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

DOI:
https://doi.org/10.1090/S0002-9939-1978-0509236-0

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 is not atomic if and only if there is an infinite group of unitary elements of the von Neumann algebra generated by , and an element of the fulfillment of , which commutes with every element of , and such that set is orthonormal. This result is then applied to gain information on the Plancherel measure of certain unimodular groups.

**[1]**William B. Arveson,*A theorem on the action of abelian unitary groups*, Pacific J. Math.**16**(1966), 205–212. MR**0188809****[2]**Carlo Cecchini and Alessandro Figà-Talamanca,*Projections of uniqueness for 𝐿^{𝑝}(𝐺)*, Pacific J. Math.**51**(1974), 37–47. MR**0394043****[3]**Alessandro Figà-Talamanca,*On the action of unitary groups on a Hilbert space*, Symposia Mathematica, Vol. XXII (Convegno sull’Analisi Armonica e Spazi di Funzioni su Gruppi Localmente Compatti, INDAM, Rome, 1976) Academic Press, London, 1977, pp. 315–319. MR**0498964****[4]**Giancarlo Mauceri,*Square integrable representations and the Fourier algebra of a unimodular group*, Pacific J. Math.**73**(1977), no. 1, 143–154. MR**0486289****[5]**Marc A. Rieffel,*Square-integrable representations of Hilbert algebras*, J. Functional Analysis**3**(1969), 265–300. MR**0244780**

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DOI:
https://doi.org/10.1090/S0002-9939-1978-0509236-0

Article copyright:
© Copyright 1978
American Mathematical Society