Representations of twisted group algebras

Busby, Robert C.; Smith, Harvey A.

doi:10.1090/S0002-9947-1970-0264418-8

Representations of twisted group algebras
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by Robert C. Busby and Harvey A. Smith PDF

Trans. Amer. Math. Soc. 149 (1970), 503-537 Request permission

Abstract:

We construct a general class of Banach algebras which include as special cases the group algebra of a locally compact group, the group algebra of a group extension (in terms of the subgroup and quotient group), and some other examples, special cases of which have been studied under the name of covariance algebras. We develop the general representation theory and generalize Mackey’s theory of induced representations.

References

Richard Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839–848. MR 45941, DOI 10.1090/S0002-9939-1951-0045941-1
Louis Auslander and Calvin C. Moore, Unitary representations of solvable Lie groups, Mem. Amer. Math. Soc. 62 (1966), 199. MR 207910
Jacques Dixmier, Les $C^{\ast }$-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars & Cie, Éditeur-Imprimeur, Paris, 1964 (French). MR 0171173

Les algèbres d’opérateurs dans l’espace hilbertien

20

Sergio Doplicher, Daniel Kastler, and Derek W. Robinson, Covariance algebras in field theory and statistical mechanics, Comm. Math. Phys. 3 (1966), 1–28. MR 205095
Edward G. Effros and Frank Hahn, Locally compact transformation groups and $C^*$-algebras, Bull. Amer. Math. Soc. 73 (1967), 222–226. MR 233213, DOI 10.1090/S0002-9904-1967-11690-2
J. Feldman and F. P. Greenleaf, Existence of Borel transversals in groups, Pacific J. Math. 25 (1968), 455–461. MR 230837
James Glimm, Families of induced representations, Pacific J. Math. 12 (1962), 885–911. MR 146297
Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
B. E. Johnson, An introduction to the theory of centralizers, Proc. London Math. Soc. (3) 14 (1964), 299–320. MR 159233, DOI 10.1112/plms/s3-14.2.299
G. Philip Johnson, Spaces of functions with values in a Banach algebra, Trans. Amer. Math. Soc. 92 (1959), 411–429. MR 107185, DOI 10.1090/S0002-9947-1959-0107185-6

Verallgemeinerte

algebren

159

Verallgemeinerte

Algebren und projektive Darstellungen lokal kompakter Gruppen

3

37

H. Leptin, Verallgemeinerte $L^{1}$-Algebren und projektive Darstellungen lokal kompakter Gruppen. I, II, Invent. Math. 3 (1967), 257–281; ibid. 4 (1967), 68–86 (German). MR 229754, DOI 10.1007/BF01402952
George W. Mackey, Imprimitivity for representations of locally compact groups. I, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 537–545. MR 31489, DOI 10.1073/pnas.35.9.537
George W. Mackey, Induced representations of locally compact groups. I, Ann. of Math. (2) 55 (1952), 101–139. MR 44536, DOI 10.2307/1969423
George W. Mackey, Unitary representations of group extensions. I, Acta Math. 99 (1958), 265–311. MR 98328, DOI 10.1007/BF02392428
Saunders Mac Lane, Homology, Die Grundlehren der mathematischen Wissenschaften, Band 114, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156879
I. E. Segal, A class of operator algebras which are determined by groups, Duke Math. J. 18 (1951), 221–265. MR 45133
Robert Schatten, A Theory of Cross-Spaces, Annals of Mathematics Studies, No. 26, Princeton University Press, Princeton, N. J., 1950. MR 0036935
Masamichi Takesaki, Covariant representations of $C^{\ast }$-algebras and their locally compact automorphism groups, Acta Math. 119 (1967), 273–303. MR 225179, DOI 10.1007/BF02392085
Takasi Turumaru, Crossed product of operator algebra, Tohoku Math. J. (2) 10 (1958), 355–365. MR 102764, DOI 10.2748/tmj/1178244669

L’intégration dans les groupes topologiques et ses applications

J. G. Wendel, Left centralizers and isomorphisms of group algebras, Pacific J. Math. 2 (1952), 251–261. MR 49911
Georges Zeller-Meier, Produits croisés d’une $C^{\ast }$-algèbre par un groupe d’automorphismes, C. R. Acad. Sci. Paris Sér. A-B 263 (1966), A20–A23 (French). MR 199734
J. Dixmier, Dual et quasi-dual d’une algèbre de Banach involutive, Trans. Amer. Math. Soc. 104 (1962), 278–283 (French). MR 139960, DOI 10.1090/S0002-9947-1962-0139960-6
N. Bourbaki, Éléments de mathématique. Fascicule XXIX. Livre VI: Intégration. Chapitre 7: Mesure de Haar. Chapitre 8: Convolution et représentations, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1306, Hermann, Paris, 1963 (French). MR 0179291
Robert J. Blattner, Group extension representations and the structure space, Pacific J. Math. 15 (1965), 1101–1113. MR 188346