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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Central twisted group algebras


Author: Harvey A. Smith
Journal: Trans. Amer. Math. Soc. 238 (1978), 309-320
MSC: Primary 46H99; Secondary 22D20, 46L99
DOI: https://doi.org/10.1090/S0002-9947-1978-0487460-8
MathSciNet review: 0487460
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Abstract: A twisted group algebra $ {L^1}(A,G;T,\alpha )$ is central iff T is trivial and A commutative. (Group algebras of central extension of G are such.) We show that if $ {H^2}(G)$ is discrete any central $ {L^1}(A,G;\alpha )$ is a direct sum of closed ideals $ {L^1}({A_i},G;{\alpha _i})$ having as duals fibre bundles over the duals of closed ideals $ {A_i}$ in A, with fibres projective duals of G, and principal $ {G^\wedge}$ bundles (where $ {G^\wedge}$ denotes the group of characters of G) satisfying the conditions which define characteristic bundles for G abelian. (If G is compact $ {H^2}(G)$ is always discrete, the direct sum is countable, and the bundles are locally trivial.) Applications are made to the duals of central extensions of groups and in particular to duals of ``central'' groups. For G commutative, $ {H^2}(G)$ discrete, and A a $ {C^\ast}$-algebra with identity, all central twisted group algebras $ {L^1}(A,G;\alpha )$ (and their duals) are classified in purely algebraic terms involving $ {H^2}(G)$, the group G, and the first Čech cohomology group of the dual of A. This result allows us, in principle, to construct all the central $ {L^1}(A,G;\alpha )$ and their duals where A is a $ {C^\ast}$-algebra with identity and G a compact commutative group.


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DOI: https://doi.org/10.1090/S0002-9947-1978-0487460-8
Article copyright: © Copyright 1978 American Mathematical Society