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

Abstract

Abstract. For a locally compact group G and a group B of topological automorphisms containing the inner automorphisms of G and being relatively compact with respect to Birkhofftopology (that is G e [FIA]j~, B ~_ I (G)) the space Gz~ of B-orbits is a commutative hypergroup (= commutative convo in JEWETT's terminology) in a natural way as JEWETT has shown. Identifying the space of hypergroup characters of G z with E (G,B) (the extreme points of B-invariant positive definite continuous functions p with p (e) = l, endowed with the topology of compact convergence) we prove that E (G, B) is a hypergroup, the "hypergroup dual" of G z. Introduction The theory of topological hypergroups was initiated by DUXKL 16'.'

Fourier algebra of a hypergroup. I

“…[15] and Ross [25]). In particular, H satisfies condition (F) with M = 1. and ||/ix,y| < M-Thus, the hypergroup H a # satisfies condition (F).…”

Fourier-Stieltjes Transforms on Hypergroups

“…If the closure Β of Β in Aut(G) is compact then the B-orbit space G_, is a commutative hypergroup with natural operations, see [20, §1]. In [13] it is shown that Gis a hypergroup with respect to pointwise Β ^^ multiplication. That (G D ) Λ is equal to G_, follows by [12].…”

The structure of hypergroup measure algebras

“…Since the characters are real-valued, the Banach algebra they span is a B*-algebra, where involution is just complex conjugation, and the unit is J(0x) = 1. Note that Hartmann, Henrichs, and Lasser [6], show a more general result that (GB) A is a hypergroup, where G is a locally compact group and B is a group of automorphisms of G which contains the inner automorphisms, and is relatively compact with respect to the Birkhoff topology. Thus A M ( i 2 £ ) spans a commutative unital I?…”