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Hypergroups with invariant metric

Author(s): Michael Voit
Journal: Proc. Amer. Math. Soc. 126 (1998), 2635-2640.
MSC (1991): Primary 43A62; Secondary 20N20, 54E35
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Abstract: The purpose of this note is to extend the following classical result from groups to hypergroups in the sense of C.F. Dunkl, R.I. Jewett, and R. Spector: If a hypergroup has a countable neighborhood base of its identity, then admits a left- or a right-invariant metric. Moreover, it admits an invariant metric if and only if there exists a countable conjugation-invariant neighborhood base of the identity.

References:

1.: G. Birkhoff: A note on topological groups. Compos. Math. 3, 427 - 430 (1936).
2.: W.R. Bloom, H. Heyer: Harmonic Analysis of Probability Measures on Hypergroups. De Gruyter (1995). MR 96a:43001
3.: E. Hewitt, K.A. Ross: Abstract Harmonic Analysis I. Springer-Verlag: Berlin - New York (1979). MR 81k:43001
4.: R.I. Jewett: Spaces with an abstract convolution of measures. Adv. Math. 18, 1 - 101 (1975). MR 52:14840
5.: S. Kakutani: Über die Metrisation der topologischen Gruppen. Proc. Imp. Acad. Tokyo 12, 82 - 84 (1936).
6.: Hm. Zeuner: One-dimensional hypergroups. Adv. Math. 76, 1 - 18 (1989). MR 90i:43002

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Additional Information:

Michael Voit
Affiliation: Mathematisches Institut, Universität Tübingen, 72076 Tübingen, Germany - Department of Mathematics, University of Virginia, Kerchof Hall, Charlottesville, Virginia 22903-3199
Email: voit@uni-tuebingen.de

DOI: 10.1090/S0002-9939-98-04366-4
PII: S 0002-9939(98)04366-4
Keywords: Hypergroups, invariant metric
Received by editor(s): December 26, 1996
Received by editor(s) in revised form: January 27, 1997
Communicated by: Palle E. T. Jorgensen
Copyright of article: Copyright 1998, American Mathematical Society


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