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Transactions of the American Mathematical Society

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Characteristic principal bundles


Author: Harvey A. Smith
Journal: Trans. Amer. Math. Soc. 211 (1975), 365-375
MSC: Primary 22D25; Secondary 55F10
DOI: https://doi.org/10.1090/S0002-9947-1975-0376953-7
MathSciNet review: 0376953
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1975-0376953-7
Article copyright: © Copyright 1975 American Mathematical Society

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