-0
Article copyright: © Copyright 1978 American Mathematical Society