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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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 $\mathbf{R}$.


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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: http://dx.doi.org/10.1090/S0002-9947-05-03702-5
PII: S 0002-9947(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