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Isomorphisms of hypergroups

Part of: Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

Walter R. Bloom  and
Martin E. Walter
Walter R. Bloom
Affiliation:
School of Mathematical and Physical SciencesMurdoch UniversityPerth Western Australia 6150, Australia
Martin E. Walter
Affiliation:
Department of MathematicsUniversity of ColoradoCampus Box 426, Boulder Colorado 80309, USA
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Abstract

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Let K1, K2 be locally compact hypergroups. It is shown that every isometric isomorphism between their measure algebras restricts to an isometric isomorphism between their L1-algebras. This result is used to relate isometries of the measure algebras to homeomorphisms of the underlying locally compact spaces.

MSC classification

Secondary: 43A62: Hypergroups 43A10: Measure algebras on groups, semigroups, etc. 43A20: $L^1$-algebras on groups, semigroups, etc. 43A22: Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 52 , Issue 3 , June 1992 , pp. 383 - 400
Copyright
Copyright © Australian Mathematical Society 1992

References

Arendt, Wolfgang and de Cannière, Jean, ‘Order isomorphisms of Fourier algebras’, J. Funct. Anal. 50 (1983), 1–7.CrossRefGoogle Scholar
Bloom, Walter R., ‘Idempotent measures on commutative hypergroups’, Probability Measures on Groups VIII (Proc. Conf., Oberwolfach Math. Res. Inst., Oberwolfach, 1985), Lecture Notes in Math. 1210, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo 1986, 13–23.CrossRefGoogle Scholar
Bloom, Walter R. and Heyer, Herbert, ‘Convergence of convolution products of probability measures on hypergroups’, Rend. Mat. Ser. VII 3(1982), 547–563.Google Scholar
Bloom, Walter R. and Heyer, Herbert, ‘Characterisation of potential kernels of transient convolution semigroups on a commutative hypergroup’, Probability Measures on Groups IX (Proc. Conf., Oberwolfach Math. Res. Inst., Oberwolfach 1988), Lecture Notes in Math. 1379, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1989, pp. 21–35.CrossRefGoogle Scholar
Dunford, Nelson and Schwartz, Jacob T., Linear Operators, Part 1: General Theory, Interscience Publishers, New York, London, 1958.Google Scholar
Dunkl, Charles F., ‘The measure algebra of a locally compact hypergroup’, Trans. Amer. Math. Soc. 179 (1973), 331–348.CrossRefGoogle Scholar
Granirer, E. E. and Leinert, M., ‘On some topologies which coincide on the unit sphere of the Fourier-Stieltjes algebra B(G) and of the measure algebra M(G),’ Rocky Mountain J. Math. 11 (1981), 459–472.CrossRefGoogle Scholar
Jewett, Robert I., ‘Spaces with an abstract convolution of measures’, Adv. in Math. 18 (1975), 1–101.CrossRefGoogle Scholar
Johnson, B. E., ‘Isometric isomorphisms of measure algebras’, Proc. Amer. Math. Soc. 15 (1964), 186–188.CrossRefGoogle Scholar
Kawada, Yukiyosi, ‘On the group ring of a topological group’, Math. Jap. 1 (1948), 1–5.Google Scholar
Ross, Kenneth A., ‘Hypergroups and centers of measure algebras’, Symp. Math. 22 (1977), 189–203.Google Scholar
Spector, R., ‘Aperçu de la théorie des hypergroupes’, Analyse Harmonique sur les Groupes de Lie (Séminaire Nancy-Strasbourg, 1973–1975), Lecture Notes in Math. 497, Springer-Verlag, Berlin, Heidelberg, New York, 1975, 643–673.CrossRefGoogle Scholar
Spector, R., ‘Mesures invariantes sur les hypergroupes’, Trans. Amer. Math. Soc. 239 (1978), 147–165.CrossRefGoogle Scholar
Strichartz, Robert S., ‘Isometric isomorphisms of measure algebras,’ Pacific J. Math. 15 (1965), 315–317.CrossRefGoogle Scholar
Walter, Martin E., ‘W*-algebras and nonabelian harmonic analysis,’ J. Funct. Acal. 11 (1972), 17–38.CrossRefGoogle Scholar
Wendel, J. G., ‘On isometric isomorphisms of group algebras’, J. Math. 1 (1951), 305–311.Google Scholar
Wendel, J. G., ‘Left centralizers and isomorphisms of group algebras’, Pacific J. Math. 2 (1952), 251–261.CrossRefGoogle Scholar