Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Vanishing of the third simplicial cohomology group of $l^1(\mathbf{Z}_+)$

Author(s): Frédéric Gourdeau; Michael C. White
Journal: Trans. Amer. Math. Soc. 353 (2001), 2003-2017.
MSC (2000): Primary 46H20, 46J40; Secondary 43A20, 16E40
Posted: January 3, 2001
MathSciNet review: 1813605
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

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:

1.: 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), 359-377. MR 88f:46098
2.: F.F. Bonsall and J. Duncan, Complete normed algebras, Springer-Verlag, New York, 1973. MR 54:11013
3.: 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 99m:46168
4.: A. Ya. Helemskii, The homology of Banach and topological algebras. Kluwer Academic Publishers, Dordrecht, 1989. MR 92d:46178
5.: A. Ya. Helemskii, Banach cyclic (co)homology and the Connes-Tzygan exact sequence, J. London Math. Soc., (2) 46, (1992), 449-462. MR 94a:46105
6.: B.E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc., 127 (1972). MR 51:11130
7.: B.E. Johnson, Higher-dimensional weak amenability, Studia Math., 123 (1997), 2, 117-134. MR 98f:46046
8.: D.G. Northcott, Introduction to homological algebra, Cambridge University Press, 1962. MR 22:9523

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|