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I am looking at a von Neumann algebra constructed from a discrete group and a 2-cocylce. Does someone know some good references (article, book)? It would be very helpful for me. To be more precise, consider a countable group $G$ and a 2-cocycle $\phi :G^2\rightarrow S^1$ where $S^1$ is the group of complex number of modulus 1. You get a representation $\pi$ of the group $G$ in the Hilbert space $l^2(G)$ defined as follow: $$\pi(g)(e_t)=\phi(g,t).e_{gt}$$, where $e_t$ is the canonical hilbert basis of $l^2(G)$. I consider $L_\phi(G)$, the von Neumann algebra generated by $\pi(G)$. I am looking for reference on those kinds of algebra. Thanks, Arnaud

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  • $\begingroup$ It might help if you give a bit more detail about how the cocycle is used to construct this vN algebra... $\endgroup$
    – Yemon Choi
    Commented May 11, 2010 at 10:01
  • $\begingroup$ How exactly are you looking at the von Neumann algebra of interest? Is it introduced in a book, a paper, by a person talking, or something else? $\endgroup$
    – S. Carnahan ♦
    Commented May 11, 2010 at 13:53

2 Answers 2

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You might find these articles helpful (although it deals with the more general case): Sutherland, Colin E. Cohomology and extensions of von Neumann algebras. I, II. Publ. Res. Inst. Math. Sci. 16 (1980), no. 1, 105--133, 135--174.

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You can consult the article

Bédos, Erik On Følner nets, Szegő's theorem and other eigenvalue distribution theorems. Exposition. Math. 15 (1997), no. 3, 193--228.

and this recent article:

http://arxiv.org/PS_cache/math/pdf/0605/0605145v2.pdf

Finally, I believe that the construction is first appeared in

Zeller-Meier, G. Produits croisés d’une C*-algèbre par un groupe d’automorphismes. (French) J. Math. Pures Appl. (9) 47 1968 101–239

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