On derivations and crossed homomorphisms

Volume 91 / 2010

Viktor Losert Banach Center Publications 91 (2010), 199-217 MSC: Primary 43A20; Secondary 22F05, 37B05, 46H99, 46L55, 47B47. DOI: 10.4064/bc91-0-12

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.

Authors

  • Viktor LosertFakultät für Mathematik
    Universität Wien
    Nordbergstr. 15
    A 1090 Wien, Austria
    e-mail

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