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."^[1][2] 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.^[3] 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.^[3][4] The alternate convention, which is used in Raeburn's CBMS book on Graph Algebras,^[5] 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^[6]
• 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).^[7] 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,^[8] 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.

See also

• k-graph C*-algebra
• Leavitt path algebra

Notes

1. ^ 2004 NSF-CBMS Conference on Graph Algebras [1]
2. ^ NSF Award [2]
3. ^ Cuntz-Krieger algebras of directed graphs, Alex Kumjian, David Pask, and Iain Raeburn, Pacific J. Math. 184 (1998), no. 1, 161–174.
4. ^ The C*-algebras of row-finite graphs, Teresa Bates, David Pask, Iain Raeburn, and Wojciech Szymański, New York J. Math. 6 (2000), 307–324.
5. ^ Graph algebras, Iain Raeburn, CBMS Regional Conference Series in Mathematics, 103. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2005. vi+113 pp. ISBN 0-8218-3660-9
6. ^ Viewing AF-algebras as graph algebras, Doug Drinen, Proc. Amer. Math. Soc., 128 (2000), pp. 1991–2000.
7. ^ The C*-algebras of arbitrary graphs, Doug Drinen and Mark Tomforde, Rocky Mountain J. Math. 35 (2005), no. 1, 105–135.
8. ^ Chapter 5 of Graph algebras, Iain Raeburn, CBMS Regional Conference Series in Mathematics, 103. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2005. vi+113 pp. ISBN 0-8218-3660-9