Integral operators and the compactness of induced representations

Busby, Robert C.; Schochetman, Irwin; Smith, Harvey A.

doi:10.1090/S0002-9947-1972-0295099-7

Integral operators and the compactness of induced representations
HTML articles powered by AMS MathViewer

by Robert C. Busby, Irwin Schochetman and Harvey A. Smith PDF

Trans. Amer. Math. Soc. 164 (1972), 461-477 Request permission

Abstract:

Integral operators are investigated which are compositions of multiplication by bounded vector valued functions and convolutions with vector valued functions. All of the functions are based on a fixed locally compact group. Conditions are given under which certain of these operators are compact. As an application of these conditions we consider induced representations of twisted group algebras (these are generalizations of representations of groups induced from closed normal sub-groups in the sense of Mackey and include these as special cases) and we give necessary and sufficient conditions for these representations to be compact (that is, to consist entirely of compact operators).

References

Louis Auslander and Calvin C. Moore, Unitary representations of solvable Lie groups, Mem. Amer. Math. Soc. 62 (1966), 199. MR 207910
Robert J. Blattner, Group extension representations and the structure space, Pacific J. Math. 15 (1965), 1101–1113. MR 188346, DOI 10.2140/pjm.1965.15.1101
Nicolas Bourbaki, Éléments de mathématique. Fasc. II. Livre III: Topologie générale. Chapitre 1: Structures topologiques. Chapitre 2: Structures uniformes, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1142, Hermann, Paris, 1965 (French). Quatrième édition. MR 0244924
Robert C. Busby and Harvey A. Smith, Representations of twisted group algebras, Trans. Amer. Math. Soc. 149 (1970), 503–537. MR 264418, DOI 10.1090/S0002-9947-1970-0264418-8
Jacques Dixmier, Les $C^{\ast }$-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars & Cie, Éditeur-Imprimeur, Paris, 1964 (French). MR 0171173
Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
J. M. G. Fell, The dual spaces of $C^{\ast }$-algebras, Trans. Amer. Math. Soc. 94 (1960), 365–403. MR 146681, DOI 10.1090/S0002-9947-1960-0146681-0
J. M. G. Fell, Weak containment and induced representations of groups, Canadian J. Math. 14 (1962), 237–268. MR 150241, DOI 10.4153/CJM-1962-016-6
J. M. G. Fell, A new proof that nilpotent groups are CCR, Proc. Amer. Math. Soc. 13 (1962), 93–99. MR 133404, DOI 10.1090/S0002-9939-1962-0133404-1
James Glimm, Locally compact transformation groups, Trans. Amer. Math. Soc. 101 (1961), 124–138. MR 136681, DOI 10.1090/S0002-9947-1961-0136681-X
George W. Mackey, Unitary representations of group extensions. I, Acta Math. 99 (1958), 265–311. MR 98328, DOI 10.1007/BF02392428
Hirosi Nagao, The extension of topological groups, Osaka Math. J. 1 (1949), 36–42. MR 30531
I. Schochetman, Topology and the duals of certain locally compact groups, Trans. Amer. Math. Soc. 150 (1970), 477–489. MR 265513, DOI 10.1090/S0002-9947-1970-0265513-X

On the equivalence of twisted group algebras and Banach

algebraic bundles