Many interesting C\*-algebras can be realized as **convolution algebras** over a groupoid, an idea introduced in 1980 by Jean Renault ([this](https://ncatlab.org/nlab/show/category+algebra) entry in nLab provides plenty of context to the general approach of attaching an algebra to a groupoid).


Perhaps due to my incompetence in this formidable field, I was not able to identify some results in the way of characterizing *which* C\*-algebras are in fact convolution algebras.


So, here is my question:



> 
> Is there a duality theorem stating that the sub-category of C\*-algebras satisfying (add here some properties) is dual to the category of locally compact groupoids ?
> 
> 
> 


**ADDENDUM**: I have changed the title to make clear what I said in my comments below: though this is a very specific question (and by no means trivial, see the excellent responses below), it is just a tassel toward a broad goal, namely get some kind of **duality theory for non commutative spaces**. I was (happily) surprised not only with the answers I have received, but also by the views and the likes this question has gotten. That means that numerous folks share this interest. There will be further questions under the same rubric, and I hope they will attract other members of MO from a broad variety of backgrounds (operator theory, higher category theory, mathematical physicists, and generally researchers with a passion for duality theory)



Non-commutative duality I: Which C*-algebras are (isomorphic to a) convolution algebra?
The main obstruction to this kind of duality is not so much that not every $C^\*$-algebra is a convolution algebra (though, at least if we don't use twisted convolution algebra, there are known obstruction as mentioned in the comment), but rather that the construction that attach a convolution $C^\*$-algebra to a groupoid is not 'injective' at all.


This is mostly due to what I want to call "Fourier isomorphisms" between $C^\*$-algebras, that exists at the analytic level without having a clear geometric origin (well, I'm sure one can find a geometric explanation for Fourier duality, but what I mean is that it cannot be interpreted as a morphisms or bibundle between the groupoids).


A typical example: $C^\*(B \mathbb{Z}) \simeq C(\mathbb{U})$


Take the groupoid $B\mathbb{Z}$ with a single object $\*$ and the additive groupe $\mathbb{Z}$ as its automorphism group. We see it as a topological groupoid for the discrete topology.


The associated $C^\*$-algebra (both maximal and reduced) $C^\*(B\mathbb{Z})$ is simply the group $C^\*$-algebra of $\mathbb{Z}$. Its a commutative $C^\*$-algebra, by Gelfand duality it is isomorphic to continuous fonction on its spectrum. Here it is the algebra of continuous function on $\mathbb{U}$ the unit circle (the elements $n \in \mathbb{Z}$ corresponds to the function $z \mapsto z^n$).


But I can also considered the groupoid with $\mathbb{U}$ as set of objects and no non-identity morphisms, that I topologize with the topology of $\mathbb{U}$. The $C^\*$-algebra attached to this groupoid are simply continuous function on $\mathbb{U}$, hence the same $C^\*$-algebra as before.


So, if you want to recover the $C^\*$-algebra you need some additional structure on it, not just a property. For example the notion of "Cartan subalgebra", which represent the subalgebra of the convolution algebra of continuous function on $G\_0$ does the trick in some cases. Have a look at [Cartan Subalgebras in $C^\*$-Algebras](https://www.idpoisson.fr/renault/pub/S6101.pdf) by Jean Renault for example, the paper also cite other similar result in different context.


