The Leavitt Path Algebras of Ultragraphs

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by  M. Imanfar

THE LEAVITT PATH ALGEBRAS OF ULTRAGRAPHS
M. IMANFAR,A. POURABBASAND H.LARKI
7
1
0 Abstract. WeintroducetheLeavittpathalgebrasofultragraphsandwecharac-
2 terize their ideal structures. We then use this notion to introduce and study the

b algebraic analogous of Exel-Laca algebras.
e
F

8
2 1. Introduction

] The Cuntz-Krieger algebras were introduced and studied in [8] for binary-valued
A
matrices with finite index. Two immediate and important extensions of the Cuntz-
R
Krieger algebras are: (1) the class of C∗-algebras associated to (directed) graphs
.
h [13, 14, 7, 10] and (2) the Exel-Laca algebras of infinite matrices with {0,1}-entries
at [9]. It is shown in [10] that if G is a graph with no sinks and sources, then the C∗-
m algebra C∗(G) is canonically isomorphic to the Exel-Laca algebra O , where A is
AG G
[ the edge matrix of G. However, there are binary valued matrices which can not be

1 1
3 considered as the edge matrix of graphs (for example A = ). So the class of
v 1 0
(cid:20) (cid:21)
3 graph C∗-algebras does not include that of Exel-Laca algebras and vise versa.
2 To study both graph C∗-algebras and Exel-Laca algebras under one theory, Tom-
3
forde [19] introduced the notion of an ultragraph and its associated C∗-algebra.
0
0 Briefly, an ultragraph is a directed graph which allows the range of each edge to
1. be a nonempty set of vertices rather than a singleton vertex. We see in [19] that for
0 each binary-valued matrix A there exists an ultragraph G so that the C∗-algebra
A
7 of G is isomorphic to the Exel-Laca algebra of A. Furthermore, every graph C∗-
A
1
algebracanbeconsideredasanultragraphC∗-algebra, whereasthereisanultragraph
:
v C∗-algebra which is not a graph C∗-algebra nor an Exel-Laca algebra.
i
X Recently many authors have discussed the algebraic versions of matrix and graph
r C∗-algebras. For example, in [3] the algebraic analogue of the Cuntz-Krieger alge-
a
bra O , denoted by CK (K), was studied for finite matrix A, where K is a field.
A A
Also the Leavitt path algebra L (G) of directed graph G was introduced in [1, 2]
K
as the algebraic version of graph C∗-algebra C∗(G). The class of Leavitt path alge-
bras includes naturally the algebras CK (K) of [3] as well as the well-known Leavitt
A
algebras L(1,n) of [17]. More recently, Tomforde defined a new version of Leavitt
path algebras with coefficients in a unital commutative ring [20]. In the case R = C,
the Leavitt path algebra LC(G) is isomorphic to a dense ∗-subalgebra of the graph
C∗-algebra C∗(G).
The purpose of this paper is to define the algebraic versions of ultragraphs C∗-
algebras and Exel-Laca algebras. For an ultragraph G and unital commutative ring
2010 Mathematics Subject Classification. 16W50; 46L55.
Key words and phrases. Ultragraph C∗-algebra, Leavitt path algebra, Exel-Laca algebra.
1
2 M.IMANFAR,A.POURABBASANDH.LARKI
R, we define the Leavitt path algebra L (G). To study the ideal structure of L (G),
R R
we use the notion of quotient ultragraphs from [16]. Given an admissible pair (H,S)
in G, we define the Leavitt path algebra L (G/(H,S)) associated to the quotient
R
ultragraph G/(H,S) and we prove two kinds of uniqueness theorems, namely the
Cuntz-Kriegerandthegraded-uniquenesstheorems,forL (G/(H,S)). Nextweapply
R
these uniqueness theorems to analyze the ideal structure of L (G). Although the
R
construction of Leavitt path algebra of ultragraph will be similar to that of ordinary
graph, but we see in Sections 4 and 5 that the analysis of its structure is more
complicated. The initial motivation for defining Leavitt path algebras of ultragraphs
is to generalize the algebras CK (K) for infinite matrices. For this, we give an
A
algebraic version of Exel-Laca algebras which coincide with the Leavitt path algebras
of ultragraphs with no singular vertices.
The article is organized as follows. we define in Section 2 the Leavitt path algebra
L (G) of an ultragraph G over a unital commutative ring R. Next, we continue by
R
considering the definition of quotient ultragraphs of [16]. For any admissible pair
(H,S) in an ultragraph G, we associate the Leavitt path algebra L G/(H,S) to
R
the quotient ultragraph G/(H,S) and we see that the Leavitt path algebras L (G)
R
(cid:0) (cid:1)
and L (G/(H,S) have a similar behavior in their structure.
R
In Section 3, we prove versions of the graded and Cuntz-Krieger uniqueness the-
(cid:1)
orems for L G/(H,S) by approximating L G/(H,S) with R-algebras of finite
R R
graphs. Since L (G) is trivially isomorphic to L G/(∅,∅) , we have these unique-
(cid:0) R (cid:1) (cid:0) R (cid:1)
ness theorems for the Leavitt path algebras of ultragraphs either. By applying the
(cid:0) (cid:1)
graded-uniqueness theorem in Section 4, we give a complete description of basic
graded ideals of L (G) in terms of admissible pairs in G. In Section 5, we use the
R
Cuntz-Krieger uniqueness theorem to show that an ultragraph G satisfies Condition
(K) if and only if every basic ideal in L (G) is graded.
R
Finally, in Section 6, we generalize the algebraic Cuntz-Krieger algebra CK (K)
A
of [3], denoted by EL (R), for every infinite matrix A with entries in {0,1} and
A
every unital commutative ring R. In the case R = C, the Leavitt path algebra
LC(G) and the Exel-Laca C-algebra ELA(C) are isomorphic to dense ∗-subalgebras
of C∗(G) and O , respectively. We prove that the class of Leavitt path algebras
A
of ultragraphs contains the Leavitt path algebras as well as the algebraic Exel-Laca
algebras. Furthermore, we give an ultragraph G such that the Leavitt path algebra
L (G) is neither a Leavitt path algebra of graph nor an Exel-Laca R-algebra.
R
2. Leavitt path algebras of ultragraphs and quotient ultragraphs
In this section, we follow the standard constructions of [1] and [20] to define the
Leavitt path algebra of an ultragraph. Since the quotient of ultragraph is not an
ultragraph, we will have to define the Leavitt path algebras of quotient ultragraphs
at the end of this section to characterize the ideal structure in Section 4.
2.1. Ultragraph. Recall from [19] that an ultragraph G = (G0,G1,r,s) consists of a
set of vertices G0, a set of edges G1, the source map s : G1 → G0 and the range map
r :G1 → P(G0)\{∅}, where P(G0) is the collection of all subsets of G0. Throughout
this work, ultragraph G will be assumed to be countable in the sense that both G0
and G1 are countable.
THE LEAVITT PATH ALGEBRAS OF ULTRAGRAPHS 3
For a set X, a subcollection C of P(X) is said to be lattice if ∅ ∈ C and it is closed
under the set operations ∩ and ∪. An algebra is a lattice C such that A\B ∈ C for
all A,B ∈ C. If G is an ultragraph, we write G0 for the algebra in P(G0) generated
by {v},r(e) :v ∈ G0,e ∈G1 .
A path in ultragraph G is a sequence α= α α ···α of edges with s(α ) ∈r(α )
1 2 n i+1 i
(cid:8) (cid:9)
for 1 ≤ i ≤ n − 1 and we say that the path α has length |α| := n. We write Gn
for the set of all paths of length n and Path(G) := ∞ Gn for the set of finite
n=0
paths. We may extend the maps r and s on Path(G) by setting r(α) := r(α ) and
|α|
S
s(α) := s(α ) for |α| ≥ 2 and r(A) = s(A) := A for A ∈ G0. Let (G1)∗ be the set
1
of ghost edges {e∗ : e ∈ G1}. We also define the ghost path α∗ := α∗α∗ ···α∗ for
n n−1 1
every α = α ···α ∈ Gn and A∗ := A for every A ∈ G0.
1 n
Definition 2.1. Let G be an ultragraph and let R be a unital commutative ring. A
Leavitt G-family in an R-algebra X is a set {pA,se,se∗ : A ∈ G0 and e ∈ G1} of
elements in X such that
(LP1) p = 0, p p = p and p = p +p −p for all A,B ∈ G0;
∅ A B A∩B A∪B A B A∩B
(LP2) ps(e)se = sepr(e) = se and pr(e)se∗ = se∗ps(e) = se∗ for all e ∈ G1;
(LP3) se∗sf = δe,fpr(e) for all e,f ∈ G1;
(LP4) pv = s(e)=vsese∗ whenever 0 < |s−1(v)| < ∞,
where p denPotes p . We usually denote s := p for A ∈ G0 and s := s ···s
v {v} A A α α1 αn
for α = α ···α ∈ Path(G).
1 n
Remark 2.2. For a nonempty set X, we write W(X) for the set of words w :=
w w ···w from the alphabet X and we say that the word w has length |w| := n.
1 2 n
The free R-algebra generated by X is denoted by F W(X) . For a definition of the
R
free R-algebra we refer the reader to [5, Section 2.3].
(cid:0) (cid:1)
Definition 2.3. Let G be an ultragraph and let R be a unital commutative ring. The
Leavitt path algebra of G, denoted by L (G), is the R-algebra F W(X) /I, where
R R
X := G0∪{e,e∗ :e ∈G1}andI istheidealofthefreeR-algebraF W(X) generated
R(cid:0) (cid:1)
by the union of the following sets:
(cid:0) (cid:1)
• ∅,A∩B −AB,A∪B+A∩B−(A+B):A,B ∈ G0 ,
• e−s(e)e,e−er(e),e∗ −e∗s(e),e∗ −r(e)e∗ :e ∈G1 ,
(cid:8) (cid:9)
• e∗f −δ r(e) :e,f ∈ G1 ,
(cid:8) e,f (cid:9)
• v− ee∗ :0 < |s−1(v)| <∞ .
(cid:8) (cid:9)
s(e)=v
(cid:8) P (cid:9)
We denote {π(A),π(e),π(e∗)} by {pA,se,se∗}. Proposition 2.6 of [5] implies that
{s,p} is a universal Leavitt G-family, that means, if {S,P} is a Leavitt G-family in
an R-algebra Y, then there exists an R-algebra homomorphism φ :L (G) → Y such
R
that φ(pA) = PA, φ(se)= Se and φ(se∗) = Se∗ for A∈ G0 and e ∈G1.
By using the Relations of Definition 2.1 we have the following lemma.
4 M.IMANFAR,A.POURABBASANDH.LARKI
Lemma 2.4. Let G be an ultragraph and let {s,p} be a Leavitt G-family. For every
α,β,µ,ν ∈ Path(G)\G0 and every A,B ∈ G0, we have
′ ′ ′
sαµ′pBsν∗ if µ = βµ , s(µ )∈ A∩r(α) and |µ |≥ 1,
(sαpAsβ∗)(sµpBsν∗) =  sαpA∩r(β)∩Bsν∗ if µ = β, ′ ′ ′
 sαpAs(νβ′)∗ if β = µβ , s(β )∈ B ∩r(ν) and |β |≥ 1,

0 otherwise.

The above lemma and [5, Proposition 2.7] imply the following theorem.
Theorem 2.5. Let G be a quotient ultragraph and let R be a unital commutative ring.
Then
LR G) = spanR sαpAsβ∗ : α,β ∈ Path(G),A ∈ G0 and r(α)∩A∩r(β) 6= ∅ .
Further(cid:0)more, LR(G)(cid:8)is a Z-graded ring by the grading (cid:9)
LR(G)n = spanR sαpAsβ∗ :α,β ∈Path(G),A ∈ G0 and |α|−|β| = n (n ∈ Z).
Theorem 2.6. Le(cid:8)t G be an ultragraph and let R be a unital commut(cid:9)ative ring. If
L (G) = L (s,p), then rp 6= 0 for every nonempty set A ∈ G0 and everyr ∈ R\{0}.
R R A
Proof. By the universality, it suffices to generate a Leavitt G-family {s˜,p˜} in which
rp˜ 6= 0 for every nonempty set A ∈ G0 and every r ∈ R\{0}. For this, we follow
A
the standard argument of [20, Proposition 3.4]. For each e ∈ G1, define Z := R
e
to be direct sum of countably many copies of R and for each v ∈ G0, let
L
Z if |s−1(v)| =6 0,
e
s(e)=v
Z :=
v  L
 R if |s−1(v)| = 0.
For every ∅ 6= A ∈ G0, definePAL: v∈AZv → v∈AZv to be the identity map.
Also, for each e ∈ G1 choose an isomorphism S : Z → Z ⊆ Z and let
L e L v∈r(e) v e s(e)
Se∗ := t−e1 : Ze ⊆ Zs(e) → v∈r(e)Zv. Now if ZL:= v∈G0Zv, then we naturally
extend all PA,Se, Se∗ to homomorphisms p˜A,s˜e,s˜e∗ ∈ HomR(Z,Z), respectively. It
L L
is straightforward to verify that {s˜,p˜} is a Leavitt G-family in Hom (Z,Z) such that
R
rp˜ 6= 0 for every ∅=6 A ∈ G0 and every r ∈ R\{0}. (cid:3)
A
Remark 2.7. Every directed graph G = (G0,G1,r,s) can be considered as an ultra-
graph G = (G0,G1,r,s), where G1 := G1 and the map r : G1 → P(G0)\{∅} is defined
by r(e) = {r(e)} for every e ∈ G1. In this case, the algebra G0 is the collection of all
finite subsets of G0. It is straightforward to see that the Leavitt path algebra L (G)
R
is naturally isomorphic to L (G) (see [1, 4, 20] for more details about the Leavitt
R
pathalgebras of directed graphs). Sotheclass of Leavitt pathalgebras ofultragraphs
contains the class of Leavitt path algebras of directed graphs.
2.2. Quotientultragraphs. Wewillusethenotionofquotientultragraphsfrom[16]
and we generalize the definition of Leavitt path algebras for quotient ultragraphs.
Definition 2.8. Let G be an ultragraph. A subcollection H ⊆ G0 is called hereditary
if satisfying the following conditions:
(1) s(e) ∈ H implies r(e) ∈ H for all e∈ G1.
THE LEAVITT PATH ALGEBRAS OF ULTRAGRAPHS 5
(2) A∪B ∈ H for all A,B ∈H.
(3) A∈ H, B ∈ G0 and B ⊆ A, imply B ∈ H.
Also, H ⊆G0 is called saturated if for any v ∈ G0 with 0 < |s−1(v)| < ∞, we have
r(e) :e ∈ G1 and s(e) = v ⊆ H implies {v} ∈ H.
For a saturat(cid:8)ed hereditary subcollection(cid:9)H ⊆ G0, the breaking vertices of H is
denoted by
B := v ∈ G0 : s−1(v) = ∞ but 0 < s−1(v)∩{e : r(e)∈/ H} < ∞ .
H
Following [12]n, an admis(cid:12)sible pa(cid:12)ir in G is a pa(cid:12)ir (H,S) of a saturated(cid:12)heredoitary set
(cid:12) (cid:12) (cid:12) (cid:12)
H ⊆ G0 and some S ⊆ B .
H
Let G = (G0,G1,r ,s ) be an ultragraph and let (H,S) be an admissible pair
G G
in G. We recall the definition of quotient ultragraph G/(H,S) form [16]. For each
A ∈ G0, we denote A := A ∪ {w′ : w ∈ A ∩ (B \ S)}. Consider the ultragraph
H
G := (G0,G1,r,s), where G1 := G1, G0 := G0 ∪{w′ : w ∈ B \S} and the maps r, s
H
are defined by r(e) := r (e) and
G
s (e)′ if s (e) ∈B \S and r (e) ∈ H,
s(e) := G G H G
s (e) otherwise,
G
(cid:26)
1 0
for every e∈ G , respectively. We write G for the algebra generated by the sets {v},
{w′} and {r(e)}.
Lemma 2.9 ([16, Lemma 3.5]). Let (H,S) be an admissible pair in an ultragraph G
0
and let ∼ be a relation on G defined by A∼ B if and only if there exists V ∈ H such
0
that A∪V = B∪V. Then ∼ is an equivalent relation on G and the operations
[A]∪[B]:= [A∪B], [A]∩[B]:= [A∩B] and [A]\[B]:= [A\B]
0
are well-defined on the equivalent classes {[A] :A ∈ G }.
Definition 2.10. Let (H,S) be an admissible pair in G. The quotient ultragraph of G
by (H,S) is the quadruple G/(H,S) := (GH)0,(GH)1,r,s , where
S S
(GH)0 := [v] : v ∈ G0\(cid:0)H ∪ [w′] :w ∈(cid:1)B \S ,
S H
(G(cid:8)H)1 := e∈ G1 =(cid:9)G1(cid:8): r(e) ∈/ H , (cid:9)
S
and s : (GH)1 → (GH)0 and r : ((cid:8)GH)1 → {[A] : A ∈ G0}(cid:9)are the maps defined by
S S S
s(e) := [s(e)] and r(e) := [r(e)] for every e ∈ (GH)1, respectively.
S
0 0
We usually denote [v] instead of [{v}] for every vertex v ∈ G . For A,B ∈ G , we
0
write [A] ⊆ [B] whenever [A]∩[B] = [A]. Also, the smallest algebra in {[A] :A ∈ G }
containing
[v],[w′]: v ∈ G0\H,w ∈ B \S ∪ r(e) :e ∈ (GH)1
H S
n o n o
is denoted by (GH)0.
S
6 M.IMANFAR,A.POURABBASANDH.LARKI
Lemma 2.11. If G/(H,S) is a quotient ultragraph, then
k nj
(GH)0 = A \B :A ,B ∈ [v],r(e) :v ∈ G0,e ∈ (GH)1 .
S ij ij ij ij S
(cid:26)j[=1i\j=1 (cid:8) (cid:9)(cid:27)
Proof. We denote by X the right hand side of the above equality. It is clear that
X ⊆ (GH)0, because (GH)0 is an algebra generated by the elements [v] and r(e). For
S S
the reverse inclusion, we note that X is a lattice. Furthermore, one can show that
X is closed under the operation \. Thus X is an algebra contains [v] and r(e) and
consequently (GH)0 ⊆X. (cid:3)
S
Remark 2.12. If A,B ∈ G0, then A∪B = A∪B, A∩B = A∩B and A\B = A\B.
Thus, by applying Lemma 2.11 for G and G, we deduce that A ∈ G0 for all A ∈ G0.
One can see that
G0 = A∪F ∪F : A∈ G0, F and F are finite subsets of G0
1 2 1 2
(cid:8) and {w′ :w ∈ BH}, respectively .
Furthermore, it follows from Lemma 2.9 that (GH)0 = [A] :A ∈ G0 . (cid:9)
S
Remark 2.13. The hereditary property of H and Rema(cid:8)rk 2.12 impl(cid:9)y that A ∼ B if
and only if both A\B and B \A belong to H.
Using Theorem 4.4(1), we define the Leavitt path algebra of a quotient ultragraph
G/(H,S) which iscorrespondingtothequotient R-algebra L (G)/I . We usethis
R (H,S)
concept to characterize the ideal structure of L (G) in Section 4.
R
Definition 2.14. Let G/(H,S) be a quotient ultragraph and let R bea unital commu-
tative ring. A Leavitt G/(H,S)-family in an R-algebra X is a set {q[A],te,te∗ : [A] ∈
(GH)0 and e ∈(GH)1} of elements in X such that
S S
(LP1) q = 0, q q = q and q = q +q −q ;
[∅] [A] [B] [A]∩[B] [A]∪[B] [A] [B] [A]∩[B]
(LP2) qs(e)te = teqr(e) = te and qr(e)te∗ = te∗qs(e) = te∗;
(LP3) te∗tf = δe,fqr(e);
(LP4) q[v] = s(e)=[v]tete∗ whenever 0 < |s−1([v])| < ∞.
The Leavitt path algebra of G/(H,S), denoted by L G/(H,S) , is the (unique up to
P R
isomorphism) universal R-algebra generated by a Leavitt G/(H,S)-family.
(cid:0) (cid:1)
Similar to ultragraphs, a path in G/(H,S) is a sequence α := α ···α of edges in
1 n
(GH)1 suchthats(α ) ⊆ r(α )for1≤ i ≤ n−1. WedenotebyPath G/(H,S) , the
S i+1 i
union of paths with finite length and we define [A]∗ := [A] and α∗ := α∗α∗ ···α∗,
(cid:0) n n−1 (cid:1) 1
for every [A] ∈ (GH)0 and α ∈ Path G/(H,S) . The maps r and s can be naturally
S
extended on Path G/(H,S) .
(cid:0) (cid:1)
From now on we denote the universal Leavitt G-family and G/(H,S)-family by
(cid:0) (cid:1)
{s,p} and {t,q}, respectively. The proof of the next theorem is similar to the argu-
ments of Theorem 2.5.
Theorem 2.15. Let G/(H,S) be a quotient ultragraph and let R be a unital commu-
tative ring. Then
LR G/(H,S) = spanR tαq[A]tβ∗ : α,β ∈ Path G/(H,S) andr(α)∩[A]∩r(β) 6= [∅] .
(cid:0) (cid:1) (cid:8) (cid:0) (cid:1) (cid:9)
THE LEAVITT PATH ALGEBRAS OF ULTRAGRAPHS 7
Furthermore, L G/(H,S) is a Z-graded ring by the grading
R
LR G(cid:0)/(H,S) n(cid:1)= spanR tαq[A]tβ∗ : |α|−|β| = n (n ∈ Z).
Note that w(cid:0)e cannot f(cid:1)ollow the ar(cid:8)gument of Theorem 2.6(cid:9)to show that rq 6= 0.
[A]
For example, suppose that G is the ultragraph
v v v ...
1 2 3
e
e e e
v
0
and let H bethe collection of all finite subsets of {v ,v ,...}, which is a hereditary
1 2
and saturated subcollection of G0. If we consider the quotient ultragraph G/(H,∅),
then {[∅] 6= [v] : [v] ⊆ r(e)} = ∅. Thus we can not define the idempotent q :
r(e)
Z → Z as in the proof of Theorem 2.6. In Section 4 we will
[v]⊆r(e) [v] [v]⊆r(e) [v]
prove this problem.
L L
3. Uniqueness Theorems
Let G/(H,S) be a quotient ultragraph. In this section, we prove the graded and
Cuntz-Krieger uniqueness theorems for L G/(H,S) and L (G). We do this by
R R
approximating the Leavitt path algebras of quotient ultragraphs with the Leavitt
(cid:0) (cid:1)
path algebras of finite graphs as in [16, Section 4] (see also [19, Section 5] and [18,
Section 2]). Ourproofinthissection is standardandwegive thedetails forsimplicity
of further results of the paper.
A vertex [v] ∈ (GH)0 is called a sink if s−1([v]) = ∅ and [v] is called an infinite
S
emitter if |s−1([v])| = ∞. A singular vertex is a vertex that is either a sink or an
infinite emitter. The set of all singular vertices is denoted by (GH)0 .
S sg
Let F ⊆ (GH)0 ∪ (GH)1 be a finite subset and write F0 := F ∩ (GH)0 and
S sg S S sg
F1 := F ∩ (GH)1 = {e ,...,e }. Following [16], we construct a finite graph G
S 1 n F
as follows. First, for every ω = (ω ,...,ω ) ∈ {0,1}n \ {0n}, we define r(ω) :=
1 n
r(e )\ r(e ) and R(ω) := r(ω)\ [v] which belong to (GH)0. We
ωi=1 i ωj=0 j [v]∈F0 S
also set
T S S
Γ := {ω ∈ {0,1}n \{0n} : there are vertices [v ],...,[v ]
0 1 m
such that R(ω) = m [v ] and [∅] 6= s−1([v ]) ⊆ F1 for 1 ≤ i≤ m},
i=1 i i
and
S
Γ := {ω ∈ {0,1}n \{0n} :R(ω) 6= [∅] and ω ∈/ Γ }.
F 0
Now we define the finite graph G = (G0,G1,r ,s ), where
F F F F F
G0 :=F0∪F1∪Γ ,
F F
G1 := (e,f) ∈ F1×F1 : s(f)⊆ r(e)
F
(cid:8)∪ (e,[v]) ∈ F1×F0 :[v] ⊆ r(e(cid:9))
∪(cid:8)(e,ω) ∈ F1×ΓF :ωi = 1 whe(cid:9)never e = ei ,
with sF(e,f) = sF(e,[v])(cid:8)= sF(e,ω) = e and rF(e,f) = f, rF(e,[v])(cid:9)= [v], rF(e,ω) =
ω.
8 M.IMANFAR,A.POURABBASANDH.LARKI
Lemma 3.1. Let G/(H,S) be a quotient ultragraph and let R be a unital commutative
ring. If L (G/(H,S)) = L (t,q), then we have the following assertion:
R R
(1) For every finite set F ⊆ (GH)0 ∪(GH)1, the elements
S sg S
Pe := tete∗, P[v] := q[v](1− tete∗), Pω := qR(ω)(1− tete∗),
e∈F1 e∈F1
S := t P , S := t P ,P S := t P , P
(e,f) e f (e,[v]) e [v] (e,ω) e ω
S(e,f)∗ := Pfte∗, S(e,[v])∗ := P[v]te∗, S(e,ω)∗ := Pωte∗,
form a Leavitt G -family which generates the subalgebra of L G/(H,S) generated
F R
by q[v],te,te∗ :[v] ∈ F0,e ∈F1 . (cid:0) (cid:1)
(2) For r ∈ R\{0}, if rq 6= 0 for all [A] 6= [∅], then rP 6= 0 for all z ∈ G0. In
(cid:8) [A] (cid:9) z F
this case, we have
LR(GF)∼= LR(S,P) = hq[v],te,te∗ : [v] ∈ F0,e ∈ F1i.
Proof. A similar argument as in the proof of [16, Proposition 4.2] implies that {S,P}
is a Leavitt GF-family that generating the subalgebra hq[v],te,te∗ : [v] ∈ F0,e ∈ F1i
of L G/(H,S) .
R
For (2), fix r ∈ R\{0}. Since rq 6= 0 for every [A] 6= [∅], we have rt 6= 0 and
[A] e
(cid:0) (cid:1)
rte∗ 6= 0 for every edge e. Thus all elements rPe and rP[v] are nonzero. Moreover, for
each ω ∈ Γ , there is a vertex [v] ⊆ R(ω) such that either [v] is a sink or there is an
F
edge f ∈ (GH)1\F1 with s(f)= [v]. In the former case, we have q (rP )= rq 6= 0
S [v] ω [v]
and in the later, tf∗(rPω) = rtf∗ 6= 0. Thus all rPω must be nonzero. Consequently,
rP 6= 0 for every vertex z ∈ G0.
z F
Now we show that L (G ) ∼= L (S,P). Note that for z ∈ G0 and g ∈ G1,
R F R F F
the degree of Pz, Sg and Sg∗ as elements in LR G/(H,S) are 0, 1 and -1, re-
spectively. So L (S,P) is a graded subalgebra of L G/(H,S) with the grading
R (cid:0) R (cid:1)
L (S,P) := L (S,P)∩L G/(H,S) . By the universal property, there is an
R n R R n (cid:0) (cid:1)
R-algebra homomorphism π : L (G ) → L (S,P) such that π(rp ) = rP 6= 0,
R F R z z
(cid:8) (cid:0) (cid:1) (cid:9)
π(sg) = Sg and π(sg∗) = Sg∗ for z ∈ G0F, g ∈ G1F and r ∈ R\{0}. Since π preserves
the degree of generators, the graded-uniqueness theorem for graphs [20, Theorem
5.3] implies that π is injective. As π is also surjective, we conclude that L (G ) is
R F
isomorphic to L (S,P) as R-algebras. (cid:3)
R
Theorem 3.2 (The graded-uniqueness theorem). Let G/(H,S) be a quotient ul-
tragraph and let R be a unital commutative ring. If S is a Z-graded ring and
π : L G/(H,S) → S is a graded ring homomorphism with π(rq ) 6= 0 for all
R [A]
[A] 6= [∅] and r ∈ R\{0}, then π is injective.
(cid:0) (cid:1)
Proof. Wefollowthesimilarargumentof[16,Theorem4.5]. Let{F }beanincreasing
n
sequence of finite subsets of (GH)0 ∪(GH)1 such that ∪∞ F = (GH)0 ∪(GH)1. For
S sg S n=1 n S sg S
each n, the graded subalgebras of LR G/(H,S) generated by {q[v],te,te∗ : [v] ∈
F0 and e ∈ F1} is denoted by X . Since π(rq ) 6= 0 for all [A] 6= [∅] and r ∈
n n n (cid:0) [(cid:1)A]
R\{0}, by Lemma 3.1, there is an graded isomorphism π : L (G ) → X . Thus
n R Fn n
π◦π :L (G )→ S is a graded homomorphism such that π◦π (rp )6= 0 for every
n R Fn n z
z ∈ G0 and r ∈ R\{0}. Hence, we may apply the graded-uniqueness theorem for
Fn
graphs [20, Theorem 5.3] to obtain the injectivity of π◦π .
n
THE LEAVITT PATH ALGEBRAS OF ULTRAGRAPHS 9
If [v] is a non-singular vertex, then we have q[v] = s(e)=[v]tete∗. Furthermore,
q = q −q q forevery[A],[B] ∈ (GH)0. Thus,byLemma2.11,L (G/(H,S))
[A]\[B] [A] [A] [B] S P R
is an R-algebra generated by
q[v],te,te∗ :[v] ∈ (GHS )0sg and e∈ (GSH)1 ,
and so ∪∞n=1Xn = LR(cid:8)(G/(H,S)). This follows that π is inje(cid:9)ctive on LR(G/(H,S)),
as desired. (cid:3)
Remark 3.3. If (H,S) = (∅,∅), then we have [A] = {A} for each A ∈ (GH)0. In
S
this case, every Leavitt G/(∅,∅)-family is a Leavitt G-family and vice versa. So,
L (G) ∼= L G/(∅,∅) and Theorem 3.2 gives the graded-uniqueness theorem for
R R
Leavitt path algebras of ultragraphs.
(cid:0) (cid:1)
Corollary 3.4. Let G be an ultragraph, R a unital commutative ring and S a Z-
graded ring. If π : L (G) → S is a graded ring homomorphism such that π(rp ) 6= 0
R A
for all ∅ =6 A∈ G0 and r ∈ R\{0}, then π is injective.
Definition 3.5. A loop in a quotient ultragraph G/(H,S) is a path α with |α| ≥ 1
and s(α) ⊆ r(α). An exit for a loop α ···α is an edge f ∈ (GH)1 with the property
1 n S
that s(f) ⊆ r(α ) but f 6= α for some 1 ≤ i ≤ n, where α := α . We say that
i i+1 n+1 1
G/(H,S) satisfies Condition (L) if every loop α:= α ···α in G/(H,S) has an exit,
1 n
or r(α )6= s(α ) for some 1 ≤ i≤ n.
i i+1
Theorem 3.6 (The Cuntz-Krieger uniqueness theorem). Let G/(H,S) be a quotient
ultragraph satisfying Condition (L) and let R be a unital commutative ring. If S is a
ring and π : L G/(H,S) → S is a ring homomorphism such that π(rq ) 6= 0 for
R [A]
every [A] 6= [∅] and r ∈ R\{0}, then π is injective.
(cid:0) (cid:1)
Proof. Choose an increasing sequence {F } of finite subsets of (GH)0 ∪(GH)1 such
n S sg S
that ∪∞ F = (GH)0 ∪(GH)1. Let X be the subalgebras of L (G/(H,S)) as in
n=1 n S sg S n R
Theorem 3.2. Since π(rq ) 6= 0 for all [A] 6= [∅] and r ∈ R\{0}, by Lemma 3.1,
[A]
there exists an isomorphism π : L (G ) → X for each n ∈ N. Furthermore,
n R Fn n
π◦π (rp )6= 0 for every z ∈ G0 and r ∈ R\{0}. Since G/(H,S) satisfies Condition
n z Fn
(L), By [16, Lemma 4.7], all finite graphs G satisfy Condition (L). So, the Cuntz-
Fn
Kriegeruniquenesstheoremforgraphs[20,Theorem6.5]impliesthatπ◦π isinjective
n
for n ≥ 1. Now by the fact ∪∞ X = L (G/(H,S)), we conclude that π is an
n=1 n R
injective homomorphism. (cid:3)
Corollary 3.7. Let G be an ultragraph satisfying Condition (L), R a unital com-
mutative ring and S a ring. If π : L (G) → S is a ring homomorphism such that
R
π(rp )6= 0 for all ∅=6 A ∈ G0 and r ∈ R\{0}, then π is injective.
A
4. Basic Graded Ideals
In this section, we apply the graded-uniqueness theorem for quotient ultragraphs
to investigate the ideal structure of L (G). As in [20], we would like to consider the
R
ideals of L (G) that are reflected in the structure of the ultragraph G. For this, we
R
give the following definition of basic ideals which is a generalization of [20, Definition
7.2] to the non-row-finite case and also it is a generalization of [15, Definition 5.2] to
the ultragraph setting.
10 M.IMANFAR,A.POURABBASANDH.LARKI
Let (H,S) be an admissible pair in an ultragraph G. For any w ∈ B , we define
H
the gap idempotent
pHw := pw − sese∗.
s(e)=w, r(e)∈/H
X
Let I be an ideal in L (G). We write H := {A ∈ G0 : p ∈ L (G)}, which is
R I A R
a saturated hereditary subcollection of G0. We say that the ideal I is basic if the
following conditions hold:
(1) rp ∈ I implies p ∈ I for A∈ G0 and r ∈R\{0}.
A A
(2) rpHI ∈ I implies pHI ∈ I for w ∈ B and r ∈ R\{0}.
w w HI
For an admissible pair (H,S) in G, the (two-sided) ideal of L (G) generated by
R
the idempotents {p :A ∈H}∪ pH : w ∈ S is denoted by I .
A w (H,S)
Lemma 4.1. If (H,S) is an adm(cid:8)issible pair(cid:9)in ultragraph G, then
I(H,S) := spanR sαpAsβ∗,sµpHwsν∗ :A ∈ H and w ∈ S
and I(H,S) is a graded basic ide(cid:8)al of LR(G). (cid:9)
Proof. We denote the right-hand side of the above equality by J. Lemma 2.4 and
the hereditary property of H imply that J is an ideal of L (G) being contained in I.
R
On the other hand, all generators of I belong to I and so we have I = J.
(H,S) (H,S)
Note that the elements sαpAsβ∗ and sµpHwsν∗ are homogeneous elements of degrees
|α|−|β| and |µ|−|ν|, respectively. Thus I is a graded ideal.
(H,S)
To show that I is a basic ideal, suppose rp ∈ I for some A ∈ G0 and
(H,S) A (H,S)
r ∈ R\{0} and write
n m
rpA = x := risαipAisβi∗ + sjsµjpHwjsνj∗,
i=1 j=1
X X
where A ∈ H, w ∈ S and r ,s ∈ R for all i,j. We first show assertion (1) of the
i j i j
definition of basic ideal in several steps.
Step I: If A = {v} ∈/ H and v ∈/ S, then 0 < |s−1(v)| < ∞.
Note that p x = x, so the assumption v ∈/ H ∪ S and the hereditary of H
v
imply that |α |,|µ | ≥ 1 and s(α ) = s(µ ) = v for every i,j. Hence v is not
i j i j
a sink. Set α = α α ···α . If |s−1(v)| = ∞, then there is an edge e 6=
i i,1 i,2 i,|α|
α1,1,...,αn,1,µ1,1,...,µm,1 with s(e) = v. So rse∗ = se∗(rpv) = se∗x = 0 which is a
contradiction. Thus 0 < |s−1(v)| < ∞.
Step II: If A∈/ H, then there exists v ∈ A such that {v} ∈/ H.
If rp = x, then A⊆ ∪ s(α )∪ s(µ ). Thus A = ∪ s(α )∩A ∪ ∪ s(µ )∩A .
A i i j j i i j j
Supposethat {v} : v ∈ A ⊆H. SinceH is Hereditary, A ∈ H, which is impossible.
(cid:0) (cid:1) (cid:0) (cid:1)
Hence there is a vertex v ∈ A such that {v} ∈/ H.
(cid:8) (cid:9)
Step III: If A = {v}, then {v} ∈ H.
We go toward a contradiction and assume {v} ∈/ H. Set v := v. If v ∈B , then
1 1 H
there is an edge e ∈ G1 such that s(e ) = v and r(e ) ∈/ H. Otherwise, we have
1 1 1 1
0 < |s−1(v)| < ∞ by Step I. The saturation of H gives an edge e with s(e ) = v
1 1 1
and r(e )∈/ H. By Step II, there is a vertex v ∈ r(e ) such that {v }∈/ H. We may
1 2 1 2
repeat the argument to choose a path γ = γ ...γ for L = max |β |,|ν | +1,
1 L i,j i j
such that s(γ) = v and s(γ ),r(γ )∈/ H for all 1 ≤ k ≤ L. Note that s(γ ) ∈/ A and
k k k i
(cid:8) (cid:9)