Properties of subhypergroups
type: Item
instance of: scholarly article
title: Properties of subhypergroups
zbMATH Open document ID: 0897.43006
DOI: 10.1007/PL00005953
author: Michael Voit
published in: Semigroup Forum
publication date: 1998-10-06T00:00:00Z
review text: In various respects, subhypergroups of hypergroups in general behave much less convenient than subgroups of locally compact groups do. It is some of these phenomena in hypergroup theory this interesting paper deals with. Let \(K\) be a hypergroup and \(H\) a subhypergroup of \(K\). The author first studies the question of when several \(H\)-translation invariance conditions on continuous functions force these functions to be constant on cosets of \(H\). Now suppose that \(K\) has a left Haar measure (the existence of Haar measure on arbitrary locally compact hypergroups still appears to be an open problem). Then every subhypergroup \(H\) of \(K\) also has a left Haar measure (Proposition 2.1). If, in addition \(H\) is normal and \(K/H\) is a hypergroup, then also \(K/H\) admits a left Haar measure (Proposition 2.9), and Haar measures on \(K,H\) and \(K/H\) are related by a Weil's formula, analogous to that in the group case. That in this situation Weil's formula holds is due to the fact that for every \(f\in C_c(H)\) the function \(x\to\int_H f(x*h)dh\) is constant on cosets of \(H\). This latter property actually turns out to be equivalent to \(K/H\) being a hypergroup (Theorems 2.3 and 2.6). Moreover, the condition that \(K/H\) be a hypergroup is also equivalent so that the trivial representation of \(H\) is inducible to \(K\) and the support of the resulting quasiregular representation consists only of irreducible representations of \(K\) which are constant on cosets of \(H\). The author also investigates certain growth conditions for Haar measure, like exponential growth and a Følner type condition \((F)\) which in the group case (but not for hypergroups) is equivalent to amenability. As a sample of the results obtained, we mention Theorem 3.4 which concerns one of the main problems in the duality theory of commutative hypergroups: If the commutative hypergroup \(K\) is not of exponential growth (thus, in particular, if \(K\) is of subexponential growth), then t…
Mathematics Subject Classification ID: 43A62
Mathematics Subject Classification ID: 20N20
zbMATH DE Number: 1156729
zbMATH Keywords: normal subhypergroup
zbMATH Keywords: quotient hypergroup
zbMATH Keywords: hypergroups
zbMATH Keywords: Haar measure
zbMATH Keywords: Weil's formula
zbMATH Keywords: growth conditions
zbMATH Keywords: Følner type condition
zbMATH Keywords: amenability
reviewed by: Eberhard Kaniuth
MaRDI profile type: Publication
full work available at URL: pl00005953
OpenAlex ID: W2013013196
description: scientific article; zbMATH DE number 1156729