Central twisted group algebras

Smith, Harvey A.

doi:10.1090/S0002-9947-1978-0487460-8

Central twisted group algebras
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Trans. Amer. Math. Soc. 238 (1978), 309-320 Request permission

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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