Characteristic principal bundles

Smith, Harvey A.

doi:10.1090/S0002-9947-1975-0376953-7

Characteristic principal bundles
HTML articles powered by AMS MathViewer

by Harvey A. Smith PDF

Trans. Amer. Math. Soc. 211 (1975), 365-375 Request permission

Abstract:

Characteristic principal bundles are the duals of commutative twisted group algebras. A principal bundle with locally compact second countable (Abelian) group and base space is characteristic iff it supports a continuous eigenfunction for almost every character measurably in the characters, also iff it is the quotient by Z of a principal E-bundle for every E in ${\operatorname {Ext}}(G,Z)$ and a measurability condition holds. If a bundle is locally trivial, n.a.s.c. for it to be such a quotient are given in terms of the local transformations and Čech cohomology of the base space. Although characteristic G-bundles need not be locally trivial, the class of characteristic G-bundles is a homotopy invariant of the base space. The isomorphism classes of commutative twisted group algebras over G with values in a given commutative ${C^\ast }$-algebra A are classified by the extensions of G by the integer first Čech cohomology group of the maximal ideal space of A.

References

Robert C. Busby, On the equivalence of twisted group algebras and Banach $^{\ast }$-algebraic bundles, Proc. Amer. Math. Soc. 37 (1973), 142–148. MR 315469, DOI 10.1090/S0002-9939-1973-0315469-4
Robert C. Busby, Centralizers of twisted group algebras, Pacific J. Math. 47 (1973), 357–392. MR 333595, DOI 10.2140/pjm.1973.47.357
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
J. Feldman and F. P. Greenleaf, Existence of Borel transversals in groups, Pacific J. Math. 25 (1968), 455–461. MR 230837, DOI 10.2140/pjm.1968.25.455
J. M. G. Fell, An extension of Mackey’s method to Banach $^{\ast }$ algebraic bundles, Memoirs of the American Mathematical Society, No. 90, American Mathematical Society, Providence, R.I., 1969. MR 0259619
László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR 0255673
Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR 0251549
Dale Husemoller, Fibre bundles, McGraw-Hill Book Co., New York-London-Sydney, 1966. MR 0229247, DOI 10.1007/978-1-4757-4008-0
Harvey B. Keynes and James B. Robertson, Eigenvalue theorems in topological transformation groups, Trans. Amer. Math. Soc. 139 (1969), 359–369. MR 237748, DOI 10.1090/S0002-9947-1969-0237748-5
Adam Kleppner, Multipliers on abelian groups, Math. Ann. 158 (1965), 11–34. MR 174656, DOI 10.1007/BF01370393
Hans Reiter, Classical harmonic analysis and locally compact groups, Clarendon Press, Oxford, 1968. MR 0306811
H. J. Reiter, Investigations in harmonic analysis, Trans. Amer. Math. Soc. 73 (1952), 401–427. MR 51341, DOI 10.1090/S0002-9947-1952-0051341-6
Harvey A. Smith, Commutative twisted group algebras, Trans. Amer. Math. Soc. 197 (1974), 315–326. MR 364538, DOI 10.1090/S0002-9947-1974-0364538-7
Norman Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951. MR 0039258, DOI 10.1515/9781400883875
J. G. Wendel, Left centralizers and isomorphisms of group algebras, Pacific J. Math. 2 (1952), 251–261. MR 49911, DOI 10.2140/pjm.1952.2.251
Ronald O. Fulp and Phillip A. Griffith, Extensions of locally compact abelian groups. I, II, Trans. Amer. Math. Soc. 154 (1971), 341-356; ibid. 154 (1971), 357–363. MR 0272870, DOI 10.1090/S0002-9947-1971-0272870-8
Ernest Michael, Convex structures and continuous selections, Canadian J. Math. 11 (1959), 556–575. MR 109344, DOI 10.4153/CJM-1959-051-9