A theorem on the action of abelian unitary groups

William Arveson

Vol. 16 (1966), No. 2, 205–212

DOI: 10.2140/pjm.1966.16.205

Abstract

Given an abelian unitary group G acting on the Hilbert space ℋ, let 𝒜 be the C^∗-algebra generated by G and let σ(𝒜) denote the maximal ideal space of this algebra. There is a natural injection α of σ(𝒜) into the compact character group Γ of the discrete group G. What conditions on G wilI ensure that α be a topological homeomorphism of σ(𝒜) on Γ?

The action of G is said to be nondegenerate if, for every finite subset F of G, there exists a vector ξ≠0 in ℋ such that Uξ ⊥ V ξ for every pair U, V of distinct elements of F. Theorem 1 contains the following answer to our question; in order that α map σ(𝒜) onto Γ, it is necessary and sufficient that the action of G be nondegenerate.

Mathematical Subject Classification

Primary: 46.65

Secondary: 47.90

Milestones

Received: 8 October 1964

Published: 1 February 1966