A characterization of nonatomic Hilbert algebras

Figà-Talamanca, Alessandro; Mauceri, Giancarlo

doi:10.1090/S0002-9939-1978-0509236-0

A characterization of nonatomic Hilbert algebras
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by Alessandro Figà-Talamanca and Giancarlo Mauceri PDF

Proc. Amer. Math. Soc. 72 (1978), 468-472 Request permission

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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