# Relation between graphs and groupoid $C^*$-algebras

In the paper "Graphs, groupoids and Cuntz-Krieger algebras" by Kumijan, Pask, Raeburn, Renault it was shown (if I understand it correctly) that whenever $G$ is a row-finite directed graph with no sinks then the corresponding graph $C^*$-algebra is isomorphic to the groupoid $C^*$-algebra w.r. to a suitable groupoid having the space of infinite paths as its unit space.

My question is whether it is possible to remove the limitation of having no sinks; I know that a typical way of "desingularizing" this sort of graphs is to add an infinite tail to any sink. Does it remain true, then, that the graph $C^*$-algebra is isomorphic to the groupoid $C^*$-algebra associated to the path groupoid?

On a more general note: what is known about functoriality of these three procedures?

\begin{aligned} \textrm{graph}&\longrightarrow C^*\textrm{-algebra}\\ \textrm{groupoid}&\longrightarrow C^*\textrm{-algebra}\\ \textrm{graph}&\longrightarrow\textrm{groupoid}\end{aligned}

• Dear Nicola Ciccoli, hoping to be helpful, I added a link to the quoted paper, with just some other minor edits.
– agt
Nov 2, 2013 at 12:13
• The only assumption I see in 4.1 and 4.2 of the paper you linked to is row-finiteness. Could you please clarify where the assumption of no sinks comes in? Nov 2, 2013 at 12:51
• @Giuseppe very nice of you, indeed thanks. Nov 2, 2013 at 13:54
• @Rasmus At the begininning of section 2 it is stated "To avoid pathological cases we will assume that [...] every vertex admits at least one edge". Since the following groupoid construction is based on this assumption I assumed that the assumption should be considered to hold in all what follows. This idea is confirmed in comments I read on other papers on the subject. Nov 2, 2013 at 13:57
• @Nicola I see. Thank you for the reference! Nov 4, 2013 at 8:29

The process of adding tails to the graph does not preserve the isomorphism class of the graph $C^*$-algebra. One simple example is to allow $E$ the be the graph consisting of one vertex and no edge. Then working from the rules for graph algebras (for graphs with sinks), you obtain that $C^*(E) \cong \mathbb{C}$. But $\tilde{E}$ will be an infinite chain of edges, and you can check that $C^*(\tilde{E})$ will be isomorphic to the compact operators on $\ell^2$, $C^*(\tilde{E}) \cong \mathcal{K}(\ell^2)$.
However, you don't lose everything by desingularizing. Desingularizing is useful because we will have $C^*(E) \sim_{\text{M. E.}} C^*(\tilde{E})$, where $\sim_{\text{M. E.}}$ denotes Morita equivalence. This is an equivalence relation between $C^*$-algebras which is weaker than isomorphism but still strong enough to ensure that many important $C^*$-algebraic properties of $C^*(E)$ are shared by $C^*(\tilde{E})$. For example, the lattice of closed two-sided ideals of $C^*(E)$ will be isomorphic to that of $C^*(\tilde{E})$ (so in particular they either are both simple as $C^*$-algebras or neither is). Many other key properties are preserved under Morita equivalence, such as being AF or purely infinite, having real rank zero, and so on.
A good place to look at this is the paper "The $C^*$-algebras of arbitrary graphs" by Drinen and Tomforde (https://projecteuclid.org/euclid.rmjm/1181069770). They develop the machinery of desingularization in some detail. In fact, their construction goes beyond removing sinks, so that infinite emitters can be removed in similar fashion from a graph without disturbing the Morita equivalence class of the affiliated $C^*$-algebra. Another route is via inverse semigroups, which Paterson carried out in this paper: http://www.theta.ro/jot/archive/2002-048-003/2002-048-003-012.pdf. Note that this other paper only addresses the infinite emitter problem, all graphs are assumed sinkless. These desingularization and inverse semigroup constructions have various generalizations to more general families of $C^*$-algebras: Cuntz-Pimsner algebras, $C^*$-algebras of higher-rank graphs, and so on.
To address your more general note: the graph $C^*$-algebra construction is not usually studied from a functorial approach. This is because in order to obtain a meaningful functor from Graphs to $C^*$-algebras, you have to be very strict as to the sort of morphisms you allow in the Graphs category. For example, suupose that $m: E \to F$ is a morphism of graphs (not saying what that means right now, but certainly it should carry vertices to vertices and edges to edges in such a way that it respects the range and source maps). Then (assuming we want the graph algebra construction to be covariant) we should get some $C^*$-algebra homomorphism $\phi=\phi_m: C^*(E) \to C^*(F)$. You might start by saying that $\phi(s_e) = t_{m(e)}$ for an edge $e \in E^1$ and $\phi(p_v) = q_{m(v)}$ for a vertex $v \in E^0$. Here $C^*(E)$ is generated by $\{s_e,p_v\}_{e \in E^1,v \in E^0}$ and $C^*(F)$ is generated by $\{t_f, q_w \}_{f \in F^1, w \in F^0}$, i.e. the universal Cuntz-Krieger families. This can cause issues if the morphism $m$ is not injective on edges. For if $e,e'$ are two distinct edges of $E$, then $s_e^* s_{e'}^* = 0$ by the definition of a graph algebra. If $m(e)=m(e')$, however, our ingenuous homomorphism will cause a problem $$0 = \phi(0) = \phi(s_e^* s_{e'}) = \phi(s_e)^* \phi(s_{e'})= t_{m(e)}^* t_{m(e')} = t_{m(e)}^* t_{m(e)} = q_{s(m(e))}$$ Since the universal generators for a graph algebra are non-zero, this causes a problem. You might start working through what the requirements are that a graph morphism must satisfy in order to define a $C^*$-algebra homomorphism in this naive fashion.
The second construction is better behaved. Of course there is a notion of isomorphism for groupoids, and the $C^*$-algebra respects it. Moreover, there is a more general notion of equivalence for groupoids. Two (sufficiently nice) topological groupoids are equivalent if they each act on a common space $Z$ in a compatible fashion. For details, see this useful paper of Muhly, "Equivalence and isomorphism for groupoid $C^*$-algebras" (http://www.theta.ro/jot/archive/1987-017-001/1987-017-001-001.pdf). The main theorem of this paper states that if two groupoids equivalent in the above sense, then their affiliated groupoid $C^*$-algebras are Morita equivalent. I am not sure if people have spent a lot of time looking at the category of topological groupoids in and of itself, or functorial properties of the groupoid $C^*$-algebra construction. One reason for this is that, most of the time, you don't work very often with morphisms between groupoids. They arise, but they are not central in many theorems. It might be that there is a category of groupoids and "one-sided equivalences" and that the equivalence relation defined above is the isomorphism relation in this category. This would line up better with the categorical construction for Cuntz-Pimsner algebras.
For the last direction you asked about, the path groupoid construction is simple enough that you can read most interesting properties directly from the parent graph. That is, the structure of the path groupoid as a topological groupoid (is it Cartan, proper, transitive, etc.) is generally straightforward to determine from the structure of the graph. If you are interested in functoriality you'd again have to decide what kind of graph morphisms you want to use and what kind of "groupoid morphisms" you'd want to use. If you have a graph morphism $m$ (of the vague kind considered in the $C^*$-algebra question), then it will certainly define a map on the infinite path spaces respecting the shift maps, so that it should it will define an algebraic groupoid homomorphism (a map $\phi: G_E \to G_F$ mapping composable pairs to composable pairs and acting like a group homomorphism, $\phi(gg')=\phi(g)\phi(g')$). The issue that could arise is that while I believe it is continuous, $\phi$ might fail to be proper (I think, this is speculative). You might look also at this paper of Putnam (http://www.math.uvic.ca/faculty/putnam/r/9901main.pdf) on Smale spaces for more ideas about functoriality for a related groupoid construction. There the desired morphism of groupoids is an injective groupoid homomorphism which is a homeomorphism onto an open subgroupoid of the codomain. But I'm far away from understanding this stuff.