I want to understand noncommutative quotients. Now the book Basic Noncommutative Geometry by M. Khalkhali gives two different constructions of the noncommutative quotient and claims they are isomorphic. One via a groupoid algebra and one through a crossed product algebra. The first is more intuitive while the second is arguably simpler to compute.
Given a group $G$ acting on a set $X$ we have an induced groupoid $O$. The groupoid algebra $C_c(O)$ of $O$ is the space of continuous compactly supported functions to $\mathbb{C}$.
Alternatively we can form the crossed product algebra $C(X)\ltimes G$ which is the vector space $C(X)\otimes\mathbb{C}G$ under the produc $(a\otimes g)(b \otimes h)=a g(b) \otimes gh$, $a,b \in C(X), g,h \in G$. There are subtleties having to do with the a suitable norm on these algebra's which I'll omit.
Now I have two related questions
- In general $O$ is a topological groupoid. But what is the exact topology on $O$?
- How are these constructions isomorphic?
Lets make it a little more concrete. Let $\mathbb{Z}$ act on $\mathbb{R}$ by translation. Then my first guess what the groupoid algebra is would be \begin{equation} C_c(O)=\{ f:\mathbb{R}\rightarrow M_{\infty}(\mathbb{C}): \textrm{f is continuous and compactly supported} \} \end{equation}
Now I don't really understand the topology on the groupoid so I'm not sure this is the right groupoid algebra. In any case it should be isomorphic to the crossed product algebra, but how?