Substitution of open subhypergroups
Hypergroups generalize locally compact groups. If the Banach space of all bounded Radon measures on a locally compact space \(X\) carries a convolution structure satisfying some properties the space \(X\) is called hypergroup. Details and many examples are contained in the recent monograph of Bloom and Heyer. In the present paper the author generalizes the construction of so-called joins. The join \(L \vee K\) of a compact hypergroup \(L\) and a discrete hypergroup \(K\) is formed by replacing the unit element of \(K\) by the hypergroup \(L\). This method is extended as follows: If \(H\) is an open subhypergroup of a hypergroup \(K\), and \(\pi\) is an open and proper hypergroup homomorphism from a further hypergroup \(L\) onto \(H\), then on the disjoint union of \(K - H\) and \(L\) is defined a natural hypergroup structure. The author calls this procedure substitution of \(H\) by \(L\) in \(K\). After the construction of hypergroups formed by substitution positive definite functions and irreducible representations are studied. Further the dual space is investigated in the commutative case. Finally the author describes some classes of hypergroups via substitution.
scientific article; zbMATH DE number 569385