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Let $\alpha:H\to G$ be a group homomorphism betwenn discrete countable groups, and assume that the kernel of $\alpha$ is an amenable group, denoted by $K$. I would like to know references for the basic questions:

  1. How does the induced group homomorphism between the group Von Neumann algebras $\mathcal{N}H\to \mathcal{N}G$ look like?

  2. Can one define via this homomorphism a functor between the (algebraic) categories of $\mathcal{N}H$ and $\mathcal{N}G$ modules? Is it exact in general in any of the approaches to homological algebra over von Neumann algebras (Farber/Lück)?

I am aware of the proof that this functor is fully faithful in the case that the kernel of the homomorphism is trivial by work of W. Lück (Dimension Theory of arbitrary modules over finite Von Neumann algebras) in J. reine. ang. Math. 495, 1998.

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    $\begingroup$ What do you mean by the "induced group homomorphism" in point 1? A group homomorphism $\alpha : H \rightarrow G$ only extends to a normal $*$-homomorphism between the associated group von Neumann algebras if the kernel of $\alpha$ is finite. $\endgroup$
    – Stefaan Vaes
    Commented Feb 3, 2021 at 12:47
  • $\begingroup$ Thanks a lot for the comment, the Group Von Neumann Algebra is only functorial for homomorphisms with finite kernel, I was asuming wrongly that the functoriality of the reduced group C* algebra goes over. $\endgroup$
    – Nicolas Boerger
    Commented Feb 5, 2021 at 8:59

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