A characterization of nonatomic Hilbert algebras

Abstract

We say that a Hilbert algebra is atomic if its fulfillment is generated by its minimal projections. We prove that Hilbert algebra A \mathcal {A} is not atomic if and only if there is an infinite group G \mathcal {G} of unitary elements of the von Neumann algebra generated by A \mathcal {A} , and an element ξ 0 {\xi _0} of the fulfillment of A \mathcal {A} , which commutes with every element of G \mathcal {G} , and such that set { U ξ 0 : U ∈ G } \{ 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.