www.elsevier.com/locate/jpaa
Purely infinite simple Leavitt path algebras
Gene Abrams
a,∗, Gonzalo Aranda Pino
baDepartment of Mathematics, University of Colorado, Colorado Springs, CO 80933, USA bDepartamento de ´Algebra, Geometr´ıa y Topolog´ıa, Universidad de M´alaga, 29071 M´alaga, Spain
Received 15 February 2005; received in revised form 25 August 2005 Available online 5 December 2005
Communicated by C.A. Weibel
Abstract
We give necessary and sufficient conditions on a row-finite graph E so that the Leavitt path algebra L(E) is purely infinite simple. This result provides the algebraic analog to the corresponding result for the Cuntz–Krieger C∗-algebra C∗(E) given in [T. Bates, D. Pask, I. Raeburn, W. Szyma´nski, The C∗-algebras of row-finite graphs, New York J. Math. 6 (2000) 307–324].
c
2005 Elsevier B.V. All rights reserved.
MSC:16S10; 16S34
An idempotentein a ring Ris calledinfiniteife Ris isomorphic as a right R-module to a proper direct summand of itself.Ris calledpurely infinitein case every nonzero right ideal of R contains an infinite idempotent. Much recent attention has been paid to the structure of purely infinite simple rings, from both an algebraic (see e.g. [3–5]) as well as an analytic (see e.g. [7,8,11]) point of view. The Leavitt path algebraL(E)of a graphE is investigated in [1].L(E)is the algebraic counterpart of the Cuntz–Krieger algebraC∗(E); furthermore, the class of algebras of the formL(E)significantly broadens the collection of algebras studied by Leavitt in his seminal papers [9] and [10]. In [1] the authors give necessary and sufficient conditions onE so thatL(E)is simple. In the current article we
∗Corresponding author.
E-mail addresses:[email protected](G. Abrams),[email protected](G. Aranda Pino).
0022-4049/$ - see front matter c2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jpaa.2005.10.010
provide necessary and sufficient conditions on E so that L(E)is purely infinite simple (Theorem 11).
We recall the definition of the Leavitt path algebraL(E).
Definitions 1. A (directed) graph E = (E0,E1,r,s) consists of two countable sets
E0,E1and functions r,s : E1 → E0. The elements of E0 are calledvertices and the elements ofE1edges. LetK be a field. Thepath K-algebra over Eis the free associative
K-algebra K[E0∪ E1] with relations given by: vivj = δi jvi for every vi, vj ∈ E0, andei = eir(ei) = s(ei)ei for every ei ∈ E1.The extended graph of E is the graph
b
E=(E0,E1∪(E1)∗,r0,s0), where(E1)∗= {e∗i :ei ∈ E1}and the functionsr0ands0are defined as:r0|E1 =r,s0|E1 =s,r0(ei∗)=s(ei), ands0(e∗i)=r(ei).We call the elements ofE1(resp.,(E1)∗) thereal edges(resp., theghost edges) ofE.
Now suppose that E is row-finite (i.e., that s−1(v) is finite for all v ∈ E0). The
Leavitt path algebra of E with coefficients in K, denoted by LK(E) (or L(E)when appropriate), is defined as the path K-algebra over the extended graph Eb, satisfying the
so-calledCuntz–Krieger relations:
(CK1) e∗iej =δi jr(ej)for everyej ∈ E1ande∗i ∈(E1)∗, and (CK2) vi =P{ej∈E1:s(ej)=vi}eje
∗
j for everyvi ∈E0for whichs
−1(v i)6= ∅.
Example 2. (i) Let E be the “finite line” graph defined by E0 = {v1, . . . , vn}, E1 = {y1, . . . ,yn−1},s(yi) = vi, andr(yi) = vi+1for i = 1, . . . ,n −1. ThenL(E) ∼=
Mn(K), via the mapvi 7→ ei i,yi 7→ ei i+1, and yi∗ 7→ ei+1i (whereei j denotes the standard(i,j)-matrix unit inMn(K)).
(ii) Let n ≥ 2. Let E be the “rose with n leaves” graph defined by E0 = {∗},
E1 = {y1, . . . ,yn}. Then L(E) ∼= L(1,n), theLeavitt algebrainvestigated in [10]. Specifically, L(E) is isomorphic to the free associative K-algebra with generators {xi,yi :1≤i ≤n}and relations
(1)xiyj =δi j for all 1≤i,j ≤n, and (2) n
X
i=1
yixi =1.
Throughout this article all graphs will be assumed to be row-finite. We briefly establish some graph-theoretic notation. For each edgee,s(e)is thesourceofeandr(e)is therange
ofe. A vertexvfor whichs−1(v)= ∅is called asink. A graphEisfiniteif E0is a finite set. Apathµin a graphEis a sequence of edgesµ=µ1. . . µnsuch thatr(µi)=s(µi+1) fori =1, . . . ,n−1. In such a case,s(µ):=s(µ1)is the source ofµandr(µ):=r(µn) is the range ofµ. For vertices we definer(v)=v =s(v). We define a preorder≤onE0
given by:v≤win the casew=vor there is a pathµsuch thats(µ)=vandr(µ)=w. Ifs(µ)=r(µ)ands(µi)6=s(µj)for everyi 6= j, thenµis a called acycle.E isacyclic if E contains no cycles. The set of paths of lengthn > 0 is denoted by En. The set of all paths (and vertices) isE∗ :=S
n≥0En. It is shown in [1] thatL(E)is aZ-gradedK -algebra, spanned as aK-vector space by{pq∗ | p,qare paths inE}. By [1, Lemma 1.6],
L(E)is unital if and only if E is finite; otherwise, L(E)is a ring with set of local units consisting of sums of distinct vertices.
Ifα∈ L(E)andd ∈ Z+, then we say thatαisrepresentable as an element of degree d in real (resp. ghost) edgesin case whenαcan be written as a sum of monomials from the aforementioned spanning set ofL(E), in such a way thatdis the maximum length of a pathp(resp.q) which appears in such monomials. We note that an element ofL(E)may be representable as an element of different degrees in real (resp. ghost) edges, depending on the particular representation used forα.
Lemma 3.Let E be a finite acyclic graph. Then L(E)is finite dimensional.
Proof.Since the graph is row-finite, the given condition onEis equivalent to the condition thatE∗is finite. The result now follows from the previous observation thatL(E)is spanned as aK-vector space by{pq∗| p,q are paths inE}.
Lemma 3is precisely the tool we need to establish the following key result.
Proposition 4.Let E be a graph. Then E is acyclic if and only if L(E)is a union of a chain of finite dimensional subalgebras.
Proof.Assume first that E is acyclic. If E is finite, then Lemma 3 gives the result. So now suppose E is infinite, and rename the vertices of E0 as a sequence {vi}∞
i=1. We now define a sequence{Fi}∞i=1 of subgraphs of E. Let Fi = (F
0 i,F
1
i ,r,s)where
Fi0 := {v1, . . . , vi} ∪r(s−1({v1, . . . , vi})),Fi1:=s−1({v1, . . . , vi}), andr,sare induced fromE. In particular,Fi ⊆Fi+1for alli. For anyi >0,L(Fi)is a subalgebra ofL(E)as follows. First note that we can constructφ:L(Fi)→ L(E)aK-algebra homomorphism because the Cuntz–Krieger relations in L(Fi)are consistent with those in L(E), in the following way. Considerva sink inFi(which need not be a sink inE), then we do not have CK2 atv inL(Fi). Ifvis not a sink inFi, then there existse ∈ Fi1 =s
−1({v
1, . . . , vi}) such that s(e) = v. Buts(e) ∈ {v1, . . . , vi} and therefore v = vj for some j, and then Fi1 = s−1({v1, . . . , vi})ensures that all the edges starting inv are in Fi, so CK2 atv is the same in L(Fi)as in L(E). The other relations offer no difficulty. Now, with a similar construction and argument to that used in [1, Proof of Theorem 3.11] we find
ψ : L(E) → L(Fi)a K-algebra homomorphism such thatψφ = I d|L(Fi), so that φ is a monomorphism, which we view as the inclusion map. By construction, each vertex inE0is in Fi for somei; furthermore, the edgeehase ∈ F1j, wheres(e) = vj. Thus we conclude that L(E) = S∞
i=1L(Fi). (We note here that the embedding of graphs
j :Fi ,→Eis a complete graph homomorphism in the sense of [6], so that the conclusion
L(E)=S∞
i=1L(Fi)can also be achieved by invoking [6, Lemma 2.1].)
SinceEis acyclic, so is eachFi. Moreover, eachFi is finite since, by the row-finiteness ofE, in each step we add only finitely many vertices. Thus, byLemma 3,L(Fi)is finite dimensional, so thatL(E)is indeed a union of a chain of finite dimensional subalgebras.
For the converse, let p ∈ E∗be a cycle inE. Then{pm}∞
m=1is a linearly independent infinite set, so thatpis not contained in any finite dimensional subalgebra ofL(E).
We note that when E is finite and acyclic thenL(E)can be shown to be isomorphic to a finite direct sum of full matrix rings overK, and, for any acyclicE,L(E)is a direct limit of subalgebras of this form. The proof follows along the same lines as that given in [8, Corollaries 2.2 and 2.3].
The description of the simple Leavitt path algebras given in [1] will play a key role here, so we briefly review the germane ideas. An edgee∈E1is anexitto the pathµ=µ1· · ·µn if there existsisuch thats(e)=s(µi)ande6=µi. A vertexw∈E0connects tov∈E0if
w≤v. A subsetH ⊆E0ishereditaryifw∈ Handw≤vimplyv∈ H;Hissaturated
if whenevers−1(v)6= ∅and{r(e):s(e)=v} ⊆ H, thenv∈ H. The main result of [1] is the following
Theorem 5 ([1, Theorem 3.11]).Let E be a graph. Then L(E)is simple if and only if:
(i) The only hereditary and saturated subsets of E0are∅and E0, and
(ii) Every cycle in E has an exit.
The following proposition will play an important role in the proof of our main result (Theorem 11).
Proposition 6.Let E be a graph with the property that every cycle has an exit. Then for every nonzeroα∈ L(E)there exist a,b∈L(E)such that aαb∈E0.
Proof. Letα be representable by an element having degree d in real edges. Ifd = 0, then by [1, Corollary 3.7] we are done. So supposed >0. By [1, Lemma 1.5], given a monomial which is not a vertex, either it begins with a real edge or all its edges are ghost edges. Thus we can write
α=
m
X
n=1
einαein +β
wherem≥ 1,einαein 6=0 for everyn, eachαein is representable as an element of degree less than that ofαin real edges, andβis a polynomial in only ghost edges (possibly zero). We will present a process by which we will findba,bbsuch thatbaαbb6=0 and is representable
as an element having degree less thandin real edges. For an arbitrary edgeej ∈E1, we have two cases: Case 1: j ∈ {i1, . . . ,im}. Thene∗jα = αej +e
∗
jβ. If this element is nonzero then by choosingba =e∗j andbba local unit forαwe would be done. For later use, we note that if
e∗jαis zero, thenαej = −e
∗
jβ, and thereforeejαej = −eje
∗
jβ.
Case 2: j 6∈ {i1, . . . ,im}. Thene∗jα=e∗jβ. Ife∗jβ 6=0, then withbbas before we would
havee∗jαbb is a nonzero polynomial which is representable as an element having degree
0<d in real edges, and again we would be done. For later use, we note that ife∗jβ =0, then in particular we have 0= −eje∗jβ.
So we may assume that we are in the latter possibilities of both Cases 1 and 2; i.e., we may assume thate∗α=0 for alle ∈ E1. We show that this situation cannot happen. First, supposevis a sink in E. Then we may assumevβ =0, as follows. Multiplying the displayed equation byvon the left givesvα =vPm
n=1einαein +vβ. Sincevis a sink we havevein = 0 for all 1 ≤ n ≤ m, so thatvα = vβ. But if vβ 6= 0 thenba = vandbb
as above would yield a nonzero element in only ghost edges and we would be done as in Case 2.
Now letS1= {vj ∈ E0:vj =s(ein)for some 1≤n≤m}, and letS2= {vk1, . . . , vkt} where(Pt
i=1vki)β=β. We note thatwβ=0 for everyw∈ E 0−S
there are no sinks inS1, and by a previous observation we may assume that there are no sinks inS2. LetS=S1∪S2. Then in particular we have(Pv∈Sv)β=β.
We now argue that in this situationαmust be zero. To this end,
α =
m
X
n=1
einαein +β= m
X
n=1 −eine
∗
inβ+β (by Case 1)
= m
X
n=1 −eine
∗
inβ−
X
j6∈{i1,...,im} s(e j)∈S
eje∗j
β
+β
(by Case 2, the newly subtracted terms equal 0)
= − X
v∈S
v !
β+β (no sinks inSimplies that CK2 applies at eachv∈S)
= −β+β =0.
As we have assumed α 6= 0 we have reached the desired contradiction. Thus we are always able to findba,bb such thatbaαbb is nonzero, and is representable in degree less
thand in real edges. By repeating this process enough times (d at most), we can find
b
ak, . . . ,ab1,bb1, . . . ,bbksuch that we can representabk. . .ab1αbb1. . .bbk 6=0 by an element of
degree zero in real edges. Thus [1, Corollary 3.7] applies, and finishes the proof.
Aclosed simple path based atvi0 is a pathµ=µ1. . . µn, withµj ∈ E
1,n ≥1 such thats(µj)6=vi0 for every j >1 ands(µ)=r(µ)=vi0. Denote byCSP(vi0)the set of all
such paths. We note that a cycle is a closed simple path based at any of its vertices, but not every closed simple path based atvi0 is a cycle. We define the following subsets ofE
0:
V0= {v∈E0:CSP(v)= ∅}
V1= {v∈E0: |CSP(v)| =1}
V2= E0−(V0∪V1).
Lemma 7.Let E be a graph. If L(E)is simple, then V1= ∅.
Proof.For any subset X ⊆ E0 we define the following subsets. H(X)is the set of all vertices that can be obtained by one application of the hereditary condition at any of the vertices ofX; that is,H(X):=r(s−1(X)). Similarly,S(X)is the set of all vertices obtained by applying the saturated condition among elements ofX, that is,S(X):= {v∈ E0: ∅ 6= {r(e): s(e)= v} ⊆ X}. We now defineG0 := X, and forn ≥ 0 we define inductively
Gn+1:= H(Gn)∪S(Gn)∪Gn. It is not difficult to show that the smallest hereditary and saturated subset ofE0containingX is the setG(X):=S
n≥0Gn.
Suppose now thatv ∈ V1, so thatCSP(v) = {p}. In this case pis clearly a cycle. By Theorem 5we can find an edgeewhich is an exit for p. Let Abe the set of all vertices in the cycle. Sincepis the only cycle based atv, andeis an exit for p, we conclude that
r(e)6∈ A. Consider then the setX = {r(e)}, and constructG(X)as described above. Then
NowTheorem 5implies thatG(X)=E0, so we can findn =min{m: A∩Gm 6= ∅}. Takew ∈ A∩Gn. We are going to show thatw ≥ r(e). First, sincer(e) 6∈ A, then
n >0 and thereforew∈ H(Gn−1)∪S(Gn−1)∪Gn−1. Here,w∈ Gn−1cannot happen by the minimality ofn. Ifw ∈ S(Gn−1)then∅ 6= {r(e) : s(e) = w} ⊆ Gn−1. Since
w is in the cycle p, there exists f ∈ E1 such that r(f) ∈ A ands(f) = w. In that caser(f) ∈ A ∪Gn−1 again contradicts the minimality ofn. So the only possibility is w ∈ H(Gn−1), which means that there existsei1 ∈ E
1 such thatr(e
i1) = w and
s(ei1)∈Gn−1.
We now repeat the process with the vertexw0 =s(ei1). Ifw
0 ∈ G
n−2then we would havew∈Gn−1, again contradicting the minimality ofn. Ifw0∈ S(Gn−2)then, as above, {r(e) : s(e) = w0} ⊆ Gn−2, so in particular would givew = r(ei1) ∈ Gn−2, which is
absurd. So thereforew0 ∈ H(Gn−2)and we can findei2 ∈ E
1such thatr(e i2)=w
0and
s(ei2)∈Gn−2.
Afternsteps we will have found a pathq =ein· · ·ei1 withr(q)=wands(q)=r(e).
In particular we havew ≥ s(e), and therefore there exists a cycle based atwcontaining the edgee. Sinceeis not in pwe get|CSP(w)| ≥2. Sincewis a vertex contained in the cycle p, we then get|CSP(v)| ≥2, contrary to the definition of the setV1.
Lemma 8. Suppose A is a union of finite dimensional subalgebras. Then A is not purely infinite. In fact, A contains no infinite idempotents.
Proof. It suffices to show the second statement. So just supposee =e2 ∈ Ais infinite. Thene Acontains a proper direct summand isomorphic toe A, which in turn, by definition and a standard argument, is equivalent to the existence of elements g,h,x,y ∈ Asuch thatg2 =g,h2=h,gh =hg=0,e=g+h,h 6=0,x ∈ e Ag,y ∈ g Aewithx y =e
and yx = g. But by hypothesis the five elementse,g,h,x,y are contained in a finite dimensional subalgebraBofA, which would yield thatBcontains an infinite idempotent, and thus contains a non-artinian right ideal, which is impossible.
Proposition 9.Let E be a graph. Suppose thatw ∈ E0has the property that, for every
v∈E0,w≤vimpliesv∈V0. Then the corner algebrawL(E)wis not purely infinite. Proof. Consider the graph H = (H0,H1,r,s) defined by H0 := {v : w ≤ v},
H1 := s−1(H0), andr,s induced by E. The only nontrivial part of showing that H is a well defined graph is verifying thatr(s−1(H0)) ⊆ H0. Takez ∈ H0ande∈ E1such thats(e)=z. But we havew≤zand thusw≤r(e)as well, that is,r(e)∈ H0.
Using that H is acyclic, along with the same argument as given inProposition 4, we have that L(H)is a subalgebra of L(E). Thus Proposition 4applies, which yields that
L(H)is the union of finite dimensional subalgebras, and therefore contains no infinite idempotents byLemma 8. AswL(H)wis a subalgebra ofL(H), it too contains no infinite idempotents, and thus is not purely infinite.
We claim thatwL(H)w = wL(E)w. To see this, givenα = P
piqi∗ ∈ L(E), then
wαw=P
pijq
∗
ij withs(pij)=w=s(qij)and thereforepij,qij ∈L(H). ThuswL(E)w is not purely infinite as desired.
We thank P. Ara for indicating the following result, which will provide the direction of proof for our main theorem. A right A-moduleT is calleddirectly infinitein the caseT
contains a proper direct summandT0 such thatT0 ∼= T. (In particular, the idempotente
is infinite precisely whene Ais directly infinite.) Recall that a ringAhaslocal unitsif for every finite subset{x1, . . . ,xn} ⊆ A there existse = e2 ∈ Awithxi ∈ e Ae for every
i =1, . . . ,n.
Proposition 10. Let A be a ring with local units. The following are equivalent:
(i) A is purely infinite simple.
(ii) A is simple, and for each nonzero finitely generated projective right A-module P, every nonzero submodule C of P contains a direct summand T of P for which T is directly infinite. (In particular, the property ‘purely infinite simple’ is a Morita invariant of the ring.)
(iii) wAwis purely infinite simple for every nonzero idempotentw∈ A.
(iv) A is simple, and there exists a nonzero idempotentwin A for whichwAwis purely infinite simple.
(v) A is not a division ring, and A has the property that for every pair of nonzero elements
α, βin A there exist elements a,b in A such that aαb=β.
Proof.(i) ⇔ (ii). Suppose A is purely infinite simple. Let P be any nonzero finitely generated projective rightA-module. Then P is a generator for Mod-A, as follows. Since
Agenerates Mod-AandPis finitely generated we have an integernsuch thatP⊕P0∼=An
as right A-modules. Again using that P is finitely generated, and using that A has local units, we have thatPis isomorphic to a direct summand of a right A-module of the form
f1A⊕· · ·⊕ftA, where each fiis idempotent. But this gives HomA(P, f1A⊕· · ·⊕ftA)6= 0, which in turn gives 06=HomA(P,At)∼= (HomA(P,A))t, so that HomA(P,A)6=0. ButP{
a ∈ A|a =g(p)for somep∈ Pand someg∈HomA(P,A)}is then a nonzero two-sided ideal of A, which necessarily equals Aas Ais simple. Now lete = e2 ∈ A. Then e = Pr
i=1gi(pi) for some pi ∈ P and gi ∈ HomA(P,A), which gives that
λe◦ ⊕gi : Pr → A → e Ais a surjection. Since P generates e Afor each idempotent
eof A, we conclude thatPgenerates Mod-A.
This observation allows us to argue exactly as in the proof of [5, Lemma 1.4 and Proposition 1.5] that if e = e2 ∈ A, then there exists a right A-module Q for which
e A∼= P ⊕Q. Since Ais purely infinite, there exists an infinite idempotente ∈ A. The indicated isomorphism yields that any submoduleC of P is isomorphic to a submodule
C0 ofe A, so that by the hypothesis that A is purely infinite we have thatC0 contains a submoduleT0which is directly infinite, and for whichT0is a direct summand ofe A. But by a standard argument, any direct summand ofe Ais equal to f Afor some idempotent
f ∈ A, so thatT0 = f Afor some infinite idempotent f of A. LetT be the preimage of
T0in P⊕Qunder the isomorphism. ThenT is directly infinite, and since f Ais a direct summand ofe Awe have thatT is a direct summand ofP ⊕Qwhich is contained in P, and henceT is a direct summand ofP.
By [2, Proposition 3.3], the lattice of two-sided ideals of Morita equivalent rings are isomorphic, so that any ring Morita equivalent to a simple ring is simple. Therefore, since the indicated property is clearly preserved by equivalence functors, we have that “purely infinite simple” is a Morita invariant.
For the converse, letIbe a nonzero right ideal ofA. We show thatIcontains an infinite idempotent. Let 0 6= x ∈ I, so that x A ≤ I. But x = ex for some e = e2 ∈ A, so
x A≤e A. So by hypothesis,x Acontains a nonzero direct summandT ofe A, whereT is directly infinite. But as noted above we have thatT = f Afor f = f2 ∈ A, where f is infinite. Thus f ∈T ≤x A≤ I and we are done.
(ii)⇒(iii). Since we have established the equivalence of (i) and (ii), we may assume
Ais purely infinite simple. Then the simplicity ofAgives thatAwA=Afor any nonzero idempotent w ∈ A, which yields by [2, Proposition 3.5] that A andwAw are Morita equivalent, so that (iii) follows immediately from (ii).
(iii) ⇒(iv). It is tedious but straightforward to show that if Ais any ring with local units, andwAwis a simple (unital) ring for every nonzero idempotentwof A, then Ais simple.
(iv) ⇒ (i). Since A is simple we get AwA = A, so that A and wAw are Morita equivalent by the previously cited [2, Proposition 3.5].
Thus we have established the equivalence of statements (i) through (iv).
(i) ⇒ (v). Suppose Ais purely infinite simple. Then Ais not left artinian, so that A
cannot be a division ring. Now choose nonzero α, β ∈ A. Then there exists a nonzero idempotent w ∈ A such that α, β ∈ wAw. But wAw is purely infinite simple by
(i)⇔ (iii), so by [5, Theorem 1.6] there exista0,b0 ∈ wAwsuch thata0αb0 = w. But then fora = a0,b = b0β we haveaαb = β. Conversely, suppose A is not a division ring and that A satisfies the indicated property. Since Ais not a division ring andA is a ring with local units, there exists a nonzero idempotentwof Afor whichwAw is not a division ring. Letα∈ wAw. Then by hypothesis there exista0,b0in Awitha0αb0 =w. But sinceα ∈ wAw, by defininga = wa0w andb = wb0wwe have aαb = w. Thus another application of [5, Theorem 1.6] (noting thatwis the identity ofwAw) gives the desired conclusion.
(v) ⇒ (iv). The indicated multiplicative property yields that any nonzero ideal of A
will contain a set of local units forA, so that Ais simple. SinceAis not a division ring and
Ahas local units there exists a nonzero idempotentwofAsuch thatwAwis not a division ring. Letα, β ∈wAw; in particular,wαw=αandwβw=β. By hypothesis there exists
a,b ∈ Asuch thataαb = β. But then(waw)α(wbw) =wβw = β, which yields that
wAwis purely infinite simple by [5, Theorem 1.6].
We now have all the necessary ingredients in hand to prove the main result of this article.
Theorem 11. Let E be a graph. Then L(E)is purely infinite simple if and only if E has the following properties.
(i) The only hereditary and saturated subsets of E0are∅and E0.
(ii) Every cycle in E has an exit.
(iii) Every vertex connects to a cycle.
Proof. First, assume (i), (ii) and (iii) hold. ByTheorem 5we have thatL(E)is simple. By Proposition 10it suffices to show that L(E)is not a division ring, and that for every pair of elementsα, βinL(E)there exist elementsa,binL(E)such thataαb=β. Conditions (ii) and (iii) easily imply that| E1 |>1, so thatL(E)has zero divisors, and thus is not a division ring.
We now applyProposition 6to finda,b∈L(E)such thataαb=w∈ E0. By condition (iii),wconnects to a vertexv6∈V0. Eitherw=vor there exists a pathpsuch thatr(p)=v
ands(p)=w. By choosinga0 =b0 =v in the former case, anda0 = p∗,b0 = pin the latter, we have produced elementsa0,b0∈L(E)such thata0wb0=v.
An application of Lemma 7 yields thatv ∈ V2, so there exist p,q ∈ CSP(v) with
p 6=q. For anym >0 letcm denote the closed path pm−1q. Using [1, Lemma 2.2], it is not difficult to show thatc∗mcn=δmnvfor everym,n>0.
Now consider any vertex vl ∈ E0. Since L(E) is simple, there exist {ai,bi ∈
L(E) | 1 ≤ i ≤ t}such thatvl = Pti=1aivbi. But by definingal = Pti=1aic∗i and
bl =Ptj=1cjbj, we get
alvbl = t
X
i=1
aic∗i
! v
t
X
j=1
cjbj
!
= t
X
i=1
aic∗ivcibi =vl.
Now letsbe a left local unit forβ(i.e.,sβ =β), and writes=P
vl∈Svlfor some finite subset of verticesS. By lettingea =P
vl∈Salc
∗
l andeb= P
vl∈Sclbl, we get
eaveb= X
vl∈S
alcl∗vclbl=
X
vl∈S
vl =s.
Finally, lettinga =
eaa
0aandb=bb0
ebβ, we have thataαb=βas desired.
For the converse, suppose thatL(E)is purely infinite simple. ByTheorem 5we have (i) and (ii). If (iii) does not hold, then there exists a vertexw ∈ E0 such that w ≤ v impliesv ∈ V0. ApplyingProposition 9we get thatwL(E)wis not purely infinite. But thenProposition 10implies thatL(E)is not purely infinite, contrary to hypothesis.
Example 12. (i) Let E be the graph defined inExample 2(i). Then L(E) ∼= Mn(K) which of course is simple, but not purely infinite since no vertex in E0connects to a cycle.
(ii) Let n ≥ 2. Let E be the graph defined in Example 2(ii). Then L(E) ∼= L(1,n), the Leavitt algebra. Sincen ≥ 2 we see that all the hypotheses ofTheorem 11are satisfied, so thatL(1,n)is purely infinite simple.
(iii) LetEbe the graph havingE0= {v, w}andE1= {e, f,g}, wheres(e)=s(f)=v,
r(e) = r(f) = w, s(g) = w,r(g) = v. Then E satisfies the hypotheses of Theorem 11, so thatL(E)is purely infinite simple.
Let L(1,n)denote the Leavitt algebra described inExample 2(ii). We complete this article by providing a realization of the purely infinite simple algebra Mm(L(1,n))as a Leavitt path algebraL(E)for a specific graphE.
Proposition 13. Let n ≥ 2 and m ≥ 1. We define the graph Emn by setting E0 := {v1, . . . , vm}, E1 := {f1, . . . , fn,e1, . . . ,em−1}, r(fi) = s(fi) = vm for 1 ≤ i ≤ n,
r(ei)=vi+1, and s(ei)=vifor1≤i ≤m−1. Then L(Enm)=∼Mm(L(1,n)). Proof.We defineΦ:K[E0∪E1∪(E1)∗] →Mm(L(1,n))on the generators by
Φ(vi)=ei i for 1≤i ≤m
Φ(ei) =ei i+1 and Φ(e∗i)=ei+1i for 1≤i≤m−1 Φ(fi)= yiemm and Φ(fi∗)=xiemm for 1≤i≤n
and extend linearly and multiplicatively to obtain aK-homomorphism. We now verify that Φfactors through the ideal of relations inL(Enm).
First,Φ(vivj−δi jvi)=ei iej j−δi jei i =0. If we consider the relationsei−eir(ei)then we haveΦ(ei−eir(ei))=Φ(ei−eivi+1)=ei i+1−ei i+1ei+1i+1=0, and analogously Φ(ei−s(ei)ei)=0. For the relations fi−fir(fi)we getΦ(fi−fir(fi))=Φ(fi−fivm)=
yiemm−yiemmemm =0, and similarlyΦ(fi −s(fi)fi)=0. With similar computations it is easy to also see thatΦ(e∗i −ei∗r(ei∗)) = Φ(e∗i −s(ei∗)e∗i)= Φ(fi∗− fi∗r(fi∗)) = Φ(fi∗−s(fi∗)fi∗)=0.
We now check the Cuntz–Krieger relations. First, Φ(ei∗ej −δi jr(ej)) = Φ(e∗iej −
δi jvj+1)=ei+1iej j+1−δi jej+1j+1=δi jei+1j+1−δi jej+1j+1=0. Second,Φ(fi∗fj−
δi jr(fj))=Φ(fi∗fj−δi jvm)=xiemmyjemm−δi jemm =0, because of the relation (1) in
L(1,n). Finally,Φ(fi∗ej−δfi,ejr(ej))=Φ(f
∗
i ej−0vj+1)=Φ(f
∗
i ej)=xiemmej j+1= 0, and similarlyΦ(e∗i fj−δei,fjr(fj))=0.
With CK2 we have two cases. First, fori <m,Φ(vi −eie∗i)=ei i −ei i+1ei+1i =0. And forvm we haveΦ(vm −Pni=1 fifi∗)= emm −Pni=1yiemmxiemm = 0, because of the relation (2) inL(1,n).
This shows that we can factor Φ to obtain a K-homomorphism of algebras Φ :
L(Enm)→Mm(L(1,n)). We will see thatΦis onto. Consider any matrix unitei jandxk∈
L(1,n). If we take the path p =ei. . .en−1fk∗e∗n−1. . .e
∗
j ∈ L(E m
n)then we getΦ(p)=
ei i+1. . .en−1n(xkenn)enn−1. . .ej+1j = xkei j. Similarly Φ(ei. . .en−1fken∗−1. . .e
∗
j) =
ykei j. In this way we get that all the generators ofMm(L(1,n))are inI m(Φ).
Finally, using the same ideas as those presented in [1, Corollary 3.13(i)], we see thatEnm
satisfies the conditions ofTheorem 5, which yields the simplicity ofL(Enm). This implies thatΦis necessarily injective, and therefore an isomorphism.
Acknowledgments
The authors are grateful to the referee for the meticulous review of the original version of this article. The authors are also grateful to P. Ara and E. Pardo for many valuable correspondences. The second author was partially supported by the MCYT and Fondos FEDER, BFM2001-1938-C02-01, the “Plan Andaluz de Investigaci´on y Desarrollo Tecnol´ogico”, 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.
References
[1] G. Abrams, G. Aranda Pino, The Leavitt path algebra of a graph, J. Algebra 293 (2) (2005) 319–334. [2] P.N. ´Anh, L. M´arki, Morita equivalence for rings without identity, Tsukuba J. Math 11 (1) (1987) 1–16. [3] P. Ara, The exchange property for purely infinite simple rings, Proc. Amer. Math. Soc. 132 (2004)
2543–2547.
[4] P. Ara, M.A. Gonz´alez-Barroso, K.R. Goodearl, E. Pardo, Fractional skew monoid rings, J. Algebra 278 (2004) 104–126.
[6] P. Ara, M.A. Moreno, E. Pardo, Nonstable K-theory for graph algebras, Algebr. Represent. Theory (in press). [7] T. Bates, D. Pask, I. Raeburn, W. Szyma´nski, TheC∗-algebras of row-finite graphs, New York J. Math. 6
(2000) 307–324.
[8] A. Kumjian, D. Pask, I. Raeburn, Cuntz–Krieger algebras of directed graphs, Pacific J. Math. 184 (1) (1998) 161–174.
[9] W.G. Leavitt, The module type of a ring, Trans. Amer. Math. Soc. 42 (1962) 113–130. [10] W.G. Leavitt, The module type of homomorphic images, Duke Math. J. 32 (1965) 305–311.
[11] N.C. Phillips, A classification theorem for nuclear purely infinite simpleC∗-algebras, Doc. Math. 5 (2000) 49–114.
