Duals of orbit spaces in groups with relatively compact inner automorphism groups are hypergroups

Abstract

For a locally compact groupG and a groupB of topological automorphisms containing the inner automorphisms ofG and being relatively compact with respect to Birkhoff topology (that isG∈[FIA]B,B\( \supseteq \)I(G)) the spaceGB of\(\bar B\)-orbits is a commutative hypergroup (=commutative convo inJewett's terminology) in a natural way asJewett has shown. Identifying the space of hypergroup characters ofGB withE(G, B) (the extreme points ofB-invariant positive definite continuous functionsp withp (e)=1, endowed with the topology of compact convergence) we prove thatE(G, B) is a hypergroup, the “hypergroup dual” ofGB.