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My question is in nature a bit vague but let me try to make it concrete. Given a Lie grupoid $G$ that is étale and proper (called an orbifold grupoid) we have an associated orbifold $X$; this is explained for instance in 1.4 here. On the other hand, from such a grupoid one can also construct a $C^\ast$ algebra $C^\ast(G)$, as explained here. So my general question is what can be said about one from the other and how should we interpret $C^\ast(G)$ in terms of the orbifold $X$.

As far as I understant when $G$ is a grupoid without morphisms except the identities (so $G_0=G_1=X$) then $C^\ast(G)$ is just $C(X)$, and in this case the orbifold $X$ is just a smooth manifold.

  1. Is this at some leve still true for orbifolds? I.e. is there a way to think (morally, maybe) of $C^\ast(G)$ as maps from $X$ to $\mathbb C$, or some modification of this?

In the smooth case the functor $C$ defines a duality between the categories of (say) locally compact Hausdorff spaces and commutative $C^\ast$ algebras. Can something similar be said about $C^\ast$ for orbifolds? Two grupoids give the same orbifold if and only if they are Morita equivalent, and if I understood correctly Morita equivalent grupoids induce Morita equivalent $C^\ast$ algebras, so we should get a duality between grupoids and Morita classes of $C^\ast$-algebras.

  1. Can this be made precise? In particular is it true that if two grupoids give Morita equivalent $C^\ast$ algebras then the grupoids are also Morita equivalent? Can we say what are the possible $C^\ast$ algebras of orbifold grupoids?

Finally, what kind of information can we get about $X$ from $C^\ast(G)$? Two particular questions we can make about this are the following:

  1. Is it true that the orbifold $K$-theory of $X$ is the operator $K$-theory of $C^\ast(G)$?

  2. Is there a way to interpret the inertia orbifold of $X$ in $C^\ast$ algebra terms? And what about Chen-Ruan cohomology?

Any related insight and reference will be very appreciated even if it doesn't answer (or does only partially) the specific questions!

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    $\begingroup$ You should think of an orbifold as a coarse reflection of the true object, which is the Lie groupoid. Even the Lie groupoid is a rigid presentation of the really true object, which is a differentiable stack. Anything you construct from a Lie groupoid that is invariant up to equivalence/isomorphism under Morita equivalence is intrinsically attached to the stack, and hence the orbifold. $\endgroup$
    – David Roberts ♦
    Commented Mar 5, 2019 at 10:56
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    $\begingroup$ Im not the right person to give an answer on this but groupoid C*-algebra is determined up to Morita equivalence by the Morita equivalence class of the groupoid by the results of Muhly, Renault and Williams. $\endgroup$
    – Benjamin Steinberg
    Commented Mar 5, 2019 at 13:05
  • $\begingroup$ I encountered this question right now, maybe of interest to see the paper ``Noncommutative geometry and diffeology: The case of orbifolds'' by Iglesias-Zemmour & Laffineur. $\endgroup$
    – ARA
    Commented Mar 6, 2023 at 18:07

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