A theorem on the action of abelian unitary groups

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 will 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 ξ Vξ 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. © 1966 by Pacific Journal of Mathematics.