Properties of subhypergroups
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 the support of the Plancherel measure is the whole dual space \(\widehat K\) of \(K\). The growth condition \((F)\) for a subhypergroup is also related to one of the above mentioned translation properties.
scientific article; zbMATH DE number 1156729