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First of all, my knowledge of operator algebras (and functional analysis) is very superficial, so sorry if the answer is actually well-known.

Let $X$ be a locally compact Hausdorff groupoid (or Lie groupoid). One can, then, define $C := C^* (X, A)$ through convolution of functions from $X_1$ to some ring $A$ (I'm ok with just $\mathbb{C}$ here).

Question: Can one recover $X$ back from $C$ (maybe up to Morita equivalence)?

The above definition seems to make sense too for increased categorical dimension, i.e., $n$-groupoids. See, for instance, https://ncatlab.org/nlab/show/category%20algebra#convolution_algebras . There are also other possible choices of geometry by taking internal ($n$)-groupoids inside other categories. For instance, one can still talk about algebraic or (internal or not) localic $n$-groupoids and maybe construct a convolution algebra as long as a notion of Haar measure makes sense.

Extra question: Any idea about the case of $n$-groupoids internal to any of the mentioned above categories (topological, differentiable, algebraic etc) even if it's just something conjectural?

Thanks in advance.

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    $\begingroup$ What is the notion of Haar measure on a topological $n$-groupoid? And what is the $C^*$-algebra associated to it? Have you any references? The page you link to was something that someone was working on at some point but never finished, never extended past $n=2$, and also claimed that "Hence the groupoid convolution algebra constructiuon is a 2-functor" (sic), except that I don't think it is functorial! That's not to say the $n=1$ case is not already an excellent question. I would focus on that, for now. $\endgroup$
    – David Roberts ♦
    Commented Mar 24, 2022 at 0:11
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    $\begingroup$ Yeah, I get that :-) But I think leading with the concrete, known version (ordinary loc. compact groupoids), and then branching to more speculative is better than posing what looks like (from the outside) a very speculative question, and then inside saying "it's ok if you can only answer for n=1". The people who know the n=1 case will not like the n-groupoid title, and the people who are comfortable with n-groupoids will likely not know the n=1 answer. Getting an n=1 answer here and then seeing where it goes it more useful imho. $\endgroup$
    – David Roberts ♦
    Commented Mar 24, 2022 at 6:48
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    $\begingroup$ The point is, the (isomorphism class of the) convolution C* algebra of a (1-)groupoid knows the number of, and dimension of, the irreducible representations of that groupoid, and not much more. So for example if $A$ is a finite abelian group, then the C* algebra of $BA$ knows the order of $A$, but not the group structure, nor does it know that $BA$ was connected. $\endgroup$
    – Theo Johnson-Freyd
    Commented Mar 24, 2022 at 23:16
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    $\begingroup$ But you can absolutely enhance the C* algebra with extra data that knows the (equivalence class of the) groupoid. For example, the C* algebra of any groupoid is naturally cocommutative-Hopfish, corresponding to the ability to tensor representations of the groupoid. (Do I mean Hopfish? Or weak Hopf? Or...? I can never remember which generalization of Hopf is which.) A version of Tannakian duality says you can recover the groupoid from its Hopfish-C*-algebra. Essentially the groupoid is the grouplike-elements in its algebra. $\endgroup$
    – Theo Johnson-Freyd
    Commented Mar 24, 2022 at 23:22
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    $\begingroup$ On the other hand, if you are a tried and true noncommutative geometer, then you will think that remembering a Hopfish structure is too much. (It is arguably a type of commutativity.) Indeed, it is an interesting feature that some invariants of a groupoid, e.g. its K-theory, factor through the C* algebra. $\endgroup$
    – Theo Johnson-Freyd
    Commented Mar 24, 2022 at 23:26

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