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Transactions of the American Mathematical Society

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The cyclic and simplicial cohomology of $l^1(\mathbf{N})$


Authors: Frédéric Gourdeau, B. E. Johnson and Michael C. White
Journal: Trans. Amer. Math. Soc. 357 (2005), 5097-5113
MSC (2000): Primary 46H20, 46J40; Secondary 43A20, 16E40
DOI: https://doi.org/10.1090/S0002-9947-05-03702-5
Published electronically: April 13, 2005
MathSciNet review: 2165399
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Abstract: Let $\mathcal{A}=l^1(\mathbf Z_+)$ be the unital semigroup algebra of $\mathbf N$. We show that the cyclic cohomology groups $\mathcal {H}C^n(\mathcal{A},\mathcal{A}')$ vanish when $n$ is odd and are one dimensional when $n$ is even ($n\ge 2$). Using Connes' exact sequence, these results are used to show that the simplicial cohomology groups $\mathcal{H}^n(\mathcal{A},\mathcal{A}')$ vanish for $n\ge 2$. The results obtained are extended to unital algebras $l^1(S)$ for some other semigroups of $\mathbf{R}$.


References [Enhancements On Off] (What's this?)

  • [BCD] W.G. Bade, P.C. Jr Curtis, and H.G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc. (3), 55 (1987), 359-377. MR 0896225 (88f:46098)
  • [BD] F.F. Bonsall and J. Duncan, Complete normed algebras, Springer-Verlag, New York, 1973. MR 0423029 (54:11013)
  • [DaDu] H. G. Dales and J. Duncan, Second order cohomology in groups of some semigroup algebras, Banach Algebras `97, Proc. 13th. Internat. Confer. on Banach Algebras, 101-117, Walter de Gruyter, Berlin, 1998. MR 1656601 (99m:46168)
  • [GW] F. Gourdeau and M. C. White, Vanishing of the third simplicial cohomology group of $l^1(\mathbf Z_+)$, Trans. AMS (353), 5 (2001), 2003-2017.MR 1813605 (2001m:46163)
  • [He0] A. Ya. Helemskii, The homology of Banach and topological algebras. Kluwer Academic Publishers, Dordrecht, 1989. MR 1093462 (92d:46178)
  • [He1] A. Ya. Helemskii, Banach cyclic (co)homology and the Connes-Tzygan exact sequence, J. London Math. Soc., (2) 46, (1992), 449-462.MR 1190429 (94a:46105)
  • [BEJ1] B.E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc., 127 (1972). MR 0374934 (51:11130)
  • [BEJ2] B.E. Johnson, Higher-dimensional weak amenability, Studia Math., 123 (1997), 2, 117-134. MR 1439025 (98f:46046)
  • [No] D.G. Northcott, Introduction to homological algebra, Cambridge University Press, 1962. MR 0118752 (22:9523)

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Additional Information

Frédéric Gourdeau
Affiliation: Département de Mathématiques et de Statistique, Université Laval, Cité Universitaire, Québec, Canada G1K 7P4
Email: Frederic.Gourdeau@mat.ulaval.ca

B. E. Johnson
Affiliation: Department of Mathematics, University of Newcastle, Newcastle upon Tyne, NE1 7RU, England

Michael C. White
Affiliation: Department of Mathematics, University of Newcastle, Newcastle upon Tyne, NE1 7RU, England
Email: Michael.White@ncl.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-05-03702-5
Keywords: Cohomology, cyclic cohomology, simplicial cohomology, Banach algebra
Received by editor(s): November 14, 2002
Received by editor(s) in revised form: April 8, 2004
Published electronically: April 13, 2005
Article copyright: © Copyright 2005 American Mathematical Society

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