Harmonic analysis of probability measures on hypergroups
type: Item
instance of: scholarly article
title: Harmonic analysis of probability measures on hypergroups
zbMATH Open document ID: 0828.43005
author: Walter R. Bloom
author: Herbert Heyer
published in: De Gruyter Studies in Mathematics
publication date: 1992-09-18T00:00:00Z
review text: Hypergroups generalize locally compact groups. They appear when the Banach space of all bounded Radon measures on a locally compact space carries a convolution having all properties of a group convolution apart from the fact that the convolution of two point measures is a probability measure with compact support and not necessarily a point measure. Discrete hypergroups are closely related to Schur rings, Hecke algebras, and association schemes. While these objects are mainly studied under algebraic and number theoretic aspects, hypergroups were introduced by \textit{C. F. Dunkl} [Trans. Am. Math. Soc. 179, 331-348 (1978; Zbl 0241.43003)], \textit{R. Jewett} [Adv. Math. 18, 1-101 (1975; Zbl 0325.42017)], and \textit{R. Spector} [Trans. Am. Math. Soc. 239, 147-166 (1978; Zbl 0428.43001)]. The intention was to unify harmonic analysis on duals of compact groups, double coset spaces \(G//H\) \((H\) a compact subgroup of a locally compact group \(G)\), and commutative convolution algebras associated with product linearization formulas of special functions. Recently it turned out that quantum Gelfand pairs as well as bifinite irreducible bimodules of \(\text{II}_1\)-factors also lead to hypergroups. Harmonic analysis and probability theory on commutative hypergroups are well developed meanwhile where many results from group theory remain valid. Besides monographs from Russian mathematicians on hypercomplex systems (which admit more complex axioms than hypergroups), the research monograph under review is the first one which deals with hypergroups from the point of view of harmonic analysis and probability theory. The monograph of Bloom and Heyer is organized as follows: The introductory chapter discusses the foundations of hypergroups, their measure algebras as well as basic constructions which lead from other mathematical objects (like groups) to hypergroups. The highlight of chapter 1 is the existence and uniqueness of Haar measures due to Jewett and Spector. Chapter 2 is…
Mathematics Subject Classification ID: 43A62
Mathematics Subject Classification ID: 43-02
Mathematics Subject Classification ID: 28A33
Mathematics Subject Classification ID: 20N20
Mathematics Subject Classification ID: 33C65
Mathematics Subject Classification ID: 60B99
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 31C99
zbMATH DE Number: 53031
zbMATH Keywords: hypergroups
zbMATH Keywords: convolution
zbMATH Keywords: probability measures
zbMATH Keywords: harmonic analysis
reviewed by: Michael Voit
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 53031