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 by Jean Renault for example, the paper also cite other similar result in different context.

As already pointed out, Buss and Sims have found an example of a $C^*$-algebra which is not isomorphic to its opposite, and hence it is not a groupoid $C^*$-algebra. However twisted groupoid $C^*$-algebras are not necessarily self-opposite so, as the authors point out, nothing so far prevents their example from being realized as a twisted convolution algebra. In fact, it appears that nobody knows an example of a $C^*$-algebra which is not a twisted convolution algebra!!

Among the algebras known to be (twisted or untwisted) convolution algebras one finds all Kirchberg algebras satisfying the UCT as well as all algebras in the Elliott classification program, including the somewhat elusive Jiang-Su algebra $\mathscr Z$.

On the other hand, the vast majority of useful tools available to the study of (untwisted) groupoid $C^*$-algebras apply just as well for twisted ones, so perhaps the more relevant question is which $C^*$-algebras are isomorphic to a twisted convolution algebra.

As mentioned by @Simon, the best results so far leading to a "Fourier-like" decomposition of a $C^*$-algebra as a convolution algebra (of an etale groupoid) are based on the assumption that we already know the abelian subalgebra of functions supported on the object space. This abelian subalgebra is sometimes called a Cartan algebra, given the similarities with the homonymous concept from the theory of Lie groups.

One of the stickiest points in this subject is whether or not there exists a conditional expectation onto the given Cartan subalgebra (one of the main assumptions in Renault's result). The reason one might prefer not to require this condition is that there are many groupoids whose convolution algebra does not admit such a conditional expectation due to the fact that the underlying groupoid is not Hausdorff. Examples of this situation are very common such as groupoids arising from foliations and certain dynamical systems.

Another relevant question is whether or not the Cartan subalgebra is maximal abelian, a crucial feature in the Lie algebra version of this concept, as well as an important assumption in Renault's theory. Maximal commutativity is closely related to topological freenes of the associated groupoid (a concept borrowed from those dynamical systems in which most points have trivial isotropy group). In particular, in the example given by @Simon of $C^*(B\mathbb Z)$, the natural Cartan subalgebra is $\mathbb C$, which is clearly not maximal commutative at the same time that the action of the group $\mathbb Z$ on a point has too much isotropy!

Working with David Pitts (Characterizing groupoid C*-algebras of non-Hausdorff étale groupoids, arXiv:1901.09683) we have found a characterization of twisted groupoid $C^*$-algebras which neither assumes conditional expectations nor maximal commutativity. Perhaps one of the simplest examples where our result applies is $$ C(S^1)\subseteq C([0,1]) $$ where we identify $C(S^1)$ as the subalgebra of periodic functions. Surprisingly, there is a twisted groupoid description of this "Cartan pair".

Significantly, the lack of a conditional expectation from $C([0,1])$ to $C(S^1)$ prevents the groupoid from being Hausdorff. In fact, the topological space underlying this groupoid is essentially the best known example of a non-Hausdorff topological space in which one takes the unit interval $[0,1]$ with a duplicate copy of the point 1, except that we instead duplicate a chosen point of $S^1$. The groupoid structure is such that the two duplicate points form a copy of the 2-element group, while all other points are considered to be objects. The twist is non trivial but it turns out to be the most natural nontrivial twist one can think of.

There are already excellent answer so I hope my small remarks here will not seem to be too trivial. They concern the geometric origin, in a way, of the Fourier morphism mentioned by Simon Henry in a very specific contect.

When quantizing Poisson manifolds into NC $C^*$-algebras on possible way is through what is called groupoid quantization. The procedure is, here, to build the symplectic groupoid integrating the original Poisson manifold (thus a very specific kind of Lie groupoid) and then performing geometric quantization on this symplectic manifold keeping in mind, in a way, its groupoid structure.

The relevant part is the choice of a polarization compatible with the groupoid structure: what is called a multiplicative polarization. Such polarizations induce a groupoid fibration from the original symplectic groupoid to a quotient groupoid (which is not Lie but only topological) and a cocycle on this quotient groupoid coming from the symplectic structure. The (twisted by the cocycle) groupoid $C^*$ algebra thus resulting is the outcome of the groupoid quantization procedure.

Different choice of polarizations may result in quotient groupoids much differing one from the other and in the quotient cocylce being trivial or not. The typical example is an invariant symplectic structure $\omega_\theta$ ($\theta\not\in \mathbb Q$) on the torus which, depending on the choice of polarization, may be either quantized by the irrational rotation algebra (groupoid $\mathbb Z\ltimes_\theta \mathbb S^1$ with trivial cocycle) or by a trivial groupoid based on $\mathbb R^2$ (0-isotropy) but with non trivial cocycle depending on $\theta$. That the two are isomorphic may be considered a quantization does not depend on polarization type of result. The relation between the two is a partial Fourier transform applied to one of the two variables, in a way, and in general whenever the symplectic groupoid can be identified with a cotangent bundle as a manifold (which happens for a wide class of Poisson manifolds) different choices of polarization defines some Fourier-type relation between different (twisted) groupoid $C^*$-algebras.

One of the subtle points here is that in some sense groupoid $C^*$-algebras behave in a contravariant functorial way with respect to the base and in a covariant functorial way with respect to the isotropy so that all relations (like this choice of polarization) that result in some interchange of variables between base and isotropy have a quite complicated functorial description.

• Eli Hawkins, A groupoid approach to quantization, J. Symplectic Geom. 6 Number 1 (2008) 61–125. Project Euclid, arXiv:math/0612363.