Graph C*-algebra

In mathematics, a graph C*-algebra is a universal C*-algebra constructed from a directed graph. Graph C*-algebras are direct generalizations of the Cuntz algebras and Cuntz-Krieger algebras, but the class of graph C*-algebras has been shown to also include several other widely studied classes of C*-algebras. As a result, graph C*-algebras provide a common framework for investigating many well-known classes of C*-algebras that were previously studied independently. Among other benefits, this provides a context in which one can formulate theorems that apply simultaneously to all of these subclasses and contain specific results for each subclass as special cases.

Although graph C*-algebras include numerous examples, they provide a class of C*-algebras that are surprisingly amenable to study and much more manageable than general C*-algebras. The graph not only determines the associated C*-algebra by specifying relations for generators, it also provides a useful tool for describing and visualizing properties of the C*-algebra. This visual quality has led to graph C*-algebras being referred to as "operator algebras we can see." Another advantage of graph C*-algebras is that much of their structure and many of their invariants can be readily computed. Using data coming from the graph, one can determine whether the associated C*-algebra has particular properties, describe the lattice of ideals, and compute K-theoretic invariants.

Graph terminology
The terminology for graphs used by C*-algebraists differs slightly from that used by graph theorists. The term graph is typically taken to mean a directed graph $$E=(E^0, E^1, r, s)$$ consisting of a countable set of vertices $$E^0$$, a countable set of edges $$E^1$$, and maps $$r, s : E^1 \rightarrow E^0$$ identifying the range and source of each edge, respectively. A vertex $$v \in E^0$$ is called a sink when $$s^{-1}(v) = \emptyset$$; i.e., there are no edges in $$E$$ with source $$v$$. A vertex $$v \in E^0$$ is called an infinite emitter when $$s^{-1}(v)$$ is infinite; i.e., there are infinitely many edges in $$E$$ with source $$v$$. A vertex is called a singular vertex if it is either a sink or an infinite emitter, and a vertex is called a regular vertex if it is not a singular vertex. Note that a vertex $$v$$ is regular if and only if the number of edges in $$E$$ with source $$v$$ is finite and nonzero. A graph is called row-finite if it has no infinite emitters; i.e., if every vertex is either a regular vertex or a sink.

A path is a finite sequence of edges $$e_1 e_2 \ldots e_n$$ with $$r(e_i) = s(e_{i+1})$$ for all $$1 \leq i \leq n-1$$. An infinite path is a countably infinite sequence of edges $$e_1 e_2 \ldots $$ with $$r(e_i) = s(e_{i+1})$$ for all $$i \geq 1$$. A cycle is a path $$e_1 e_2 \ldots e_n$$ with $$r(e_n) = s(e_1)$$, and an exit for a cycle $$e_1 e_2 \ldots e_n$$ is an edge $$f \in E^1$$ such that $$s(f) = s(e_i)$$ and $$f \neq e_i$$ for some $$1 \leq i \leq n$$. A cycle $$e_1 e_2 \ldots e_n$$ is called a simple cycle if $$s(e_i) \neq s(e_1)$$ for all $$2 \leq i \leq n$$.

The following are two important graph conditions that arise in the study of graph C*-algebras.

Condition (L): Every cycle in the graph has an exit.

Condition (K): There is no vertex in the graph that is on exactly one simple cycle. That is, a graph satisfies Condition (K) if and only if each vertex in the graph is either on no cycles or on two or more simple cycles.

The Cuntz-Krieger Relations and the universal property
A Cuntz-Krieger $$E$$-family is a collection $$\left\{ s_e, p_v : e \in E^1, v \in E^0 \right\}$$ in a C*-algebra such that the elements of $$\left\{ s_e : e \in E^1 \right\}$$ are partial isometries with mutually orthogonal ranges, the elements of $$\left\{ p_v : v \in E^0 \right\}$$ are mutually orthogonal projections, and the following three relations (called the Cuntz-Krieger relations) are satisfied:


 * 1) (CK1) $$s_e^*s_e = p_{r(e)}$$ for all $$e \in E^1$$,
 * 2) (CK2) $$p_v = \sum_{s(e)=v} s_e s_e^*$$ whenever $$v$$ is a regular vertex, and
 * 3) (CK3) $$s_e s_e^* \le p_{s(e)}$$ for all $$e \in E^1$$.

The graph C*-algebra corresponding to $$E$$, denoted by $$C^*(E)$$, is defined to be the C*-algebra generated by a Cuntz-Krieger $$E$$-family that is universal in the sense that whenever $$\left\{ t_e, q_v : e \in E^1, v \in E^0 \right\}$$ is a Cuntz-Krieger $$E$$-family in a C*-algebra $$A$$ there exists a $*$-homomorphism $$\phi : C^*(E) \to A$$ with $$\phi(s_e) = t_e$$ for all $$e \in E^1$$ and $$\phi(p_v)=q_v$$ for all $$v \in E^0$$. Existence of $$C^*(E)$$ for any graph $$E$$ was established by Kumjian, Pask, and Raeburn. Uniqueness of $$C^*(E)$$ (up to $*$-isomorphism) follows directly from the universal property.

Edge Direction Convention
It is important to be aware that there are competing conventions regarding the "direction of the edges" in the Cuntz-Krieger relations. Throughout this article, and in the way that the relations are stated above, we use the convention first established in the seminal papers on graph C*-algebras. The alternate convention, which is used in Raeburn's CBMS book on Graph Algebras, interchanges the roles of the range map $$r$$ and the source map $$s$$ in the Cuntz-Krieger relations. The effect of this change is that the C*-algebra of a graph for one convention is equal to the C*-algebra of the graph with the edges reversed when using the other convention.

Row-Finite Graphs
In the Cuntz-Krieger relations, (CK2) is imposed only on regular vertices. Moreover, if $$v \in E^0$$ is a regular vertex, then (CK2) implies that (CK3) holds at $$v$$. Furthermore, if $$v \in E^0$$ is a sink, then (CK3) vacuously holds at $$v$$. Thus, if $$E$$ is a row-finite graph, the relation (CK3) is superfluous and a collection $$\left\{ s_e, p_v : e \in E^1, v \in E^0 \right\}$$ of partial isometries with mutually orthogonal ranges and mutually orthogonal projections is a Cuntz-Krieger $$E$$-family if and only if the relation in (CK1) holds at all edges in $$E$$ and the relation in (CK2) holds at all vertices in $$E$$ that are not sinks. The fact that the Cuntz-Krieger relations take a simpler form for row-finite graphs has technical consequences for many results in the subject. Not only are results easier to prove in the row-finite case, but also the statements of theorems are simplified when describing C*-algebras of row-finite graphs. Historically, much of the early work on graph C*-algebras was done exclusively in the row-finite case. Even in modern work, where infinite emitters are allowed and C*-algebras of general graphs are considered, it is common to state the row-finite case of a theorem separately or as a corollary, since results are often more intuitive and transparent in this situation.

Examples
The graph C*-algebra has been computed for many graphs. Conversely, for certain classes of C*-algebras it has been shown how to construct a graph whose C*-algebra is $*$-isomorphic or Morita equivalent to a given C*-algebra of that class.

The following table shows a number of directed graphs and their C*-algebras. We use the convention that a double arrow drawn from one vertex to another and labeled $$\infty$$ indicates that there are a countably infinite number of edges from the first vertex to the second.

The class of graph C*-algebras has been shown to contain various classes of C*-algebras. The C*-algebras in each of the following classes may be realized as graph C*-algebras up to $*$-isomorphism:


 * Cuntz algebras
 * Cuntz-Krieger algebras
 * finite-dimensional C*-algebras
 * stable AF algebras

The C*-algebras in each of the following classes may be realized as graph C*-algebras up to Morita equivalence:


 * AF algebras
 * Kirchberg algebras with free K1-group

Correspondence between graph and C*-algebraic properties
One remarkable aspect of graph C*-algebras is that the graph $$E$$ not only describes the relations for the generators of $$C^*(E)$$, but also various graph-theoretic properties of $$E$$ can be shown to be equivalent to C*-algebraic properties of $$C^*(E)$$. Indeed, much of the study of graph C*-algebras is concerned with developing a lexicon for the correspondence between these properties, and establishing theorems of the form "The graph $$E$$ has a certain graph-theoretic property if and only if the C*-algebra $$C^*(E)$$ has a corresponding C*-algebraic property." The following table provides a short list of some of the more well-known equivalences.

The gauge action
The universal property produces a natural action of the circle group $$\mathbb{T} := \{ z \in \Complex : |z| = 1 \}$$ on $$C^*(E)$$ as follows: If $$\left\{ s_e, p_v : e \in E^1, v \in E^0 \right\}$$ is a universal Cuntz-Krieger $$E$$-family, then for any unimodular complex number $$z \in \mathbb{T}$$, the collection $$\left\{ zs_e, p_v : e \in E^1, v \in E^0 \right\}$$ is a Cuntz-Krieger $$E$$-family, and the universal property of $$C^*(E)$$ implies there exists a $*$-homomorphism $$\gamma_z : C^*(E) \to C^*(E)$$ with $$\gamma_z (s_e) = zs_e$$ for all $$e \in E^1$$ and $$\gamma_z(p_v) = p_v$$ for all $$v \in E^0$$. For each $$z \in \mathbb{T}$$ the $*$-homomorphism $$\gamma_\overline{z}$$ is an inverse for $$\gamma_z$$, and thus $$\gamma_z$$ is an automorphism. This yields a strongly continuous action $$\gamma: \mathbb{T} \to \operatorname{Aut} C^*(E)$$ by defining $$\gamma(z) := \gamma_z$$. The gauge action $$\gamma$$ is sometimes called the canonical gauge action on $$C^*(E)$$. It is important to note that the canonical gauge action depends on the choice of the generating Cuntz-Krieger $$E$$-family $$\left\{ s_e, p_v : e \in E^1, v \in E^0 \right\}$$. The canonical gauge action is a fundamental tool in the study of $$C^*(E)$$. It appears in statements of theorems, and it is also used behind the scenes as a technical device in proofs.

The uniqueness theorems
There are two well-known uniqueness theorems for graph C*-algebras: the gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem. The uniqueness theorems are fundamental results in the study of graph C*-algebras, and they serve as cornerstones of the theory. Each provides sufficient conditions for a $*$-homomorphism from $$C^*(E)$$ into a C*-algebra to be injective. Consequently, the uniqueness theorems can be used to determine when a C*-algebra generated by a Cuntz-Krieger $$E$$-family is isomorphic to $$C^*(E)$$; in particular, if $$A$$ is a C*-algebra generated by a Cuntz-Krieger $$E$$-family, the universal property of $$C^*(E)$$ produces a surjective $*$-homomorphism $$\phi : C^*(E) \to A$$, and the uniqueness theorems each give conditions under which $$\phi$$ is injective, and hence an isomorphism. Formal statements of the uniqueness theorems are as follows:

The Gauge-Invariant Uniqueness Theorem: Let $$E$$ be a graph, and let $$C^*(E)$$ be the associated graph C*-algebra. If $$A$$ is a C*-algebra and $$\phi : C^*(E) \to A$$ is a $*$-homomorphism satisfying the following two conditions:


 * 1) there exists a gauge action $$\beta : \mathbb{T} \to \operatorname{Aut} A$$ such that $$\phi \circ \beta_z = \gamma_z \circ \phi$$ for all $$z \in \mathbb{T}$$, where $$\gamma$$ denotes the canonical gauge action on $$C^*(E)$$, and
 * 2) $$\phi(p_v) \neq 0$$ for all $$v \in E^0$$,

then $$\phi$$ is injective.

The Cuntz-Krieger Uniqueness Theorem: Let $$E$$ be a graph satisfying Condition (L), and let $$C^*(E)$$ be the associated graph C*-algebra. If $$A$$ is a C*-algebra and $$\phi : C^*(E) \to A$$ is a $*$-homomorphism with $$\phi(p_v) \neq 0$$ for all $$v \in E^0$$, then $$\phi$$ is injective.

The gauge-invariant uniqueness theorem implies that if $$\left\{ s_e, p_v : e \in E^1, v \in E^0 \right\}$$ is a Cuntz-Krieger $$E$$-family with nonzero projections and there exists a gauge action $$\beta$$ with $$\beta_z (p_v) = p_v$$ and $$\beta_z (s_e) = zs_e$$ for all $$v \in E^0$$, $$e \in E^1$$, and $$z \in \mathbb{T}$$, then $$\{ s_e, p_v : e \in E^1, v \in E^0 \}$$ generates a C*-algebra isomorphic to $$C^*(E)$$. The Cuntz-Krieger uniqueness theorem shows that when the graph satisfies Condition (L) the existence of the gauge action is unnecessary; if a graph $$E$$ satisfies Condition (L), then any Cuntz-Krieger $$E$$-family with nonzero projections generates a C*-algebra isomorphic to $$C^*(E)$$.

Ideal structure
The ideal structure of $$C^*(E)$$ can be determined from $$E$$. A subset of vertices $$H \subseteq E^0 $$ is called hereditary if for all $$e \in E^1$$, $$s(e) \in H$$ implies $$r(e) \in H$$. A hereditary subset $$H$$ is called saturated if whenever $$v$$ is a regular vertex with $$\{r(e): e \in E^0, s(e) = v\} \subseteq H$$, then $$v \in H$$. The saturated hereditary subsets of $$E$$ are partially ordered by inclusion, and they form a lattice with meet $$H_1 \wedge H_2 := H_1 \cap H_2$$ and join $$H_1 \vee H_2$$ defined to be the smallest saturated hereditary subset containing $$H_1 \cup H_2$$.

If $$H$$ is a saturated hereditary subset, $$I_H$$ is defined to be closed two-sided ideal in $$C^*(E)$$ generated by $$\{ p_v : v \in H \}$$. A closed two-sided ideal $$I$$ of $$C^*(E)$$ is called gauge invariant if $$\gamma_z(a) \in C^*(E)$$ for all $$a \in I$$ and $$z \in \mathbb{T}$$. The gauge-invariant ideals are partially ordered by inclusion and form a lattice with meet $$I_1 \wedge I_2 := I_1 \cap I_2$$ and joint $$I_1 \vee I_2$$ defined to be the ideal generated by $$I_1 \cup I_2$$. For any saturated hereditary subset $$H$$, the ideal $$I_H$$ is gauge invariant.

The following theorem shows that gauge-invariant ideals correspond to saturated hereditary subsets.

Theorem: Let $$E$$ be a row-finite graph. Then the following hold:
 * 1) The function $$H \mapsto I_H$$ is a lattice isomorphism from the lattice of saturated hereditary subsets of $$E$$ onto the lattice of gauge-invariant ideals of $$C^*(E)$$ with inverse given by $$I \mapsto \left\{ v \in E^0 : p_v \in I \right\}$$.
 * 2) For any saturated hereditary subset $$H$$, the quotient $$C^*(E)/I_H$$ is $$*$$-isomorphic to $$C^*(E \setminus H)$$, where $$E \setminus H$$ is the subgraph of $$E$$ with vertex set $$(E \setminus H)^0 := E^0 \setminus H$$ and edge set $$(E \setminus H)^1 := E^1 \setminus r^{-1}(H)$$.
 * 3) For any saturated hereditary subset $$H$$, the ideal $$I_H$$ is Morita equivalent to $$C^*(E_H)$$, where $$E_H$$ is the subgraph of $$E$$ with vertex set $$E_H^0 := H$$ and edge set $$E_H^1 := s^{-1}(H)$$.
 * 4) If $$E$$ satisfies Condition (K), then every ideal of $$C^*(E)$$ is gauge invariant, and the ideals of $$C^*(E)$$ are in one-to-one correspondence with the saturated hereditary subsets of $$E$$.

Desingularization
The Drinen-Tomforde Desingularization, often simply called desingularization, is a technique used to extend results for C*-algebras of row-finite graphs to C*-algebras of countable graphs. If $$E$$ is a graph, a desingularization of $$E$$ is a row-finite graph $$F$$ such that $$C^*(E)$$ is Morita equivalent to $$C^*(F)$$. Drinen and Tomforde described a method for constructing a desingularization from any countable graph: If $$E$$ is a countable graph, then for each vertex $$v_0$$ that emits an infinite number of edges, one first chooses a listing of the outgoing edges as $$s^{-1}(v_0) = \{ e_0, e_1, e_2, \ldots \}$$, one next attaches a tail of the form



to $$E$$ at $$v_0$$, and finally one erases the edges $$e_0, e_1, e_2, \ldots$$ from the graph and redistributes each along the tail by drawing a new edge $$f_i$$ from $$v_i$$ to $$r(e_i)$$ for each $$i = 0, 1, 2, \ldots$$.

Here are some examples of this construction. For the first example, note that if $$E$$ is the graph



then a desingularization $$F$$ is given by the graph



For the second example, suppose $$E$$ is the $$\mathcal{O}_\infty$$ graph with one vertex and a countably infinite number of edges (each beginning and ending at this vertex). Then a desingularization $$F$$ is given by the graph



Desingularization has become a standard tool in the theory of graph C*-algebras, and it can simplify proofs of results by allowing one to first prove the result in the (typically much easier) row-finite case, and then extend the result to countable graphs via desingularization, often with little additional effort.

The technique of desingularization may not work for graphs containing a vertex that emits an uncountable number of edges. However, in the study of C*-algebras it is common to restrict attention to separable C*-algebras. Since a graph C*-algebra $$C^*(E)$$ is separable precisely when the graph $$E$$ is countable, much of the theory of graph C*-algebras has focused on countable graphs.

K-theory
The K-groups of a graph C*-algebra may be computed entirely in terms of information coming from the graph. If $$E$$ is a row-finite graph, the vertex matrix of $$E$$ is the $$E^0 \!\times\! E^0$$ matrix $$A_E$$ with entry $$A_E(v,w)$$ defined to be the number of edges in $$E$$ from $$v$$ to $$w$$. Since $$E$$ is row-finite, $$A_E$$ has entries in $$\mathbb{N} \cup \{ 0 \}$$ and each row of $$A_E$$ has only finitely many nonzero entries. (In fact, this is where the term "row-finite" comes from.) Consequently, each column of the transpose $$A_E^t$$ contains only finitely many nonzero entries, and we obtain a map $A_E^t : \bigoplus_{E^0} \mathbb{Z} \to \bigoplus_{E^0} \mathbb{Z}$  given by left multiplication. Likewise, if $$I$$ denotes the $$E^0 \!\times\! E^0$$ identity matrix, then $I - A_E^t : \bigoplus_{E^0} \mathbb{Z} \to \bigoplus_{E^0} \mathbb{Z}$ provides a map given by left multiplication.

Theorem: Let $$E$$ be a row-finite graph with no sinks, and let $$A_E$$ denote the vertex matrix of $$E$$. Then $$I - A_E^t : \bigoplus_{E^0} \mathbb{Z} \to \bigoplus_{E^0} \mathbb{Z}$$ gives a well-defined map by left multiplication. Furthermore, $$K_0(C^*(E)) \cong \operatorname{coker} (I- A_E^t) \quad\text{ and }\quad K_1(C^*(E)) \cong \ker (I - A_E^t).$$ In addition, if $$C^*(E)$$ is unital (or, equivalently, $$E^0$$ is finite), then the isomorphism $$K_0(C^*(E)) \cong \operatorname{coker} (I- A_E^t)$$ takes the class of the unit in $$K_0(C^*(E))$$ to the class of the vector $$(1, 1, \ldots, 1)$$ in $$\operatorname{coker} (I- A_E^t)$$.

Since $$K_1(C^*(E))$$ is isomorphic to a subgroup of the free group $\bigoplus_{E^0} \mathbb{Z}$, we may conclude that $$K_1(C^*(E))$$ is a free group. It can be shown that in the general case (i.e., when $$E$$ is allowed to contain sinks or infinite emitters) that $$K_1(C^*(E))$$ remains a free group. This allows one to produce examples of C*-algebras that are not graph C*-algebras: Any C*-algebra with a non-free K1-group is not Morita equivalent (and hence not isomorphic) to a graph C*-algebra.