Second cohomology group of group algebras with coefficients in iterated duals

Pourabbas, A.

doi:10.1090/S0002-9939-03-07219-8

Second cohomology group of group algebras with coefficients in iterated duals
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Proc. Amer. Math. Soc. 132 (2004), 1403-1410 Request permission

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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