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Harmonic analysis on compact commutative hypergroups: The role of the maximum subgroup

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Abstract

The article is written in two parts: Part I develops basic results within a new axiomatic development introduced by one of the authors in a previous work. It allows for a larger category of measure algebras to be included than did the former axioms (DJS-hypergroups). While new and independent proofs are given, this part of the article is expository.

Part II is the core of the paper: its focus is the study of the spectral properties of compact commutative hypergroup measure algebras. These share commonL 1 andL 2 theories; but the spectra of the measure algebras are much more diverse, and we pursue a classification based on the maximal ideal spaces of these algebras. Important features depend on the existence and structure of a non-trivial maximum subgroup. We investigate symmetry, idempotents, and Sidon sets with respect to these considerations. The results generalize earlier studies of K. Ross, Y. Kanjin, and the authors.

In the context of theL 1 theory, we use Dixmier's symbolic calculus to construct special approximate identities. Concrete examples are provided. The two parts may be read independently of each other, but the whole paper is self-contained.

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Authors and Affiliations

  1. Département de Mathématique Institut de Recherche Mathématique Avancée, Université Louis Pasteur, 67100, Strasbourg, France

    Oliver Gebuhrer

  2. Department of Mathematics and Computer Science, University of Missouri-St. Louis, 63121, St. Louis, MO, USA

    Alan L. Schwartz

Authors
  1. Oliver Gebuhrer
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  2. Alan L. Schwartz
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Correspondence to Oliver Gebuhrer.

Additional information

The second author was supported by the National Science Foundation, Grant No. DMS 9706965.

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Gebuhrer, O., Schwartz, A.L. Harmonic analysis on compact commutative hypergroups: The role of the maximum subgroup. J. Anal. Math. 82, 175–206 (2000). https://doi.org/10.1007/BF02791226

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  • DOI: https://doi.org/10.1007/BF02791226

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