Commutative twisted group algebras
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- by Harvey A. Smith PDF
- Trans. Amer. Math. Soc. 197 (1974), 315-326 Request permission
Abstract:
A twisted group algebra ${L^1}(A,G;T,\alpha )$ is commutative iff A and G are, T is trivial and $\alpha$ is symmetric: $\alpha (\gamma ,g) = \alpha (g,\gamma )$. The maximal ideal space ${L^1}(A,G;\alpha )\hat \emptyset$ of a commutative twisted group algebra is a principal $G\hat \emptyset$ bundle over $A\hat \emptyset$. A class of principal $G\hat \emptyset$ bundles over second countable locally compact M is defined which is in 1-1 correspondence with the (isomorphism classes of) ${C_\infty }(M)$-valued commutative twisted group algebras on G. If G is finite only locally trivial bundles can be such duals, but in general the duals need not be locally trivial.References
- Louis Auslander and Calvin C. Moore, Unitary representations of solvable Lie groups, Mem. Amer. Math. Soc. 62 (1966), 199. MR 207910
- 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
- C. M. Edwards and J. T. Lewis, Twisted group algebras. I, II, Comm. Math. Phys. 13 (1969), 119–130; ibid. 131–141. MR 253064, DOI 10.1007/BF01649871
- 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
- Dale Husemoller, Fibre bundles, McGraw-Hill Book Co., New York-London-Sydney, 1966. MR 0229247, DOI 10.1007/978-1-4757-4008-0
- 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
- 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 H. Leptin, Verallgemeinerte ${L^1}$-Algebren, Math. Ann. 159 (1965), 51-76. MR 39 #1909. —, Verallgemeinerte ${L^1}$-Algebren und projektive Darstellungen lokal kompakter Gruppen. I, Invent. Math. 3 (1967), 257-281. MR 37 #5328.
- 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 M. A. Naĭmark, Normed rings, GITTL, Moscow, 1956; English transl., Noordhoff, Groningen, 1959. MR 19, 870; 22 #1824.
- 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
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 197 (1974), 315-326
- MSC: Primary 22D15; Secondary 46J20
- DOI: https://doi.org/10.1090/S0002-9947-1974-0364538-7
- MathSciNet review: 0364538