Vanishing of the third simplicial cohomology group of $l^1(\mathbf {Z}_+)$
HTML articles powered by AMS MathViewer
- by Frédéric Gourdeau and Michael C. White PDF
- Trans. Amer. Math. Soc. 353 (2001), 2003-2017 Request permission
Abstract:
We show that $\mathcal {H}^3(l^1(\mathbf {Z}_+),l^1(\mathbf {Z}_+)’)=0$. We first use the Connes-Tzygan exact sequence to prove that this is equivalent to the vanishing of the third cyclic cohomology group $\mathcal {H}C^3(\mathcal {I},\mathcal {I}’)$, where $\mathcal {I}$ is the non-unital Banach algebra $l^1(\mathbf {N})$, and then prove that $\mathcal {H}C^3(\mathcal {I},\mathcal {I}’)=0$.References
- W. G. Bade, P. C. Curtis Jr., and H. G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc. (3) 55 (1987), no. 2, 359–377. MR 896225, DOI 10.1093/plms/s3-55_{2}.359
- Frank F. Bonsall and John Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80, Springer-Verlag, New York-Heidelberg, 1973. MR 0423029
- H. Garth Dales and John Duncan, Second-order cohomology groups of some semigroup algebras, Banach algebras ’97 (Blaubeuren), de Gruyter, Berlin, 1998, pp. 101–117. MR 1656601, DOI 10.1111/j.1439-0531.1998.tb01324.x
- A. Ya. Helemskii, The homology of Banach and topological algebras, Mathematics and its Applications (Soviet Series), vol. 41, Kluwer Academic Publishers Group, Dordrecht, 1989. Translated from the Russian by Alan West. MR 1093462, DOI 10.1007/978-94-009-2354-6
- A. Ya. Helemskii, Banach cyclic (co)homology and the Connes-Tzygan exact sequence, J. London Math. Soc. (2) 46 (1992), no. 3, 449–462. MR 1190429, DOI 10.1112/jlms/s2-46.3.449
- Barry Edward Johnson, Cohomology in Banach algebras, Memoirs of the American Mathematical Society, No. 127, American Mathematical Society, Providence, R.I., 1972. MR 0374934
- B. E. Johnson, Higher-dimensional weak amenability, Studia Math. 123 (1997), no. 2, 117–134. MR 1439025, DOI 10.4064/sm-123-2-117-134
- D. G. Northcott, An introduction to homological algebra, Cambridge University Press, New York, 1960. MR 0118752
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
- Michael C. White
- Affiliation: Department of Mathematics, University of Newcastle, Newcastle upon Tyne, NE1 7RU, England
- Email: Michael.White@ncl.ac.uk
- Received by editor(s): September 27, 1999
- Published electronically: January 3, 2001
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 2003-2017
- MSC (2000): Primary 46H20, 46J40; Secondary 43A20, 16E40
- DOI: https://doi.org/10.1090/S0002-9947-01-02738-6
- MathSciNet review: 1813605