1
$\begingroup$

Let $G$ be a discrete group and $\mathcal{G}$ a groupoid, that is, a small category in which every arrow is an isomorphism. Wolfgang Lück explains how we can construct a $C^*$-category from $\mathcal{G}$ this way. This construction is very useful, because if we consider a discrete group $G$ as a groupoid with one element $e$ and morphisms $Mor_G(e,e)=G$, then $Mor_{C_r(G)}(e,e)$ agrees with the reduced group $C^*$ -algebra $C_r(G)$, as the notation implies. Now, I want to find a way to extend this principle to the reduced crossed product.

Let $A$ be a separable $G$-$C^*$-algebra. I want to find a way to construct a similar functor $F$ to that described by Lück, only that if I consider $G$ or a subgroup $H$ of $G$ to be a groupoid, I actually have $Mor_{F(H)}(e,e)$ agreeing with the reduced cross product of $H$ and $A$. I think it's easy if $G$ acts trivially on $A$: I would just take the (minimal) tensor product of the construction described by Lück and A. But what if the action isn't trivial?

My overall aim is to formulate the Baum-Connes conjecture with coefficients using spectra as introduced by Lück.

Thank you

$\endgroup$
3
$\begingroup$

This has been documented in the literature, see Paul D. Mitchener, C*-categories, Groupoid Actions, Equivariant KK-theory, and the Baum-Connes Conjecture (arXiv:math/0204291) and works of Michel Matthey.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.