www.elsevier.com/locate/jalgebra
The Leavitt path algebra of a graph
Gene Abrams
a,∗, Gonzalo Aranda Pino
baDepartment of Mathematics, University of Colorado, Colorado Springs, CO 80933, USA
bDepartamento de Álgebra, Geometría y Topología, Universidad de Málaga, 29071 Málaga, Spain
Received 23 September 2004 Available online 13 September 2005
Communicated by Kent R. Fuller
Abstract
For any row-finite graphEand any fieldKwe construct the Leavitt path algebraL(E)having coefficients inK. WhenKis the field of complex numbers, thenL(E)is the algebraic analog of the Cuntz–Krieger algebraC∗(E)described in [I. Raeburn, Graph algebras, in: CBMS Reg. Conf. Ser. Math., vol. 103, Amer. Math. Soc., 2005]. The matrix ringsMn(K)and the Leavitt algebrasL(1, n) appear as algebras of the formL(E)for various graphsE. In our main result, we give necessary and sufficient conditions onEwhich imply thatL(E)is simple.
2005 Elsevier Inc. All rights reserved.
Keywords: Path algebra; Leavitt algebra; Cuntz–KriegerC∗-algebra
Introduction
Throughout this articleKwill denote an arbitrary field. In his seminal paper [6], Leavitt describes a class ofK-algebras (nowadays denoted byL(m, n)) which are universal with respect to an isomorphism property between finite rank free modules. In [7], Leavitt goes on to show that the algebras of the formL(1, n)are simple. More than a decade later, Cuntz [3] constructed and investigated theC∗-algebrasOn(nowadays called the Cuntz algebras),
showing, among other things, that eachOnis (algebraically) simple. WhenKis the field
* Corresponding author.
E-mail addresses: [email protected] (G. Abrams), [email protected] (G. Aranda Pino).
0021-8693/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2005.07.028
Cof complex numbers, thenOncan be viewed as the completion, in an appropriate norm,
ofL(1, n). Soon after the appearance of [3], Cuntz and Krieger [4] described the signifi-cantly more general notion of theC∗-algebra of a (finite) matrixA, denotedOA. Among
this class ofC∗-algebras one can find, for any finite graphE, the Cuntz–Krieger algebra
C∗(E), defined originally in [5]. TheseC∗-algebras, as well as those arising from various infinite graphs, have been the subject of much investigation (see, e.g., [2,8,9]). Recently, the ‘algebraic analogs’ of theC∗-algebrasOAhave been presented in [1]; these are denoted
byCKA(K). By restricting attention to a specific set of allowable matrices, the simplicity
of the algebraCKA(K)for some subset of these allowable matrices has been determined
(although the condition for simplicity is not explicitly given in terms of the matrixA). The goal of this article is to ‘complete the algebraic picture.’ Specifically, we give the definition of the Leavitt path algebraL(E)corresponding to any row-finite graphEand fieldK. When E is finite without sources and sinks, then L(E) can be realized as an algebra of the formCKA(K)for some matrixA. Analogous to the relationship that exists
betweenL(1, n)andOn,L(E) has the property that whenK=C, thenC∗(E)can be
viewed as the completion, in an appropriate norm, ofL(E)[8, Proposition 1.20].
In our main result, Theorem 3.11, we give necessary and sufficient conditions on the row-finite graphEwhich imply thatL(E)is simple. These results extend those presented in [1], in that: they apply also to some important algebras which are explicitly not consid-ered in [1]; they apply also to algebras which arise from infinite matrices; and they provide necessary conditions onEfor the simplicity ofL(E). The statement of Theorem 3.11 par-allels a similar theorem forC∗-algebras of the formC∗(E)given in [8, Theorem 4.9 and subsequent remarks]. However, the techniques utilized here are significantly different than those used in the analytic setting.
We begin by establishing some notational conventions. A (directed) graph E = (E0, E1, r, s) consists of two countable sets E0, E1 and functions r, s:E1→E0. The elements ofE0are called vertices and the elements ofE1edges. For each edgee,s(e)is the source ofeandr(e)is the range ofe. Ifs(e)=vandr(e)=w, then we also say that
v emitseand thatwreceivese, or thatepoints tow.
A vertex which does not receive any edges is called a source. A vertex which emits no edges is called a sink. A graph is called row-finite if s−1(v) is a finite set for each vertexv. In this paper, we will only be concerned with row-finite graphs. Of course, under this hypothesis, the edge set ofE,E1, is finite if its set of vertices,E0, is finite. Thus,
we will say a graphEis finite if E0is a finite set. A pathµin a graphEis a sequence of edgesµ=µ1. . . µn such thatr(µi)=s(µi+1)for i=1, . . . , n−1. In such a case, s(µ):=s(µ1)is the source ofµandr(µ):=r(µn)is the range ofµ. Ifs(µ)=r(µ)and
s(µi)=s(µj)for everyi=j, thenµis a called a cycle.
1. Leavitt path algebras
In this section we define the algebraic structures under investigation. We begin by re-minding the reader of the construction of the standard path algebra of a graph.
1.1. Definition. LetKbe a field andEbe a graph. The pathK-algebra overEis defined as the freeK-algebraK[E0∪E1]with the relations:
(1) vivj=δijvi for everyvi, vj∈E0.
(2) ei=eir(ei)=s(ei)ei for everyei∈E1.
This algebra is denoted byA(E).
1.2. Definition. Given a graph E we define the extended graph of E as the new graph
E=(E0, E1∪(E1)∗, r, s)where(E1)∗= {e∗i: ei∈E1}and the functionsrandsare
defined as
r|E1=r, s|E1=s, r
ei∗=s(ei) and s
ei∗=r(ei).
1.3. Definition. LetK be a field andEbe a row-finite graph. The Leavitt path algebra of
E with coefficients inK is defined as the path algebra over the extended graphE, with relations:
(CK1) ei∗ej=δijr(ej)for everyej∈E1andei∗∈(E1)∗.
(CK2) vi={ej∈E1:s(e
j)=vi}eje ∗
j for everyvi∈E0which is not a sink.
This algebra is denoted byLK(E)(or more commonly simply byL(E)).
The conditions (CK1) and (CK2) are called the Cuntz–Krieger relations. In particular condition (CK2) is the Cuntz–Krieger relation atvi. Ifvi is a sink, we do not have a (CK2)
relation atvi. Note that the condition of row-finiteness is needed in order to define the
equation (CK2).
1.4. Examples. Many well-known algebras are of the formL(E)for some graphE:
(i) Matrix algebrasMn(K): Consider the graphE defined byE0= {v1, . . . , vn},E1= {e1, . . . , en−1}ands(ei)=vi andr(ei)=vi+1fori=1, . . . , n−1. ThenMn(K)∼=
L(E), via the mapvi→e(i, i), ei→e(i, i+1), ande∗i →e(i+1, i)(wheree(i, j )
denotes the standard(i, j )-matrix unit inMn(K)).
(ii) Laurent polynomial algebrasK[x, x−1]: Consider the graphEdefined byE0= {∗},
E1= {x}. Then clearlyK[x, x−1] ∼=L(E).
(iii) Leavitt algebras A=L(1, n) for n2 investigated in [7]: Consider the graph E
defined byE0= {∗},E1= {y1, . . . , yn}. ThenL(1, n)∼=L(E).
1.5. Lemma. Every monomial inL(E)is of the following form:
(a) kivi withki∈Kandvi∈E0, or
(b) kei1. . . eiσe∗j1. . . e
∗
jτ wherek∈K;σ, τ0, σ+τ >0,eis ∈E
1ande∗
jt ∈(E 1)∗ for
Proof. The proof is almost identical to the proof of [8, Corollary 1.15] (a straightforward
induction argument on the length of the monomialkx1. . . xnwithxi∈E0∪E1∪(E1)∗),
and so is omitted. 2
1.6. Lemma. IfE0is finite thenL(E)is a unitalK-algebra. IfE0is infinite, thenL(E)is
an algebra with local units (specifically, the set generated by finite sums of distinct elements ofE0).
Proof. First assume thatE0 is finite: we will show thatni=1vi is the unit element of
the algebra. First we compute(ni=1vi)vj =ni=1δijvj=vj. Now if we takeej∈E1
we may use the equations (2) in the definition of path algebra together with the previous computation to get(ni=1vi)ej =(
n
i=1vi)s(ej)ej =s(ej)ej=ej. In a similar manner
we see that(ni=1vi)e∗j =ej∗. Since L(E)is generated byE0∪E1∪(E1)∗, then it is
clear that(ni=1vi)α=αfor everyα∈L(E), and analogouslyα(
n
i=1vi)=αfor every
α∈L(E). Now suppose thatE0is infinite. Consider a finite subset{ai}ti=1ofL(E)and
use Lemma 1.5 to writeai=
ni
s=1ksivis+
mi
l=1cilpli wherekis, cli∈K− {0}, andpil are
monomials of type (b). Then with the same ideas as above it is not difficult to prove that forV =ti=1{vis, s(pil), r(pil): s=1, . . . , ni;l=1, . . . , mi}, thenα=v∈Vvis a finite
sum of vertices such thatαai=aiα=ai for everyi. 2
1.7. Lemma.L(E)is aZ-graded algebra, with grading induced by
deg(vi)=0 for allvi∈E0;
deg(ei)=1 and deg
e∗i= −1 for allei∈E1.
That is,L(E)=n∈ZL(E)n, whereL(E)0=KE0+A0,L(E)n=Anforn=0 where
An= kei1. . . eiσe∗j1. . . e
∗
jτ: σ+τ >0, eis ∈E 1, e
jt∈
E1∗, k∈K, σ−τ =n .
Proof. The fact thatL(E)=n∈ZL(E)nfollows from Lemma 1.5. The grading onL(E)
follows directly from the fact that A(E) is Z-graded, and that the relations (CK1) and (CK2) are homogeneous in this grading. 2
Note that by virtue of Lemma 1.7 we can define the degree of an arbitrary polynomial in
L(E)as the maximum of the degrees of its monomials. We say that a monomial inL(E)
is a real path (respectively a ghost path) if it contains no terms of the forme∗i (respec-tivelyei); we say thatp∈L(E)is a polynomial in only real edges (respectively in only
ghost edges) if it is a sum of real (respectively ghost) paths.
For a path q=q1. . . qn, we denote by q∗ the ghost path qn∗. . . q1∗. If α∈L(E) and d∈Z+, then we say that αis representable as an element of degreed in real (respec-tively ghost) edges in caseαcan be written as a sum of monomials from the spanning set
{pq∗: p, q are paths inE}given by Lemma 1.5, in such a way thatd is the maximum length of a pathp(respectivelyq) which appears in such monomials. We note that an ele-ment ofL(E)may be representable as an element of different degrees in real (respectively
ghost) edges, depending on the particular representation used forα. For instance, forEas in Example 1.4(ii),xx−1is representable as an element of degree 0 in real edges inL(E), asxx−1=1.
2. Closed paths
Certain paths in the graphEwill play a central role in the structure of the Leavitt path algebraL(E).
2.1. Definition. An edgeeis an exit to the pathµ=µ1. . . µn if there existsisuch that
s(e)=s(µi)ande=µi.
A closed path based atvis a pathµ=µ1. . . µn, withµj∈E1,n1, and such that
s(µ)=r(µ)=v. Denote by CP(v)the set of all such paths.
A closed simple path based atvis a closed path based atv,µ=µ1. . . µn, such that
s(µj)=vfor everyj >1. Denote by CSP(v)the set of all such paths.
Remark. Note that a cycle is a closed simple path based at any of its vertices, but not every
closed simple path based atvis a cycle because a closed simple path may visit some of its vertices (but notv) more than once. Moreover, every closed simple path is in particular a closed path, while the converse is false.
2.2. Lemma. Letµ, ν∈CSP(v). Thenµ∗ν=δµ,νv.
Proof. We first assume α and β are arbitrary paths and write α=ei1. . . eiσ andβ = ej1. . . ejτ.
Case 1. deg(α)=deg(β)butα=β. Defineb1 the subindex of the first edge where the pathsαandβdiffer. That is,eia=eja for everya < bbuteib=ejb. Then
α∗β=e∗i σ. . . e
∗
i1ej1. . . ejτ =e∗iσ. . . e ∗
i2r(ej1)ej2. . . ejτ =δr(ej1),s(ej2)e∗iσ. . . e
∗
i2ej2. . . ejτ = · · · =δr(ej1),s(ej2). . . δr(ejb−1),s(ejb)ei∗σ. . . e ∗
ibejb. . . ejτ =0.
Case 2. α=β. Proceeding as above,α∗β=δr(ei1),s(ei2). . . δr(eiσ−1),s(eiσ)r(eiσ)=r(α).
Case 3. Now letµ, ν∈CSP(v)with deg(µ) <deg(ν). Writeν=ν1ν2where deg(ν1)=
deg(µ),deg(ν2) >0. Now ifµ=ν1then we have thatv=r(µ)=r(ν1)=s(ν2),
contra-dicting thatν∈CSP(v), soµ=ν1and thus Case 1 applies to obtainµ∗ν=µ∗ν1ν2=0.
The case deg(µ) >deg(ν)is analogous to Case 3 by changing the roles ofµandν. 2
2.3. Lemma. For everyp∈CP(v)there exist uniquec1, . . . , cm∈CSP(v)such thatp=
Proof. Writep=ei1. . . ein. LetT = {t∈ {1, . . . , n}: r(eit)=v}and listt1<· · ·< tm=n
all the elements ofT. Thenc1=ei1. . . eit1 andcj=eitj−1. . . eitj forj >1 give the desired
decomposition.
To prove the uniqueness, writep=c1. . . cr=d1. . . ds withci, dj∈CSP(v). Multiply
byc∗1on the left and use Lemma 2.2 to obtain 0=vc2. . . cr=c∗1d1. . . ds, and therefore by
Lemma 2.2 againc1=d1. Now an induction process finishes the proof. 2
2.4. Definition. Forp∈CP(v)we define the return degree (atv) ofpto be the number
m1 in the decomposition above. (So, in particular, CSP(v)is the subset of CP(v)having return degree equal one.) We denote it by RD(p)=RDv(p)=m. We extend this notion to
vertices by setting RDv(v)=0, and to nonzero linear combinations of the form
ksps,
withps∈CP(v)∪ {v}andks∈K− {0}by: RD(
ksps)=max{RD(ps)}.
2.5. Lemma. For a graphEthe following conditions are equivalent:
(i) Every cycle has an exit. (ii) Every closed path has an exit. (iii) Every closed simple path has an exit.
(iv) For everyvi∈E0, if CSP(vi)= ∅, then there existsc∈CSP(vi)having an exit.
Proof. (ii)⇒(iii)⇒(i) is trivial by definition, and (iii)⇒(iv) is obvious.
(i)⇒(ii). Considerµ∈CP(vi). First by Lemma 2.3 we can factorµ=c(1). . . c(m),
wherec(j )∈CSP(vi), and we examinec(m). If it is a cycle then we can find an exit for it,
and therefore forµ, by hypothesis. If not,c(m)visits a vertex (different fromvi) more than
once. Writec(m)=c(m)1 . . . cs(m)with eachci(m)∈E1and letc (m)
s0 be the last edge for which
s(cj(m))∈ {s(ci(m)): 1is, i=j}. Thus, there existss1< s0such thats(c(m)s0 )=s(c (m) s1 ).
We have several possibilities:
Case 1. c(m)s0 =c (m)
s1 and s0< s. Then r(c (m) s0 )=r(c
(m)
s1 ); that is, s(c (m)
s0+1)=s(c (m) s1+1),
which contradicts the choice ofc(m)s0 .
Case 2. c(m)s0 =c (m)
s1 ands0=s. This means thatr(c (m) s1 )=r(c
(m)
1 )=vi, which is
impos-sible becausec(m)∈CSP(vi).
Case 3. cs(m)0 =c (m)
s1 . In this casec (m)
s1 is an exit forc
(m), and then forµ.
In each case we reach a contradiction or we find an exit forµ, as needed.
(iv) ⇒(iii). Consider c(1) ∈CSP(vi). By hypothesis we find c(2)∈CSP(vi)having
an exit. Ifc(1)=c(2) we are done. If not, we writec(1)=ei1. . . eis,c
(2)=e
j1. . . ejr and
proceed by steps:
Step 1. Ifei1=ej1, sinces(ei1)=s(ej1)=vi, thenej1 is an exit forc (1).
Step 3. Ifei2=ej2, then as in Step 1,ej2is an exit forc (1).
Step 4. Ifei2=ej2, then continue as in Step 2.
With this process, we either find an exit or we run out of edges in one path but not in the other (becausec(1)=c(2)). Thus:
Case 1. c(1)=c(2)eit. . . eis forts. But this is impossible becauses(eit)=r(c
(2))=v i
andc(1)∈CSP(vi).
Case 2. c(2)=c(1)e
jq. . . ejr forqr, which is similarly impossible.
In any case, we reach a contradiction or we are able to find an exit forc(1), and this finishes the proof. 2
3. Simplicity ofL(E)
In this final section we build the algebraic machinery necessary to obtain our main result, Theorem 3.11.
3.1. Proposition. LetEbe a graph with the property that every cycle has an exit. Ifα∈ L(E)is a polynomial in only real edges with deg(α) >0, then there exista, b∈L(E)such thataαb=0 is a polynomial in only real edges and deg(aαb) <deg(α).
Proof. Write α=ei∈E1eiαei +
vl∈E0klvl, where αei are polynomials in only real
edges, and deg(αei) <deg(α)=m.
Case (A). kl=0 for everyl. Sinceα=0, there existsi0 such thatei0αei0 =0. Letb∈ L(E)haveαb=α; such exists by Lemma 1.6. Thena=ei∗
0,bgivee
∗
i0αb=αei0 =0 is a
polynomial in only real edges and deg(αei0) <deg(α).
Case (B). There existskl0=0. Then we can write
vl0αvl0=kl0vl0+
p∈CP(vl0)
kpp, kp∈K.
Note that this is a polynomial in only real edges, and is nonzero becausekl0 is nonzero.
Case (B.1). deg(vl0αvl0) <deg(α). Then we are done witha=vl0 andb=vl0.
Case (B.2). deg(vl0αvl0)=deg(α)=m >0. Then there exists p0∈CP(vl0)such that
kp0p0=0. Now by Lemma 2.3, we can writep0=c1. . . cσ,σ1 and thus CSP(vl0)= ∅.
We apply now Lemma 2.5 to findcs0∈CSP(vl0)which hasei0 as an exit, that is, ifcs0=
s(eij)=s(ei0)we can therefore build the path given byz=ei1. . . eij−1ei0. This path has
c∗s
0z=0 because c
∗
s0z=e
∗
is0. . . e ∗
i1ei1. . . eij−1ei0 = · · · =e
∗
is0. . . e ∗
ijei0 =0. (We will use
this observation later on.) Again Lemma 2.3 allows us to write
vl0αvl0=kl0vl0+
cs∈CSP(vl0)
csα(c1s), (†)
where γ =RD(vl0αvl0) > 0, and α (1)
cs are polynomials in only real edges satisfying
RD(αc(1s)) < γ.
We now present a process in which we decrease the return degree of the polynomials by multiplying on both sides by appropriate elements inL(E). In the sequel we will often make use of Lemma 2.2 without mentioning it explicitly. In particular, multiplying(†)on the left byc∗s
0 gives
c∗s
0(vl0αvl0)=kl0c∗s0+α (1)
cs0. (‡)
Case 1. αc(1s0)=0. ThenA=c∗s0 andB=cs0 are such thatA(vl0αvl0)B=kl0vl0 =0 is a
polynomial in only real edges and RD(A(vl0αvl0)B)=0< γ =RD(vl0αvl0).
Case 2. αc(1s0)=0 but RD(α
(1)
cs0)=0. Then α
(1) cs0 =k
(2)v
l0 for some 0=k
(2)∈K. Using
the pathzwith an exit forc∗s
0 we have:z
∗c∗
s0(vl0αvl0)z=z∗(kl0c∗s0+k (2)v
l0)z=z∗(0+
k(2)z)=k(2)r(z)=0. So we have A=z∗c∗s
0 andB=zsuch that A(vl0αvl0)B=0 is a
polynomial in only real edges and RD(A(vl0αvl0)B)=0< γ =RD(vl0αvl0).
Case 3. RD(α(c1s0)) >0. We can write
αc(1) s0 =k
(2)v l0+
cs∈CSP(vl0) csα(c2s),
whereα(c2s)are polynomials in only real edges with return degree less than the return degree
ofαc(1s0). Now 0<RD(α
(1)
cs0) < γ impliesγ2. Multiply(‡)byc∗s0 to get
c∗s
0
2
(vl0αvl0)=kl0
cs∗
0
2
+k(2)c∗s
0+α (2)
cs0. (§)
We are now in position to proceed in a manner analogous to that described in Cases 1, 2, and 3 above.
Case 3.1. αc(2s0)=0. Then(c∗s0)
2(v
l0αvl0)(cs0)
2=k
l0vl0+k (2)c
s0and hence we have found
A=(c∗s
0)
2andB=(c
s0)
2 such thatA(v
l0αvl0)B=0 is a polynomial in only real edges
Case 3.2. αc(2s0) =0 but RD(α
(2)
cs0)=0. Then α
(2) cs0 =k
(3)v
l0 for some 0=k
(3)∈K, and
thenz∗(c∗s
0)
2(v
l0αvl0)z=z∗(kl0(cs∗0)
2+k(2)c∗ s0+k
(3)v
l0)z=z∗(0+k
(3)z)=k(3)r(z)=0.
Thus, we getA=z∗(c∗s
0)
2andB=zsuch thatA(v
l0αvl0)B=0 is a polynomial in only
real edges and RD(A(vl0αvl0)B)=0< γ =RD(vl0αvl0).
Case 3.3. RD(αc(2s0)) >0. We write
α(c2) s0 =k
(3)v l0+
cs∈CSP(vl0) csαc(3s),
whereαc(3s)are polynomials in only real edges with return degree less than the return degree
ofα(c2s0). Now 0<RD(α
(2)
cs0) <RD(α
(1)
cs0) < γ impliesγ3. And by multiplying(§)byc∗s0
we get(c∗s
0)
3(v
l0αvl0)=kl0(cs∗0)
3+k(2)(c∗ s0)
2+k(3)c∗ s0+α
(3) cs0.
We continue the process of analyzing each such equation by considering three cases. If at any stage either of the first two cases arise, we are done. But since at each stage the third case can occur only by producing elements of subsequently smaller return degree, then after at mostγ stages we must have one of the first two cases.
Thus, by repeating this process at mostγ times we are guaranteed to findA,Bsuch thatA(v l0αvl0)B=0 is a polynomial in only real edges and RD(A(vl0αvl0)B)=0. But
this then gives 0=deg(A(vl0αvl0)B) <deg(α). Soa=Av l0 andb=vl0Bare the desired
elements. 2
3.2. Corollary. LetEbe a graph with the property that every cycle has an exit. Ifα=0 is
a polynomial in only real edges then there exista, b∈L(E)such thataαb∈E0.
Proof. Apply Proposition 3.1 as many times as needed (deg(α) at most) to find a, b
such thataαb is a nonzero polynomial in only real edges with deg(aαb)=0; that is,
aαb=ti=1kivi=0. So there existsj withkj=0, and finallya=k−j1aandb=bvj
give thataαb=vj∈E0. 2
3.3. Corollary. LetEbe a graph with the property that every cycle has an exit. IfJ is an ideal ofL(E)and contains a nonzero polynomial in only real edges, thenE0∩J= ∅.
Proof. Straightforward by Corollary 3.2. 2
In order to extend all the previous results of this section to analogous results about polynomials in only ghost edges, we define an involution inL(E).
3.4. Lemma.L(E)can be equipped with an involutionx→x defined in the monomials by:
(b) kei1. . . eiσe∗j1. . . e
∗
jτ=kejτ. . . ej1e ∗
iσ. . . e ∗
i1wherek∈K;σ, τ0,σ+τ >0,eis ∈E
1
andejt ∈(E 1)∗,
and extending linearly toL(E).
Proof. The proposed map is well defined by Lemma 1.5, and it is linear by definition. It is
easily shown to satisfyxy=yxandx=xfor everyx, y∈L(E). It is also straightforward to check that the map is compatible with the relations definingL(E). 2
3.5. Remark. Note that the involution transforms a polynomial in only real edges into
a polynomial in only ghost edges and vice versa. IfJ is an ideal ofL(E)then so isJ. We note here that while Leavitt path algebras behave somewhat like theirC∗-algebra sib-lings, they are indeed different in many respects. For instance, whereas in C∗-algebras every two-sided ideal J is self-adjoint (i.e., J =J), this is not the case in the Leav-itt path algebras setting. For instance, letL(E)=K[x, x−1] as in Example 1.4(ii), and let J be the ideal 1+x +x3 of L(E). Then J is not self-adjoint, as follows: if
J =J, then f (x)=1+x−1+x−3∈J and thus x3f (x)=1+x2+x3∈J. Now
K[x, x−1]being a unital commutative ring implies that there existsp=∞i=−∞aixi with
p(1+x+x3)=1+x2+x3. A degree argument on the highest power on the left-hand side of the previous equation leads toai=0 for everyi1. By reasoning in a similar fashion
on the lowest power we also getai=0 for everyi−1, that is,p=a0, which is absurd.
We can define sets and quantities for ghost paths analogous to those given for real paths. Using the involution given in Lemma 3.4 we can then analogously prove the following three results.
3.6. Proposition. LetEbe a graph with the property that every cycle has an exit. Ifα∈ L(E)is a polynomial in only ghost edges with deg(α) >0 then there exist a, b∈L(E)
such thataαb=0 is a polynomial in only ghost edges and deg(aαb) <deg(α).
3.7. Corollary. LetEbe a graph with the property that every cycle has an exit. Ifα=0 is
a polynomial in only ghost edges then there exista, b∈L(E)such thataαb∈E0.
3.8. Corollary. LetEbe a graph with the property that every cycle has an exit. IfJ is an ideal ofL(E)and contains a nonzero polynomial in only ghost edges, thenE0∩J= ∅.
For a graphEwe define a preorderon the vertex setE0given by:
vwif and only ifv=wor there is a pathµsuch thats(µ)=vandr(µ)=w.
We say that a subsetH⊆E0is hereditary ifw∈H andwvimplyv∈H. We say that
H is saturated if whenevers−1(v)= ∅and{r(e): s(e)=v} ⊆H, thenv∈H. (In other words,H is saturated if, for any vertexvinE, if all of the range verticesr(e)for those edgesehavings(e)=vare inH, thenvmust be inH as well.)
3.9. Lemma. IfJ is an ideal ofL(E), thenJ∩E0 is a hereditary and saturated subset ofE0.
Proof. We first show thatJ∩E0is hereditary. Considerv, w∈E0such thatv∈J and
v w. By the definition of the preorder we can find a path µ=µ1. . . µn such that
s(µ1)=vandr(µn)=w. Apply thatJ is an ideal to get thatµ∗1vµ1=µ∗1µ1=r(µ1)= s(µ2)∈J. Repeating this argumentntimes, we get thatr(µn)=w∈J.
Now we see that J ∩E0 is saturated: consider a vertex v with s−1(v)= ∅ and
{r(e): s(e)=v} ⊆J. The first condition implies thatv is not a sink, so (CK2) applies and we obtainv={e
j∈E1:s(ej)=v}eje ∗
j. If we takeej such thats(ej)=v, then by
hy-pothesis we have thatr(ej)∈J and thereforeej=ejr(ej)∈J. Now applying (CK2) we
conclude thatv∈J. 2
3.10. Corollary. LetEbe a graph with the following properties:
(i) the only hereditary and saturated subsets ofE0are∅andE0.
(ii) every cycle has an exit.
IfJ is a nonzero ideal ofL(E)which contains a polynomial in only real edges (or a polynomial in only ghost edges), thenJ=L(E).
Proof. Apply Corollaries 3.3 or 3.8 to get thatJ ∩E0= ∅. Now by Lemma 3.9 and (i) we haveJ∩E0=E0. ThereforeJ contains a set of local units by Lemma 1.6, and hence
J=L(E). 2
We are now in position to prove the main result of this article.
3.11. Theorem. LetEbe a row-finite graph. Then the Leavitt path algebraL(E)is simple if and only ifEsatisfies the following conditions:
(i) The only hereditary and saturated subsets ofE0are∅andE0; and (ii) Every cycle inEhas an exit.
Proof. First we assume that (i) and (ii) hold and we will show thatL(E)is simple. Suppose thatJ is a nonzero ideal ofL(E). Choose 0=α∈J representable as an element having minimal degree in the real edges. If this minimal degree is 0, then αis a polynomial in only ghost edges, so that by Corollary 3.10 we haveJ=L(E). So suppose this degree in real edges is at least 1. Then we can write
α=
m
n=1
einαein+β,
wherem1,einαein=0 for everyn, and eachαeinis representable as an element of degree
Supposev is a sink inE. Then we may assume vβ=0, as follows. Multiplying the displayed equation byvon the left givesvα=vmn=1einαein+vβ. But sincevis a sink
we havevein=0 for all 1nm, so that vα=vβ∈J. Butvβ=0 would then yield
a nonzero element of J in only ghost edges, so that again by Corollary 3.10 we have
J=L(E).
For an arbitrary edgeej∈E1, we have two cases:
Case 1. j ∈ {i1, . . . , im}. Thene∗jα=αej+e∗jβ∈J. If this element is nonzero it would be
representable as an element with smaller degree in the real edges than that ofα, contrary to our choice. So it must be zero, and henceαej = −e∗jβ, so thatejαej= −eje∗jβ.
Case 2. j /∈ {i1, . . . , im}. Thene∗jα=e∗jβ∈J. Ife∗jβ=0, then as before we would have a
nonzero element ofJ in only ghost edges, so thatJ =L(E)and we are done. So we may assume thatej∗β=0, so that in particular we have 0= −eje∗jβ.
Now let S1= {vj∈E0: vj =s(ein)for some 1nm}, and letS2= {vk1, . . . , vkt}
where(ti=1vki)β=β. (Such a setS2exists by Lemma 1.6.) We note that wβ=0 for
everyw∈E0−S2. Also, by definition there are no sinks inS1, and by a previous
obser-vation we may assume that there are no sinks inS2. LetS=S1∪S2. Then in particular we
have(v∈Sv)β=β.
We now argue that in this situationαmust be zero, which will contradict our original choice ofαand thereby complete the proof. To this end,
α=
m
n=1
einαein+β=
m
n=1
−eine∗inβ+β (by Case 1)
= m
n=1
−eine∗inβ−
j /∈{i1,...,im} s(ej)∈S
eje∗j
β+β
(by Case 2, the newly subtracted terms equal 0)
= − v∈S
v
β+β (no sinks inSimplies that (CK2) applies at eachv∈S)
= −β+β=0.
Thus we have shown that ifEsatisfies the two indicated properties, thenL(E)is simple. For the converse, first suppose that there is a cyclep having no exit. We will prove that
L(E)cannot be simple. Letvbe the base of that cycle. We will show that forα=v+p,
α is a nontrivial ideal ofL(E)becausev /∈ α. Writep=ei1. . . eiσ. Since this cycle
does not have an exit, for everyeij there is no edge with sources(eij)other thaneij itself,
so that the (CK2) relation at this vertex yieldss(eij)=eijei∗j. This easily impliespp ∗=v
(we recall here thatp∗p=valways holds), and that CSP(v)= {p}.
Now suppose thatv∈ α. So there exist nonzero monic monomialsan, bn∈L(E)and
cn∈Kwithv=
m
n=1cnanαbn. Sincevαv=α, by multiplying byvif necessary we may
We claim that for eachan (respectivelybn) there exists an integeru(an)0
(respec-tively u(bn)0) such that an=pu(an) or an=(p∗)u(an) (respectively bn=pu(bn) or
bn=(p∗)u(bn)).
Nowa1is of the formek1. . . ekcej∗1. . . e
∗
jd withc, d1. (Otherwise we are in a simple
case that will be contained in what follows.) Sincea1starts and ends invwe can consider
the elements:g=min{z: r(ej∗
z)=v}andf =max{z: s(ekz)=v}, and we will focus on a1=ekf. . . ekcej∗1. . . e
∗
jg.
First, sincev=r(e∗j
g)=s(ejg)andei1 is the only edge coming fromv, then we have
ejg=ei1. Now,s(ejg−1)=r(e∗jg−1)=s(e
∗
jg)=r(ejg)=r(ei1)=s(ei2), and again the only
edge coming froms(ei2)isei2 and thereforeejg−1=ei2. This process must stop before we
run out of edges ofpbecause by our choice ofgwe have thatv /∈ {r(e∗j
z): z < g}. So in
the end there existsγ < σsuch thatej∗
1. . . e
∗
jg=e ∗
iγ. . . e ∗
i1.
With the same (reversed) ideas in the paragraph above we can find δ < σ such that
ekf. . . ekc=ei1. . . eiδ. Thus,a1 =ei1. . . eiδe∗iγ. . . e ∗
i1, and we have two cases:
Case 1. δ=γ. We know thatpis a cycle, so thatr(eiδ)=r(eiγ)=s(e∗iγ), soeiδe ∗
iγ =0,
which is absurd becausea1=0.
Case 2. δ=γ. In this casea1 =p0p0∗for a certain subpathp0ofp, and by using again
the argument of the (CK2) relation in this case, we obtainp0p∗0=v.
Hence, we geta1=ek1. . . ekf−1e∗jg+1. . . e
∗
jd =xy
∗, withx, y∈CP(v). (Obviously, the
casec1, d=0 yieldsa1=x, the casec=0,d1 yieldsa1=y∗andc=d=0 yields a1=v.) Using Lemma 2.3 we havex=c(1). . . c(ν)for somec(µ)∈CSP(v)= {p}, and
the same happens withy. In this way we havea1=pu(p∗)vfor someu, v0, and taking
into account thatpp∗=v we finally obtain thata1 is of the formpu or (p∗)u for some u0 as claimed. An identical argument holds for the other coefficientsanandbn.
Now since bothp andp∗ commute with p, p∗ andα, we use the conclusion of the previous paragraph to write the sumv=mn=1cnanαbnasv=αP (p, p∗)for some
poly-nomialP having coefficients inK. Specifically,P (p, p∗)can be written asP (p, p∗)= k−m(p∗)m+ · · · +k0v+ · · · +knpn∈
n
j=−mL(E)σj, wherem, n0. First, we claim
thatk−i=0 for everyi >0, as follows. If not, letm0be the maximumihavingk−i =0.
ThenαP (p, p∗)=k−m0(p∗)
m0+terms of greater degree=v, and sincem
0>0 we get
that k−m0 =0, which is absurd. In a similar way we obtainki=0 for everyi >0, and
thereforeP (p, p∗)=k0v. But this would yieldv=αP (p, p∗)=αk0v=k0α, which is
impossible.
Thus we have shown that ifE contains a cycle which has no exit, thenL(E) is not simple. Now we will consider the situation whereE0contains a nontrivial hereditary and saturated subsetH, and conclude in this case as well thatL(E)is not simple. To do so, we construct a new graphF =(F0, F1, r
F, sF)=(E0−H, r−1(E0−H ), r|E0−H, s|E0−H).
In other words, F is the graph consisting of all vertices not in H, together with all edges whose range is not in H. To ensure that F is well defined, we must check that
s(e)∈F0, since otherwise we haves(e)∈H; but sincer(e)s(e)andH is hereditary, we getr(e)∈H, which contradictse∈F1. SoF is a well defined graph.
We now produce a K-algebra homomorphism Ψ:L(E)→L(F ). To do so, we de-fine Φ on the generators of the free K-algebra B =K[E0∪E1∪(E1)∗] by setting
Φ(vi)=χF0(vi)vi,Φ(ei)=χF1(ei)ei andΦ(e∗i)=χ(F1)∗(ei∗)e∗i (whereχXdenotes the
usual characteristic function of a setX), and extending toB. In order to factorΦ through
A(E) we need to check that
vivj−δijvi: vi, vj∈E0 ∪
ei−eir(ei), ei−s(ei)ei: ei∈E1 ⊆Ker(Φ).
This is a straightforward computation done by cases, with the only nontrivial situation arising whenei∈F1. But thenr(ei) /∈H, and thereforeΦ(ei−eir(ei))=ei−eir(ei)=0
inL(F ). Now, sinces(ei)r(ei) /∈H andH is hereditary, we haves(ei) /∈H, so that
Φ(ei−s(ei)ei)=ei−s(ei)ei=0 inL(F ).
Now to produce the desired ring homomorphismΨ:L(E)→L(F )we need only check thatΦfactors through the relations ideal
ei∗ej−δijr(ej): ej∈E1, e∗i ∈
E1∗ ∪
vi−
{ej∈E1:s(ej)=vi}
eje∗j: vi∈s
E1
ofA(E). ThatΦ(ei∗ej−δijr(ej))=0 inL(F )is straightforward. So now considervi∈
s(E1); i.e., consider a vertexvi which is not a sink inE.
Case 1. Supposevi∈H. Then for everyej ∈E1 withs(ej)=vi we have thatei∈/F1
(otherwiseei∈F1impliesr(ei) /∈H and by hereditarinesss(ej)=vi ∈/H). So,Φ(vi−
{ej∈E1:s(ej)=vi}eje ∗
j)=0−
{ej∈E1:s(ej)=vi}0·0=0.
Case 2. Supposevi ∈/ H and vi ∈/s(F1). Since vi ∈s(E1)we have s−1(vi)= ∅. But
sinceH is saturated there must existei ∈E1 such thats(ei)=vi, butr(ei) /∈H. That
meansei∈F1withs(ei)=vi, which contradicts the hypothesis thatvi∈/s(F1). Thus the
saturated condition onH implies that Case 2 configuration cannot occur.
Case 3. Supposevi∈/Hbutvi∈s(F1). Then we have a (CK2) relation inL(F )atvi:
vi=
{ej∈F1:s(ej)=vi} eje∗j.
Considerej∈E1such thats(ej)=vi. Ifej ∈F1 thenΦ(ejej∗)=eje∗j. Ifej∈/F1then
Φ(eje∗j)=0. Thus we get
Φ
vi−
{ej∈E1:s(e
j)=vi} eje∗j
=vi−
{ej∈F1:s(ej)=vi}
eje∗j=0
Thus we have shown that there exists aK-algebra homomorphismΨ:L(E)→L(F ). Now consider Ker(Ψ )PL(E). SinceH= ∅there existsv∈H, so 0=v∈Ker(Ψ ). Since
H=E0there existsw∈E0−Hand in this caseΨ (w)=w=0 soΨ =0. In other words, 0=Ker(Ψ )=L(E), so thatL(E)is not simple.
Thus we conclude that the negation of either condition (i) or condition (ii) yields that
L(E)is not simple, which completes the proof of the theorem. 2
3.12. Remark. If we start with a finite and row-finite graphE=(E0, E1, r, s)withE0= {v1, . . . , vn}, E1= {e1, . . . , em}, there exist algorithms that decide, in a finite number of
steps, whether or not the graph satisfies conditions (i) and/or (ii), and therefore whether or notL(E)is simple.
3.13. Corollary. We re-establish the simplicity (or non-simplicity) of the algebras given in
Examples 1.4 above.
(i) Matrix algebrasMn(K): Since there are clearly no cycles inE, we need only verify
condition (i) in Theorem 3.11. To this end, letH = ∅be a set of vertices which is hereditary and saturated. Pickvi∈H. By hereditariness we have thatvi+1, . . . , vn∈
H. Now if we use the condition of being saturated at vi−1 we get that vi−1∈H,
and inductively vi−1, . . . , v1∈H and thereforeH=E0. Hence there are no
non-trivial hereditary and saturated subsets ofE0, and Theorem 3.11 applies to give that
Mn(K)=L(E)is simple.
(ii) Laurent polynomial algebrasK[x, x−1]: The cyclexdoes not have an exit, so by
The-orem 3.11L(E)∼=K[x, x−1]is not simple. (Indeed, similar to the argument which arises in the proof of Theorem 3.11, it is easy to show that 1∈ / 1+x.)
(iii) Leavitt algebrasL(1, n)forn2: The conditions in Theorem 3.11 are clearly
satis-fied here, soL(1, n)is simple, as was established in [7, Theorem 2].
3.14. Example. LetCndenote the graph havingnvertices andnedges, where the edges
form a single cycle. (In particular, the graph described in Example 1.4(ii) is the graphC1.)
ThenL(Cn)is not simple for alln, since the single cycle contains no exit.
3.15. Example. The Cuntz–Krieger algebraCKA(K)of a finite matrixAis defined in [1,
Example 2.5]. For a finite graphEwe can define the edge matrixAEassociated toE;AEis
then×nmatrix with entriesaij=δr(ei),s(ej), wheren= |E
1|. It is long but straightforward
to show that if a finite graphEhas no sinks nor sources, thenL(E)∼=CKAE(K).
In [1, Theorem 4.1] the authors provide sufficient conditions on A which yield the simplicity ofCKA(K), in caseAis a finite matrix which has no row or column of zeros,
and in case Ais not a permutation matrix. (There is also an additional condition on an associated functionαwhich must be satisfied in order to yield the simplicity ofCKA(K).)
But these conditions onAeliminate both the simple algebrasMn(K)and the non-simple
algebrasL(Cn)from consideration in [1, Theorem 4.1], since the edge matrix for the graph
0 1 0 · · · 0 0 0 1 · · · 0
..
. ... ... . .. ...
0 0 0 · · · 1 0 0 0 · · · 0
,
which contains both a zero column and a zero row, while the edge matrix for the cycle graphCngiven in Example 3.14 is
0 1 0 · · · 0 0 0 1 · · · 0
..
. ... ... . .. ...
0 0 0 · · · 1 1 0 0 · · · 0
,
which is a permutation matrix.
Acknowledgments
The authors are grateful to E. Pardo for many valuable correspondences, and to the ref-eree for a very careful review (especially the comments regarding the relationship between Leavitt path algebras andC∗-algebras). The first author thanks P. Muhly for providing the opportunity to attend the NSF–CBMS conference onC∗-graph algebras held in Iowa City, Iowa, in May/June 2004. The second author was partially supported by the MCYT and Fondos FEDER, BFM2001-1938-C02-01, MTM2004-06580-C02-02, the “Plan Andaluz de Investigación y Desarrollo Tecnológico,” FQM 336 and by a FPU fellowship by the MEC (AP2001-1368). This work was done while the second author was a Research Scholar at the University of Colorado at Colorado Springs supported by a “Estancias breves” FPU grant. The second author thanks this host center for its warm hospitality.
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