View source for 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 algebra]]s and Cuntz-Krieger algebras, but the class of graph C*-algebras has been shown to also include several other widely studied classes of [[C*-algebra]]s.  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 [[theorem]]s 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 algebra]]s we can see."<ref>2004 NSF-CBMS Conference on Graph Algebras [https://www.math.uh.edu/~tomforde/CBMS2004/cbms.html]</ref><ref>NSF Award [https://www.nsf.gov/awardsearch/showAward?AWD_ID=0332279&HistoricalAwards=false]</ref>  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-theory|K-theoretic]] invariants.

==Graph terminology==
The terminology for graphs used by C*-algebraists differs slightly from that used by [[graph theory|graph theorists]].  The term '''graph''' is typically taken to mean a [[directed graph]] <math>E=(E^0, E^1, r, s)</math> consisting of a countable set of vertices <math>E^0</math>, a countable set of edges <math>E^1</math>, and maps <math>r, s : E^1 \rightarrow E^0</math> identifying the range and source of each edge, respectively.  A vertex <math>v \in E^0</math> is called a '''sink''' when <math>s^{-1}(v) = \emptyset</math>; i.e., there are no edges in <math>E</math> with source <math>v</math>.  A vertex <math>v \in E^0</math> is called an '''infinite emitter''' when <math>s^{-1}(v)</math> is infinite; i.e., there are infinitely many edges in <math>E</math> with source <math>v</math>.  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 <math>v</math> is regular [[if and only if]] the number of edges in <math>E</math> with source <math>v</math> 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 <math>e_1 e_2 \ldots e_n</math> with <math>r(e_i) = s(e_{i+1})</math> for all <math>1 \leq i \leq n-1</math>.  An '''infinite path''' is a countably infinite sequence of edges <math>e_1 e_2 \ldots </math> with <math>r(e_i) = s(e_{i+1})</math> for all <math>i \geq 1</math>.  A '''cycle''' is a path <math>e_1 e_2 \ldots e_n</math> with <math>r(e_n) = s(e_1)</math>, and an '''exit''' for a cycle <math>e_1 e_2 \ldots e_n</math> is an edge <math>f \in E^1</math> such that <math>s(f) = s(e_i)</math> and <math>f \neq e_i</math> for some <math>1 \leq i \leq n</math>.  A cycle <math>e_1 e_2 \ldots e_n</math> is called a '''simple cycle''' if <math>s(e_i) \neq s(e_1)</math> for all <math>2 \leq i \leq n</math>.

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 <math>E</math>-family''' is a collection <math>\left\{ s_e, p_v : e \in E^1, v \in E^0 \right\}</math> in a C*-algebra such that the elements of <math>\left\{ s_e : e \in E^1 \right\}</math> are [[partial isometry|partial isometries]] with mutually orthogonal ranges, the elements of <math>\left\{ p_v : v \in E^0 \right\}</math> are mutually orthogonal projections, and the following three relations (called the '''Cuntz-Krieger relations''') are satisfied:

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

The graph C*-algebra corresponding to <math>E</math>, denoted by <math>C^*(E)</math>, is defined to be the C*-algebra generated by a Cuntz-Krieger <math>E</math>-family that is '''universal''' in the sense that whenever <math>\left\{ t_e, q_v : e \in E^1, v \in E^0 \right\}</math> is a Cuntz-Krieger <math>E</math>-family in a C*-algebra <math>A</math> there exists a {{nowrap|[[C*-algebra#Abstract_characterization|<math>*</math>-homomorphism]]}} <math>\phi : C^*(E) \to A</math> with <math>\phi(s_e) = t_e</math> for all <math>e \in E^1</math> and <math>\phi(p_v)=q_v</math> for all <math>v \in E^0</math>.  Existence of <math>C^*(E)</math> for any graph <math>E</math> was established by Kumjian, Pask, and Raeburn.<ref name="KPR">Cuntz-Krieger algebras of directed graphs, Alex Kumjian, David Pask, and Iain Raeburn, Pacific J. Math. 184 (1998), no. 1, 161–174.</ref>  Uniqueness of <math>C^*(E)</math> (up to {{nowrap|<math>*</math>-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.<ref name="KPR" /><ref>The C*-algebras of row-finite graphs, Teresa Bates, David Pask, Iain Raeburn, and Wojciech Szymański, New York J. Math. 6 (2000), 307–324.</ref>  The alternate convention, which is used in Raeburn's CBMS book on Graph Algebras,<ref>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}}</ref> interchanges the roles of the range map <math>r</math> and the source map <math>s</math> 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 <math>v \in E^0</math> is a regular vertex, then (CK2) implies that (CK3) holds at <math>v</math>.  Furthermore, if <math>v \in E^0</math> is a sink, then (CK3) vacuously holds at <math>v</math>. Thus, if <math>E</math> is a row-finite graph, the relation (CK3) is superfluous and a collection <math>\left\{ s_e, p_v : e \in E^1, v \in E^0 \right\}</math> of partial isometries with mutually orthogonal ranges and mutually orthogonal projections is a Cuntz-Krieger <math>E</math>-family if and only if the relation in (CK1) holds at all edges in <math>E</math> and the relation in (CK2) holds at all vertices in <math>E</math> 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 [[C*-algebra#Abstract_characterization|<math>*</math>-isomorphic]] or [[Morita equivalence|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 <math>\infty</math> indicates that there are a countably infinite number of edges from the first vertex to the second.

{| class="wikitable" style="text-align: center;"
|-
! Directed Graph <math>E</math> !! Graph C*-algebra <math>C^*(E)</math>
|-
| <math>\bullet</math> || <math>\mathbb{C}</math>, the [[complex number]]s
|-
| [[File:Graph-one-edge-one-vertex.svg|80px]] || <math>C(\mathbb{T})</math>, the complex-valued [[continuous function]]s on the [[circle group|circle]] <math>\mathbb{T}</math>
|-
| <math>v_1\longrightarrow v_2\longrightarrow\cdots\longrightarrow v_{n-1}\longrightarrow v_n</math> <!--<math>v_1 \longrightarrow v_2 \longrightarrow \dots \longrightarrow v_{n - 1} \longrightarrow v_n</math>--> || <math>M_n(\mathbb{C})</math>, the <math>n \times n</math> [[matrix (mathematics)|matrices]] with entries in <math>\mathbb{C}</math>
|-
| <math>\bullet\longrightarrow\bullet\longrightarrow\bullet\longrightarrow\cdots</math> <!--<math>\bullet \longrightarrow \bullet \longrightarrow \bullet \longrightarrow \cdots</math>--> || <math>\mathcal{K}</math>, the [[compact operator]]s on a separable infinite-dimensional [[Hilbert space]]
|-
| [[File:C-M-n-graph.svg|200px]] || <math>M_n(C(\mathbb{T}))</math>, the <math>n \times n</math> matrices with entries in <math>C(\mathbb{T})</math>
|-
| [[File:O-n-graph.jpg|130px]]  || <math>\mathcal{O}_n</math>, the [[Cuntz algebra]] generated by <math>n</math> isometries
|-
| [[File:O-infinity-graph.svg|90px]] || <math>\mathcal{O}_\infty</math>, the Cuntz algebra generated by a countably infinite number of isometries
|-
| [[File:K-unitization-graph.svg|100px]] <!--<math>\bullet \overset{\infty}{\Longrightarrow} \bullet</math>--> || <math>\mathcal{K}^1</math>, the unitization of the algebra of compact operators <math>\mathcal{K}</math>
|-
| [[File:Toeplitz-graph.svg|150px]] || <math>\mathcal{T}</math>, the [[Toeplitz algebra]]
|}


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 {{nowrap|<math>*</math>-isomorphism}}:

* [[Cuntz algebra]]s
* [[Cuntz-Krieger algebra]]s
* 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<ref>''[https://www.ams.org/proc/2000-128-07/S0002-9939-99-05286-7/S0002-9939-99-05286-7.pdf Viewing AF-algebras as graph algebras]'', Doug Drinen, Proc. Amer. Math. Soc., 128 (2000), pp. 1991–2000.</ref> 
* Kirchberg algebras with free K<sub>1</sub>-group

==Correspondence between graph and C*-algebraic properties==
One remarkable aspect of graph C*-algebras is that the graph <math>E</math> not only describes the relations for the generators of <math>C^*(E)</math>, but also  various graph-theoretic properties of <math>E</math> can be shown to be equivalent to {{nowrap|C*-algebraic}} properties of <math>C^*(E)</math>.  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 <math>E</math> has a certain graph-theoretic property if and only if the C*-algebra <math>C^*(E)</math> has a corresponding {{nowrap|C*-algebraic}} property."  The following table provides a short list of some of the more well-known equivalences.

{| class="wikitable" style="text-align: center;"
|-
! Property of <math>E</math> !! Property of <math>C^*(E)</math>
|-
| <math>E</math> is a finite graph and contains no cycles. || <math>C^*(E)</math> is finite-dimensional.
|-
| The vertex set <math>E^0</math> is finite. || <math>C^*(E)</math> is unital (i.e., <math>C^*(E)</math> contains a multiplicative identity).
|-
| <math>E</math> has no cycles. || <math>C^*(E)</math> is an AF algebra.
|-
|  style="text-align:left;" | <math>E</math> satisfies the following three properties:
# Condition (L), 
# for each vertex <math>v</math> and each infinite path <math>\alpha</math> there exists a directed path from <math>v</math> to a vertex on <math>\alpha</math>, and 
#for each vertex <math>v</math> and each singular vertex <math>w</math> there exists a directed path from <math>v</math> to <math>w</math>
|| <math>C^*(E)</math> is simple.
|-
| style="text-align:left;" | <math>E</math> satisfies the following three properties:
# Condition (L), 
# for each vertex <math>v</math> in <math>E</math> there is a path from <math>v</math> to a cycle. 
|| Every hereditary subalgebra of <math>C^*(E)</math> contains an infinite projection.  <br>(When <math>C^*(E)</math> is simple this is equivalent to <math>C^*(E)</math> being purely infinite.)
|}

==The gauge action==
The universal property produces a natural action of the [[circle group]] <math>\mathbb{T} := \{ z \in \Complex : |z| = 1 \}</math> on <math>C^*(E)</math> as follows: If <math>\left\{ s_e, p_v : e \in E^1, v \in E^0 \right\}</math> is a universal Cuntz-Krieger <math>E</math>-family, then for any unimodular complex number <math>z \in \mathbb{T}</math>, the collection <math>\left\{ zs_e, p_v : e \in E^1, v \in E^0 \right\}</math> is a Cuntz-Krieger <math>E</math>-family, and the universal property of <math>C^*(E)</math> implies there exists a {{nowrap|<math>*</math>-homomorphism}} <math>\gamma_z : C^*(E) \to C^*(E)</math> with <math>\gamma_z (s_e) = zs_e</math> for all <math>e \in E^1</math> and <math>\gamma_z(p_v) = p_v</math> for all <math>v \in E^0</math>.  For each <math>z \in \mathbb{T}</math> the {{nowrap|<math>*</math>-homomorphism}} <math>\gamma_\overline{z}</math> is an inverse for <math>\gamma_z</math>, and thus <math>\gamma_z</math> is an [[automorphism]].  This yields a strongly continuous action <math>\gamma: \mathbb{T} \to \operatorname{Aut} C^*(E)</math> by defining <math>\gamma(z) := \gamma_z</math>.  The gauge action <math>\gamma</math> is sometimes called the '''canonical gauge action''' on <math>C^*(E)</math>.  It is important to note that the canonical gauge action depends on the choice of the generating Cuntz-Krieger <math>E</math>-family <math>\left\{ s_e, p_v : e \in E^1, v \in E^0 \right\}</math>.  The canonical gauge action is a fundamental tool in the study of <math>C^*(E)</math>.  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 {{nowrap|<math>*</math>-homomorphism}} from <math>C^*(E)</math> into a C*-algebra to be [[injective function|injective]].  Consequently, the uniqueness theorems can be used to determine when a C*-algebra generated by a Cuntz-Krieger <math>E</math>-family is 
isomorphic to <math>C^*(E)</math>; in particular, if <math>A</math> is a C*-algebra generated by a Cuntz-Krieger <math>E</math>-family, the universal property of <math>C^*(E)</math> produces a [[surjective]] {{nowrap|<math>*</math>-homomorphism}} <math>\phi : C^*(E) \to A</math>, and the uniqueness theorems each give conditions under which <math>\phi</math> is injective, and hence an isomorphism.  Formal statements of the uniqueness theorems are as follows:

'''The Gauge-Invariant Uniqueness Theorem:'''  Let <math>E</math> be a graph, and let <math>C^*(E)</math> be the associated graph C*-algebra.  If <math>A</math> is a C*-algebra and <math>\phi : C^*(E) \to A</math> is a {{nowrap|<math>*</math>-homomorphism}} satisfying the following two conditions:

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

then <math>\phi</math> is injective.

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

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

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

If <math>H</math> is a saturated hereditary subset, <math>I_H</math> is defined to be closed two-sided ideal in <math>C^*(E)</math> generated by <math>\{ p_v : v \in H \}</math>.  A closed two-sided ideal <math>I</math> of <math>C^*(E)</math> is called '''gauge invariant''' if <math>\gamma_z(a) \in C^*(E)</math> for all <math>a \in I</math> and <math>z \in \mathbb{T}</math>.  The gauge-invariant ideals are partially ordered by inclusion and form a lattice with meet <math>I_1 \wedge I_2 := I_1 \cap I_2</math> and joint <math>I_1 \vee I_2</math> defined to be the ideal generated by <math>I_1 \cup I_2</math>.  For any saturated hereditary subset <math>H</math>, the ideal <math>I_H</math> is gauge invariant.

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

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

==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 <math>E</math> is a graph, a desingularization of <math>E</math> is a row-finite graph <math>F</math> such that <math>C^*(E)</math> is Morita equivalent to <math>C^*(F)</math>.<ref>The C*-algebras of arbitrary graphs, Doug Drinen and Mark Tomforde, Rocky Mountain J. Math. 35 (2005), no. 1, 105–135.</ref>  Drinen and Tomforde described a method for constructing a desingularization from any countable graph:  If <math>E</math> is a countable graph, then for each vertex <math>v_0</math> that emits an infinite number of edges, one first chooses a listing of the outgoing edges as <math>s^{-1}(v_0) = \{ e_0, e_1, e_2, \ldots \}</math>, one next attaches a '''tail''' of the form

[[File:Tail-added-to-graph.jpg|300px|center]]
<!--<math>v_0 \longrightarrow v_1 \longrightarrow v_2 \longrightarrow v_3 \longrightarrow \cdots</math>-->

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

Here are some examples of this construction.  For the first example, note that if <math>E</math> is the graph

[[File:Desingularization-1.jpg|400px|center]]
<!--<math>\cdots \longrightarrow \cdot \longrightarrow \cdot \longrightarrow v_0 \; \overset{\infty}{\Longrightarrow} \; \cdot \longrightarrow \cdot \longrightarrow \cdots</math>-->

then a desingularization <math>F</math> is given by the graph

[[File:Desingularization-2.jpg|400px|center]]

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

[[File:Desingularization-3.jpg|400px|center]]

Desingularization has become a standard tool in the theory of graph C*-algebras,<ref>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}}</ref>  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 [[uncountably infinite|uncountable]] number of edges.  However, in the study of C*-algebras it is common to restrict attention to [[Separable space|separable C*-algebras]].  Since a graph C*-algebra <math>C^*(E)</math> is separable precisely when the graph <math>E</math> 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 <math>E</math> is a row-finite graph, the '''vertex matrix''' of <math>E</math> is the <math>E^0 \!\times\! E^0</math> matrix <math>A_E</math> with entry <math>A_E(v,w)</math> defined to be the number of edges in <math>E</math> from <math>v</math> to <math>w</math>.  Since <math>E</math> is row-finite, <math>A_E</math> has entries in <math>\mathbb{N} \cup \{ 0 \}</math> and each row of <math>A_E</math> has only finitely many nonzero entries.  (In fact, this is where the term "row-finite" comes from.)  Consequently, each column of the [[transpose]] <math>A_E^t</math> contains only finitely many nonzero entries, and we obtain a map <math display="inline">A_E^t : \bigoplus_{E^0} \mathbb{Z} \to \bigoplus_{E^0} \mathbb{Z}</math> given by left multiplication.  Likewise, if <math>I</math> denotes the <math>E^0 \!\times\! E^0</math> [[identity matrix]], then <math display="inline">I - A_E^t : \bigoplus_{E^0} \mathbb{Z} \to \bigoplus_{E^0} \mathbb{Z}</math> provides a map given by left multiplication.


'''Theorem:''' Let <math>E</math> be a row-finite graph with no sinks, and let <math>A_E</math> denote the vertex matrix of <math>E</math>.  Then 
<math display="block">I - A_E^t : \bigoplus_{E^0} \mathbb{Z} \to \bigoplus_{E^0} \mathbb{Z}</math> 
gives a well-defined map by left multiplication.  Furthermore, 
<math display="block">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).</math>  
In addition, if <math>C^*(E)</math> is unital (or, equivalently, <math>E^0</math> is finite), then the isomorphism <math>K_0(C^*(E)) \cong \operatorname{coker} (I- A_E^t)</math> takes the class of the unit in <math>K_0(C^*(E))</math> to the class of the vector <math>(1, 1, \ldots, 1)</math> in <math>\operatorname{coker} (I- A_E^t)</math>.


Since <math>K_1(C^*(E))</math> is isomorphic to a [[subgroup]] of the [[free group]] <math display="inline">\bigoplus_{E^0} \mathbb{Z}</math>, we may conclude that <math>K_1(C^*(E))</math> is a free group.  It can be shown that in the general case (i.e., when <math>E</math> is allowed to contain sinks or infinite emitters) that <math>K_1(C^*(E))</math> 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 K<sub>1</sub>-group is not Morita equivalent (and hence not isomorphic) to a graph C*-algebra.

== See also ==
{{portal|Mathematics}}
* [[k-graph C*-algebra]]
* [[Leavitt path algebra]]

==Notes==
{{Reflist}}

{{DEFAULTSORT:Graph C}}
[[Category:C*-algebras]]