On derivations and crossed homomorphisms
Volume 91 / 2010
Abstract
We discuss some results about derivations and crossed homomorphisms arising in the context of locally compact groups and their group algebras, in particular, $L^1(G)$, the von Neumann algebra $VN(G)$ and actions of $G$ on related algebras. We answer a question of Dales, Ghahramani, Grønbæk, showing that $L^1(G)$ is always permanently weakly amenable. Then we show that for some classes of groups (e.g. IN-groups) the homology of $L^1(G)$ with coefficients in $VN(G)$ is trivial. But this is no longer true, in general, if $VN(G)$ is replaced by other von Neumann algebras, like ${\cal B}(L^2(G))$. Finally, as an example of a non-discrete, non-amenable group, we investigate the case of $G=SL(2,\mathbb R)$ where the situation is rather different.
