Non-commutative versions of X/G

Question

Let $X$ be a Riemannian manifold and let $G$ be a (at most countable, if that matters) discrete group acting properly and by isometries on $X$. Let $\mathcal{O}$ be the sheaf of analytic functions on $X$. By analogy with what happens for finite groups acting on vector spaces, one is tempted to study a sheaf written $\mathcal{O} \rtimes G$. What is a good reference for an algebraist to learn about the various convergence conditions one might impose to define this sheaf, and their relationship with the quotient X/G?

[I'm fine with an answer that works in a different category---e.g. complex analytic spaces, but I want there to be some convergence conditions imposed at some point. The ideal reference is a short survey paper with precise definitions.]

Answer 1

Noncommutative versions of sheaves and holomorphic functions are not very well understood. Better understood are noncommutative versions of measurable, continuous, or smooth functions. I generally work with the continuous functions, i.e. $C^* $-algebras, or various subalgebras that deserve to be called smooth. I'll describe things in the $C^*$-framework.

What came to mind immediately for me is the notion of strong Morita equivalence, due to Rieffel. It works like this: suppose you have a locally compact group $G$ acting on a $C^* $- algebra $A$ (think of $A$ as $C(X)$ here). You can form what is called the crossed product algebra, which is a $C^*$-algebra containing $A$ and $G$, and where the action of $G$ on $A$ is implemented via conjugation by $G$; i.e. if $a \in A$ and $g \in G$, then $g a g^* = \alpha_g(a)$, where $\alpha$ is the action.

This can be done when $A$ is unital or not, and $G$ can be discrete or not. The resulting algebra, which I would denote $A \times_\alpha G$, is unital if and only if $A$ is unital and $G$ is discrete.

Now suppose that $X$ is a compact Hausdorff space with an action of $G$. Then $G$ also acts on $A = C(X)$, and so we can make the crossed product algebra $C(X) \times_\alpha G$. Here's the punchline: when the action of $G$ on $X$ is free and proper, so that the quotient $X/G$ is well-behaved, then the crossed product algebra is strongly Morita equivalent to the algebra $C(X/G)$ of functions on the quotient.

When the action is not free and proper, the quotient may be very bad (e.g. the integers acting on the circle by rotation by an irrational angle) and so the algebra $C(X/G)$ may be reduced to nothing more than scalars, and so be useless for obtaining any information about the quotient. In this case, one uses the crossed-product algebra as a sort of substitute for the algebra of functions on the quotient.

A reference for this is the paper "Applications of Strong Morita Equivalence to Transformation Group $C^*$-algebras, by Rieffel, which is available on his website. Unfortunately it doesn't have the definitions of crossed products (which he calls transformation group algebras), but the wikipedia page is ok, although phrased just for von Neumann algebras.