Abstract
We give a set of axioms for the notion of locally compact hypergroupoids, as an extension of both groupoids and hypergroups, and study their basic properties. We show that, adding a natural condition on the continuity of support, one of the axioms assumed by Renault on the left Haar system automatically follows. We show that an irreducible representation of a compact hypergroupoid is fiberwise finite dimensional.
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References
M. Amini, Tannaka-Krein duality for compact groupoids I, representation theory, Adv. Math., 214(1) (2007), 78–91.
W. R. Bloom and H. Heyer, Harmonic analysis of probability measures on hypergroups, De Gruyter, Berlin (1995).
J. B. Conway, A course in functional analysis, 2nd Edition, Springer, New York (1990).
C. F. Dunkl and D. E. Ramirez, A family of countably compact P*-hypergroups, Trans. Amer. Math. Soc., 202 (1975), 339–356.
G. B. Folland, A course in abstract harmonic analysis, CRC Press, London (1995).
R. I. Jewett, Spaces with an abstract convolution of measures, Adv. Math., 18 (1975), 1–101.
A. A. Kalyuzhnyi, G. B. Podkolzin, and Yu A. Chapovski, Harmonic analysis on a locally compact hypergroup, Methods of Functional Analysis and Topology, 16 (2010), 304–332.
E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc., 71 (1951), 152–182.
A. L. T. Paterson, Groupoids, inverse semigroups, and their operator algebras, Birkhauser, New York (1999).
A. Ramsay, Topologies for measured groupoids, J. Func. Anal., 47 (1982), 314–343.
J. Renault, A groupoid approach to C*-algebras, Lecture Notes in Mathematics, 793, Springer, Berlin, (1980).
J. Renault, Induced representations and hypergroupoids, Symmetry, Integrability and Geometry: Methods and Applications, 10 (2014), 1–18.
L. Székelyhidi, S. M. Tabatabaie, and B. H. Sadathoseyni, Convolution operators on measure algebras of KPC-hypergroups, Advances in Pure and Applied Math., 10(1) (2019), 1–6.
R. C. Vrem, Harmonic analysis on compact hypergroups, Pac. J. Math., 85(1) (1979), 239–251.
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We would like to thank the referee(s) of this paper for a careful reading and helpful suggestions.
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Tabatabaie, S.M., Pourgholamhossein, M., Amini, M. et al. Locally Compact Hypergroupoids. Indian J Pure Appl Math 51, 39–54 (2020). https://doi.org/10.1007/s13226-020-0383-y
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DOI: https://doi.org/10.1007/s13226-020-0383-y