Classification of one-dimensional hypergroups

Author:
Alan L. Schwartz

Journal:
Proc. Amer. Math. Soc. **103** (1988), 1073-1081

MSC:
Primary 43A10; Secondary 22A30, 33A45

DOI:
https://doi.org/10.1090/S0002-9939-1988-0954986-1

MathSciNet review:
954986

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Abstract: is a hypergroup if is a locally compact space and * is a binary operation with respect to which , the bounded Borel measures on , becomes a Banach algebra with a number of additional properties so that * generalizes group convolution.

The case when is a one-dimensional set, that is, a circle or an interval (possibly unbounded), includes a large number of examples: the classical group algebras and ( is the unit circle), the subalgebra of consisting of even measures, the subalgebra of consisting of rotation invariant measures, and the subalgebra of ( is the unit sphere in ) consisting of zonal measures. In addition to these, there are several continua of measure algebras unrelated to groups or geometry such as the measure algebras which arise in connection with Hankel transforms, ultraspherical and Jacobi polynomial series, and Sturm-Liouville expansions.

The main result of this article is a classification of the one-dimensional hypergroups. It is shown that if a certain amount of "regularity" for * is assumed, then every one-dimensional hypergroup is commutative and, up to a change of variables, must be one of the following types:

(i) the classical group algebra ,

(ii) the classical group algebra ,

(iii) , where or , is an identity for *, and if and only if .

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DOI:
https://doi.org/10.1090/S0002-9939-1988-0954986-1

Article copyright:
© Copyright 1988
American Mathematical Society