Graph C*-algebra explained

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

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} → 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 ≠ 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)} ≠ 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 ^{*s} _{e} = p _{r(e)} for all e \in E ^{1} ,

1. (CK2)

p _{v} = \sum _{s(e)=v} s _{e} s ^{*} _{e} whenever v is a regular vertex, and

1. (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 \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) 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.^[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

inf ty 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 :

• 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 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 property." The following table provides a short list of some of the more well-known equivalences.

Property of E	Property of C ^{*(E)}
E is a finite graph and contains no cycles.	C ^{*(E)} is finite-dimensional.
The vertex set E ^{0} is finite.	C ^{*(E)} is unital (i.e., C ^{*(E)} contains a multiplicative identity).
E has no cycles.	C ^{*(E)} is an AF algebra.
E satisfies the following three properties: Condition (L), for each vertex v and each infinite path \alpha there exists a directed path from v to a vertex on \alpha , and for each vertex v and each singular vertex w there exists a directed path from v to w	C ^{*(E)} is simple.
E satisfies the following three properties: Condition (L), for each vertex v in E there is a path from v to a cycle.	Every hereditary subalgebra of C ^{*(E)} contains an infinite projection. (When C ^{*(E)} is simple this is equivalent to C ^{*(E)} being purely infinite.)

The gauge action

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 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 \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 T the \gamma _{\overline{z}} is an inverse for \gamma _{z} , and thus \gamma _{z} is an automorphism. This yields a strongly continuous action \gamma: 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 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 \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 satisfying the following two conditions:

1. there exists a gauge action

\beta : T \to \operatorname{Aut} A such that \phi \circ \beta _{z} = \gamma _{z} \circ \phi for all z \in T , where \gamma denotes the canonical gauge action on C ^{*(E)} , and\phi(p _{v)} ≠ 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 with \phi(p _{v)} ≠ 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 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 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\} .

1. 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) .

1. 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 ^{0} _{H} : = H and edge set E ^{1} _{H} : = s ^{-1} (H) .

1. 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 l{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} x 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 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 ^{t} _{E} contains only finitely many nonzero entries, and we obtain a map $A\_E^t\; :\; \backslash bigoplus\_\; \backslash mathbb\; \backslash to\; \backslash bigoplus\_\; \backslash mathbb$ given by left multiplication. Likewise, if I denotes the E ^{0} x E ^{0} identity matrix, then $I\; -\; A\_E^t\; :\; \backslash bigoplus\_\; \backslash mathbb\; \backslash to\; \backslash bigoplus\_\; \backslash mathbb$ 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 $$I\; -\; A\_E^t\; :\; \backslash bigoplus\_\; \backslash mathbb\; \backslash to\; \backslash bigoplus\_\; \backslash mathbb$$ gives a well-defined map by left multiplication. Furthermore, $$K\_0(C^*(E))\; \backslash cong\; \backslash operatorname\; (I-\; A\_E^t)\; \backslash quad\backslash text\backslash quad\; K\_1(C^*(E))\; \backslash cong\; \backslash ker\; (I\; -\; A\_E^t).$$ In addition, if C ^{*(E)} is unital (or, equivalently, E ^{0} is finite), then the isomorphism K ^{*(E))} _{0 (C} \cong \operatorname{coker} (I - A ^{t)} _{E} takes the class of the unit in K ^{*(E))} _{0 (C} to the class of the vector ( 1 , 1 , \ldots, 1 ) in \operatorname{coker} (I - A ^{t)} _{E} .

Since

K ^{*(E))} _{1 (C} is isomorphic to a subgroup of the free group $\backslash bigoplus\_\; \backslash mathbb$, we may conclude that K ^{*(E))} _{1 (C} 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 ^{*(E))} _{1 (C} 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.

Notes and References

1. 2004 NSF-CBMS Conference on Graph Algebras https://www.math.uh.edu/~tomforde/CBMS2004/cbms.html
2. NSF Award https://www.nsf.gov/awardsearch/showAward?AWD_ID=0332279&HistoricalAwards=false
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.
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.