Article
Kyungpook Mathematical Journal 2020; 60(1): 21-43
Published online March 31, 2020
Copyright © Kyungpook Mathematical Journal.
The Leavitt Path Algebras of Ultragraphs
Mostafa Imanfar and Abdolrasoul Pourabbas∗, Hossein Larki
Faculty of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Avenue, 15914 Tehran, Iran e-mail : m.imanfar@aut.ac.ir and arpabbas@aut.ac.ir
Department of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Iran e-mail : h.larki@scu.ac.ir
Abstract
- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References
We introduce the Leavitt path algebras of ultragraphs and we characterize their ideal structures. We then use this notion to introduce and study the algebraic analogy of Exel-Laca algebras.
Keywords: ultragraph C∗,-algebra, Leavitt path algebra, Exel-Laca algebra.
1. Introduction
- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References
The Cuntz-Krieger algebras were introduced and studied in [6] for binary-valued matrices with finite index. Two immediate and important extensions of the Cuntz-Krieger algebras are: (1) the class of , where
To study both graph so that the
is isomorphic to the Exel-Laca algebra of
Recently many authors have discussed the algebraic versions of matrix and graph , denoted by
, was studied for finite matrix
of [3] as well as the well-known Leavitt algebras
The purpose of this paper is to define the algebraic versions of ultragraphs and unital commutative ring
. To study the ideal structure of
, we use the notion of quotient ultragraphs from [13]. Given an admissible pair (
, we define the Leavitt path algebra
associated to the quotient ultragraph
and we prove two kinds of uniqueness theorems, namely the Cuntz-Krieger and the graded-uniqueness theorems, for
. Next we apply these uniqueness theorems to analyze the ideal structure of
. Although the construction of Leavitt path algebra of ultragraph will be similar to that of ordinary graph, we see in Sections 3 and 4 that the analysis of its structure is more complicated. The aim of the definition of ultragraph Leavitt path algebras can be summarized as follows:
Every Leavitt path algebra of a directed graph can be embedded as an subalgebra in a unital ultragraph Leavitt path algebra. Also, the ultragraph Leavitt path algebra
is isomorphic to a dense *-subalgebra of
.
By using the definition of ultragraph Leavitt path algebras, we give an algebraic version of Exel-Laca algebras.
The class of ultragraph Leavitt path algebras is strictly larger than the class of Leavitt path algebras of directed graphs.
The article is organized as follows. We define in Section 2 the Leavitt path algebra of an ultragraph
over a unital commutative ring
, we associate the Leavitt path algebra
to the quotient ultragraph
and we see that the Leavitt path algebras
and
have a similar behavior in their structure. Next, we prove versions of the graded and Cuntz-Krieger uniqueness theorems for
by approximating
with
By applying the graded-uniqueness theorem in Section 3, we give a complete description of basic graded ideals of in terms of admissible pairs in
. In Section 4, we use the Cuntz-Krieger uniqueness theorem to show that an ultragraph
satisfies Condition (K) if and only if every basic ideal in
is graded.
In Section 5, we generalize the algebraic Cuntz-Krieger algebra of [3], denoted by ℰℒ
. We prove that the class of Leavitt path algebras of ultragraphs contains the Leavitt path algebras as well as the algebraic Exel-Laca algebras. Furthermore, we give an ultragraph
such that the Leavitt path algebra
is neither a Leavitt path algebra of graph nor an Exel-Laca
2. Leavitt Path Algebras
- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References
In this section, we follow the standard constructions of [1] and [15] 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 and prove the uniqueness theorems for them to characterize the ideal structure in Section 3.
2.1. Ultragraphs
Recall from [15] that an consists of a set of vertices
, the source map
and the range map
, where ℘(
will be assumed to be countable in the sense that both
are countable.
For a set of ℘(
and it is closed under the set operations ∩ and ∪. An
such that
for all
. If
is an ultragraph, we write
for the algebra in ℘(
.
A is a sequence
for 1 ≤
for the set of all paths of length
and
on
by setting
and
for |
for
. For every edge
Let be an ultragraph and let
and
} of elements in
p ∅︀ = 0,pApB =pA∩B andpA∪B =pA +pB −pA∩B for all;
and
for all
;
for all
e ,;
for every vertex
v with,
where -family {
We say that the Leavitt -family {
-family in
and every
. The
, is the (unique up to isomorphism)
-family.
2.2. Quotient Ultragraphs
We will use the notion of quotient ultragraphs and we generalize the definition of Leavitt path algebras for quotient ultragraphs.
Let be an ultragraph. A subcollection
is called
implies
for all
e ∈.
A ∪B ∈H for allA, B ∈H .A ∈H ,and
B ⊆A , implyB ∈H .
Also, is called
For a saturated hereditary subcollection , the
An is a pair (
and some
In order to define the quotient of ultragraphs we need to recall and introduce some notations from [13, Section 3]. Let (. For each
, we denote
, where
,
for every , respectively. We write
for the algebra generated by the sets {
.
We usually denote [, we write [
Let (. The
, where
and and
are the maps defined by
and
, respectively.
We denote by , because
is an algebra generated by the elements [
.
Remark 2.6
If , then
and
, we deduce that
for all
. One can see that
For example we have
and
Furthermore, it follows from Lemma 2.3 that .
Remark 2.7
The hereditary property of
Similar to ultragraphs, a is a sequence
such that
to be paths of length zero. We denote by
, the union of paths with finite length. The maps
. Let
be the set of
. We also define the
and [
.
Using Theorem 3.4, we define the Leavitt path algebra of a quotient ultragraph which is corresponding to the quotient
. We use this concept to characterize the ideal structure of
in Section 3. The following definition is the algebraic version of [13, Definition 3.8].
Definition 2.8
Let be a quotient ultragraph and let
and
of elements in
q [∅︀] = 0,q [A ]q [B ] =q [A ]∩ [B ] andq [A ]∪ [B ] =q [A ] +q [B ] −q [A ]∩ [B ];qs (e )te =teqr (e ) =te andqr (e )te * =te * qs (e ) =te * ;te * tf =δe, f qr (e );q [v ] = ∑s (e )=[v ]tete * for every [v ] ∈ Φ(G 0) with 0< |s −1([v ])|< ∞.
The -family {
, is the (unique up to isomorphism)
-family (the definition of universal Leavitt
-family is similar to ultragraph case).
If (. In this case, every Leavitt
-family is a Leavitt
-family and vice versa. So, we can consider the ultragraph Leavitt path algebra
as
.
Let . For definition of the free
Now, we show that for every quotient ultragraph , there exists a universal Leavitt
-family. Suppose that
and
generated by the union of the following sets:
{[∅︀], [
A ][B ] − [A ∩B ],},
{
e –s (e )e, e –er (e ),e * –e *s (e ),},
{
e *f –δe, f r (e ) :e ,},
{
v – ∑s (e )=[v ]ee * : 0< |s −1([v ])|< ∞}.
If is the quotient map, then it is easy to check that the collection {
} is a Leavitt
-family. For our convenience, we denote
and
, and we show that the Leavitt
-family {
Assume that { -family in an
such that
-family,
by
. Furthermore, we have
From now on we denote the universal Leavitt -family and
-family by {
and
, respectively.
Theorem 2.9
If , then
which proves the first statement. Let and consider
as defined before. If we define a degree map
and
, then by [4, Proposition 2.7],
is a ℤ-graded ring with the grading
Theorem 2.10
By the universality, it suffices to generate a Leavitt -family {
and every
, we define a disjoint copy
where ⊕ , define
choose an isomorphism
and let
-family in Hom
and every
Note that we cannot follow the argument of Theorem 2.10 to show that is the ultragraph
and let . If we consider the quotient ultragraph
, then {[∅︀] ≠ [
Remark 2.11
Every directed graph , where
and the map
is defined by
for every
. In this case, the algebra
is the collection of all finite subsets of
(see [1, 18] for more details about the Leavitt path algebras of directed graphs). So the class of ultragraph Leavitt path algebras contains the class of Leavitt path algebras of directed graphs.
Lemma 2.12
.
If , then the Relations of Definition 2.1 imply that
.
Conversely, suppose that is unital and write
where and
. Let
, then we can choose an element
this contradicts Theorem 2.10 and it follows , as desired.
We note that, for directed graph associated to
is unital and
. More precisely, Consider the ultragraph
where
and
. Since
, by Lemma 2.12,
is unital. Define
2.3. Uniqueness Theorems
- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References
Let be a quotient ultragraph. We prove the graded and Cuntz-Krieger uniqueness theorems for
and
. We do this by approximating the Leavitt path algebras of quotient ultragraphs with the Leavitt path algebras of finite graphs. Our proof in this section is standard (see [13, Section 4]), and we give the details for simplicity of further results of the paper.
A vertex [
Let be a finite subset and write
. Following [13], we construct a finite graph
. We also set
and
Now we define the finite graph
with
Lemma 2.13
For every finite set ,
the elements form a Leavitt EF -family which generates the subalgebra of generated by {q [v ],te, te * : [v ] ∈F 0,e ∈F 1}.For r ∈R {0},if rq [A ] ≠ 0for all [A ] ≠ [∅︀]in ,
then rPz ≠ 0for all . In this case, we have
The statement (i) follows from the fact that {-family, or see the similar [13, Proposition 4.2].
For (ii), fix , then
such that
with
Now we show that are 0, 1 and −1, respectively. So
with the grading
Let {
Theorem 2.14. (The graded-uniqueness theorem)
Let { such that
generated by {
and
Fix , the elements
such that
with
If [. Thus, by Lemma 2.5,
is an
and so , as desired.
Corollary 2.15
Definition 2.16
A is a path
with the property that
satisfies
has an exit, or
Theorem 2.17. (The Cuntz-Krieger uniqueness theorem)
Choose an increasing sequence { such that
as in Theorem 2.14. Since
and
satisfies Condition (L), By [13, Lemma 4.8], all finite graphs
Corollary 2.18
3. Basic Graded Ideals
- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References
In this section, we apply the graded-uniqueness theorem for quotient ultragraphs to investigate the ideal structure of . We would like to consider the ideals of
that are reflected in the structure of the ultragraph
. For this, we give the following definition of basic ideals.
Let (. For any
Let . We write
, which is a saturated hereditary subcollection of
. Also, we set
rpA ∈I impliespA ∈I forand
r ∈R {0}.implies for w ∈BH I andr ∈R {0}.
For an admissible pair (, the (two-sided) ideal of
generated by the idempotents
Lemma 3.1. (cf. [12, Lemma 3.9])
,
.
We denote the right-hand side of the above equality by being contained in
Note that the elements
To show that and
where
If
Note that for every
. So
If
Let . In this case, we have
, then
. Suppose that {{
If
We go toward a contradiction and assume { such that
and
. If
and
. By Step II, there is a vertex
such that {
and
for all 1 ≤
and so
for all
If
If
Now, we show that
where and
. If
Remark 3.2
- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References
Let ( and let
Lemma 3.3. (cf. [13 , Proposition 3.3])
- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References
,
.
Proof
Suppose that {-family. If we define
then it is straightforward to see that {-family in
. Note that
. Since
and
.
We show that . For
,
for every . Thus for
for
Also,
Thus, by Remark 2.6, . Since
. Consequently,
.
Theorem 3.4. (cf. [12 , Theorem 3.10])
- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References
For any admissible pair (H, S )in ,
we have .
The map (H, S ) ↦I (H, S )is a bijection from the set of all admissible pairs of to the set of all graded basic ideals of .
Proof
(1) Let {-family and let
, where {
-family as defined in 3.1. Define
It can be shown that the family is a Leavitt
-family that generates
. Furthermore, by Remark 3.2,
and
Now we use the universal property of to get an
such that
and
is graded. Moreover, the elements
, respectively and thus
is generated by
, we deduce that
.
(2) The injectivity of the map (. Then
with respect to admissible pair (
is graded. Let
be the quotient map. For (
,
We show that and
. Let
. If [
. Furthermore,
As we have seen in the proof of Theorem 3.4, if (, then there is a Leavitt
-family {
such that
and
Proposition 3.5
- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References
4. Condition (K)
- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References
In this section we recall Condition (K) for ultragraphs and we consider the ultragraph that satisfy Condition (K) to describe that all basic ideals of
are graded.
Let be an ultragraph and let
is a loop
and
for
Definition 4.1. ([9, Section 7])
An ultragraph satisfies
Let be a quotient ultragraph. By rewriting Definition 2.2 for
, one can define the hereditary property for the subcollections of
. More precisely, a subcollection
is called
s (e ) ∈K impliesr (e ) ∈K for all.
[
A ] ∪ [B ] ∈K for all [A ], [B ] ∈K .[
A ] ∈K ,and [
B ] ⊆ [A ], imply [B ] ∈K .
For any hereditary subcollection , the ideal
generated by {
Suppose that contains a closed path
generated by {
The subalgebra
Set
By [18, Lemma 7.16],
Claim 2
.
We show that
Suppose that there exist such that
of , then
which implies that
Let
and set .
Claim 3
If and some
such that
Suppose that
where . If there is no such path, then we can choose an edge
Claim 4
.
Suppose that . For
we have
Since . Hence
Finally, we show that and
. Since
.
Theorem 4.3. (cf. [12 , Theorem 3.18], [18 , Theorem 7.17])
- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References
Proof
Suppose that satisfies Condition (K). If
, then by Theorem 3.4, we have the quotient map
such that and all
Conversely, if does not satisfy Condition (K), then [13, Proposition 6.2] and Lemma 4.2 imply that there exist
contains a non-graded ideal
and all
is the quotient map, then
. Similar to the last paragraph of the proof of Theorem 3.4, we have
. Also,
for satisfies Condition (K).
Corollary 4.4
- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References
.
5. Exel-Laca R -algebras
- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References
The Exel-Laca algebras are generated by partial isometries whose relations are determined by a countable {0, 1}-valued matrix
Definition 5.1
Let
By [15, Theorem 4.5], the Exel-Laca algebra is canonically isomorphic to
. Motivated by this fact we give the following definition.
Definition 5.2
Let .
Example 5.3
Let
xiyixi =xi andyixiyi =yi for alli ,yixj = 0 for alli ≠j ,for all i ,.
The Cuntz-Krieger . We will show that
. Note that
is a finite ultragraph. If we define
-family.
On the other hand, if {-family for
, then the elements
and
conclude that
.
Note that for any {0, 1}-valued matrix contains no singular vertices. So, each Exel-Laca algebra is the Leavitt path algebra of a non-singular ultragraph. In Theorem 5.7 below, we will prove the converse. For this, given any ultragraph
with no singular vertices, we first reform
to an ultragraph
as follows. Associated to each
. Next we define the ultragraph
, where
, the source map
Remark 5.4
We notice that each vertex emits exactly one edge
Lemma 5.5
Let {-family for
. Since
is row-finite, for every
. Define
By Remark 5.4, for each , the idempotent
-family in
. For example, to verify condition (2) of Definition 2.1 suppose that
. Then
Thus there is an such that
for
and
. As
Definition 5.6. ([15, Definition 2.4])
Let be an ultragraph. The edge matrix of an ultragraph
is the
matrix
given by
We can check that, if is an ultragraph with no singular vertices, then
. So, by Lemma 5.5 we have the following.
Theorem 5.7
.
Example 5.8
Suppose that
Remark 5.9
Theorem 5.7 shows that the Leavitt path algebras of ultragraphs with no singular vertices are precisely the Exel-Laca
Remark 5.10
We know that for every directed graph is isomorphic to a dense *-subalgebra of the
introduced in [15]. In particular, for any countable {0, 1}-valued matrix
[7].
Last part of this section is an example to emphasize that the class of the Leavitt path algebras of ultragraphs is strictly larger than the class of the Leavitt path algebras of directed graphs as well as the class of algebraic Exel-Laca algebras.
Example 5.11
Let be the ultragraph
with one ultraedge satisfies Condition (K) and so all ideals of
are basic and graded by Theorem 4.3. Moreover,
is unital, because by Lemma 2.12
. We claim that the algebra
is not isomorphic to the Leavitt path algebra of a graph. Suppose on the contrary that there is a such graph
does not include such member. Thus
has infinitely many pairwise orthogonal ideals
Now assume for some matrix
be an isomorphism. We may consider the ideal
. Since
and so | has no sinks. Hence the graded ideal
includes infinitely many elements, which is a contradiction.
Acknowledgements
- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References
The authors are grateful to the referee for carefully reading the paper, pointing out a number of misprints and for some helpful comments.
References
- Abstract
- 1. Introduction
- 2. Leavitt Path Algebras
- 2.3. Uniqueness Theorems
- 3. Basic Graded Ideals
- Remark 3.2
- Lemma 3.3. (cf. [
13 , Proposition 3.3]) - Theorem 3.4. (cf. [
12 , Theorem 3.10]) - Proposition 3.5
- 4. Condition (K)
- Theorem 4.3. (cf. [
12 , Theorem 3.18], [18 , Theorem 7.17]) - Corollary 4.4
- 5. Exel-Laca
R -algebras - Acknowledgements
- References
- G. Abrams, and G. Aranda Pino.
The Leavitt path algebra of a graph . J Algebra.,293 (2005), 319-334. - G. Abrams, and G. Aranda Pino.
The Leavitt path algebras of arbitrary graphs . Houston J Math.,34 (2008), 423-442. - P. Ara, MA. González-Barroso, KR. Goodearl, and E. Pardo.
Fractional skew monoid rings . J Algebra.,278 (2004), 104-126. - G. Aranda Pino, J. Clark, A. an Huef, and I. Raeburn.
Kumjian-Pask algebras of higher rank graphs . Trans Amer Math Soc.,365 (2013), 3613-3641. - T. Bates, D. Pask, I. Raeburn, and W. Szymański.
The C*-algebras of row-finite graphs . New York J Math.,6 (2000), 307-324. - J. Cuntz, and W. Krieger.
A class of C*-algebras and topological Markov chains . Invent Math.,56 (1980), 251-268. - R. Exel, and M. Laca.
Cuntz-Krieger algebras for infinite matrices . J Reine Angew Math.,512 (1999), 119-172. - N. Fowler, M. Laca, and I. Raeburn.
The C*-algebras of infinite graphs . Proc Amer Math Soc.,128 (2000), 2319-2327. - T. Katsura, PS. Muhly, A. Sims, and M. Tomforde.
Utragraph C*-algebras via topological quivers . Studia Math.,187 (2008), 137-155. - A. Kumjian, D. Pask, and I. Raeburn.
Cuntz-Krieger algebras of directed graphs . Pacific J Math.,184 (1998), 161-174. - A. Kumjian, D. Pask, I. Raeburn, and J. Renault.
Graphs, groupoids and Cuntz-Krieger algebras . J Funct Anal.,144 (1997), 505-541. - H. Larki.
Ideal structure of Leavitt path algebras with coefficients in a unital commutative ring . Comm Algebra.,43 (2015), 5031-5058. - H. Larki.
Primitive ideals and pure infiniteness of ultragraph C*-algebras . J Korean Math Soc.,56 (2019), 1-23. - WG. Leavitt.
Modules without invariant basis number . Proc Amer Math Soc.,8 (1957), 322-328. - M. Tomforde.
A unified approach to Exel-Laca algebras and C*-algebras associated to graphs . J Operator Theory.,50 (2003), 345-368. - M. Tomforde.
Simplicity of ultragraph algebras . Indiana Univ Math J.,52 (2003), 901-925. - M. Tomforde.
Uniqueness theorems and ideal structure for Leavitt path algebras . J Algebra.,318 (2007), 270-299. - M. Tomforde.
Leavitt path algebras with coefficients in a commutative ring . J Pure Appl Algebra.,215 (2011), 471-484.