Abstract
For a locally compact groupG and a groupB of topological automorphisms containing the inner automorphisms ofG and being relatively compact with respect to Birkhoff topology (that isG∈[FIA] B,B \( \supseteq \) I(G)) the spaceG B of\(\bar B\)-orbits is a commutative hypergroup (=commutative convo inJewett's terminology) in a natural way asJewett has shown. Identifying the space of hypergroup characters ofG B withE(G, B) (the extreme points ofB-invariant positive definite continuous functionsp withp (e)=1, endowed with the topology of compact convergence) we prove thatE(G, B) is a hypergroup, the “hypergroup dual” ofG B.
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References
Chilana, A. K., andK. A. Ross: Spectral synthesis in hypergroups. Pacific J. Math.76, 313–328 (1978).
Dunkl, C. F.: The measure algebra of a locally compact hypergroup. Trans. Amer. Math. Soc.179, 331–348 (1973).
Fell, J. M. G.: Weak containment and induced representations of groups II. Trans. Amer. Math. Soc.110, 424–447 (1964).
Greenleaf, F. P.: Amenable actions of locally compact groups. J. Funct. Anal.4, 295–315 (1969).
Grosser, S., andM. Moskowitz: On central topological groups. Trans. Amer. Math. Soc.127, 317–340 (1969).
Grosser, S., andM. Moskowitz: Compactness conditions in topological groups. J. Reine Angew. Math.246, 1–40 (1971).
Henrichs, R. W.: Über Fortsetzung positiv definiter Funktionen. Math. Ann.232, 131–150 (1978).
Henrichs, R. W.: Weak Frobenius reciprocity and compactness conditions in topological groups. Pacific J. Math. (To appear.)
Jewett, R. I.: Spaces with an abstract convolution of measures. Adv. Math.18, 1–101 (1975).
Kaniuth, E.: Zur harmonischen Analyse klassenkompakter Gruppen. Math. Z.110, 297–305 (1969).
Kaniuth, E., andG. Schlichting: Zur harmonischen Analyse klassenkompakter Gruppen II. Inventiones math.10, 332–345 (1970).
Kaniuth, E., andD. Steiner: On complete regularity of group algebras. Math. Ann.204, 305–329 (1973).
Liukkonen, J., andR. Mosak: Harmonic analysis and centers of group algebras. Trans. Amer. Math. Soc.195, 147–163 (1974).
Michael, E.: Topologies on spaces of subsets. Trans. Amer. Math. Soc.71, 152–182 (1951).
Mosak, R.: TheL 1- andC *-algebras of[FIA] −B groups and their representations. Trans. Amer. Math. Soc.163, 277–310 (1972).
Ross, K. A.: Centers of hypergroups. Trans. Amer. Math. Soc.243, 251–269 (1978).
Spector, R.: Aperçu de la théorie des hypergroupes. Lecture Notes Math. 497 (Analyse Harmonique sur les Groupes de Lie, Sem. Nancy-Strasbourg 1973–1975). Berlin-Heidelberg-New York: Springer. 1975.
Steiner, D.: Zur harmonischen Analyse von klassenkompakten Gruppen. Dissertation. Technische Universität München. 1973.
Thoma, E.: Über unitäre Darstellungen abzählbarer, diskreter Gruppen. Math. Ann.153, 111–138 (1964).
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Hartmann, K., Henrichs, R.W. & Lasser, R. Duals of orbit spaces in groups with relatively compact inner automorphism groups are hypergroups. Monatshefte für Mathematik 88, 229–238 (1979). https://doi.org/10.1007/BF01295237
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DOI: https://doi.org/10.1007/BF01295237