Duality theory for locally compact groups with precompact conjugacy classes. I. The character space

Abstract

Let G be a locally compact group, and let X ( G ) \mathcal {X}(G) consist of the nonzero extreme points of the set of continuous, G-invariant, positive definite functions f on G such that f ( e ) ≤ 1 f(e) \leq 1 . X ( G ) \mathcal {X}(G) is called the character space, and is given the topology of uniform convergence on compacta. The purpose of the present paper is to extend the main results from the duality theory of abelian groups and [Z] groups to the class of [ F C ] − {[FC]^ - } groups (i.e., groups with precompact conjugacy classes), letting X ( G ) \mathcal {X}(G) play the role of the character group in the abelian theory. Some of our theorems are only proved for the class [ F D ] − ( ⊂ [ F C ] − ) {[FD]^ - }\;( \subset {[FC]^ - }) . If G ∈ [ F C ] − G \in {[FC]^ - } then X ( G ) ≈ X ( H ) \mathcal {X}(G) \approx \mathcal {X}(H) where H is a certain [ F I A ] − {[FIA]^ - } quotient group. Hence there is no loss of generality to study character spaces of [ F I A ] − {[FIA]^ - } groups.