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Second cohomology group of group algebras with coefficients in iterated duals

Author: A. Pourabbas
Journal: Proc. Amer. Math. Soc. 132 (2004), 1403-1410
MSC (2000): Primary 43A20; Secondary 46M20
Published electronically: August 28, 2003
MathSciNet review: 2053346
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Abstract: In this paper we show that the first cohomology group $\mathcal{H}^1(\ell^1(G),(\ell^1(S))^{(n)})$ is zero for every odd $n\in\mathbb{N}$ and for every $G$-set $S$. In the case when $G$ is a discrete group, this is a generalization of the following result of Dales et al.: for any locally compact group $G$, $L^1(G)$ is $(2n+1)$-weakly amenable.

Next we show that the second cohomology group $\mathcal{H}^2(\ell^1(G),(\ell^1(S))^{(n)})$ is a Banach space. Finally, for every locally compact group $G$ we show that $\mathcal{H}^2(L^1(G),(L^1(G))^{(n)})$ is a Banach space for every odd $n\in\mathbb{N}$.

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Additional Information

A. Pourabbas
Affiliation: Faculty of Mathematics and Computer Science, Amirkabir University, 424 Hafez Avenue, Tehran 15914, Iran

Received by editor(s): January 14, 2002
Received by editor(s) in revised form: December 31, 2002
Published electronically: August 28, 2003
Additional Notes: This research was supported by a grant from Amir Kabir University. The author would like thank the Institute for their kind support.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2003 American Mathematical Society

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