The Leavitt Path Algebras of Ultragraphs

Mostafa Imanfar and Abdolrasoul Pourabbas∗; Hossein Larki

doi:10.5666/KMJ.2020.60.1.21

Article

Kyungpook Mathematical Journal 2020; 60(1): 21-43

Published online March 31, 2020

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

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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

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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 C^*-algebras associated to (directed) graphs [5, 8, 10, 11] and (2) the Exel-Laca algebras of infinite matrices with {0, 1}-entries [7]. It is shown in [8] that if E is a graph with no sinks and sources, then the C^*-algebra C^*(E) is canonically isomorphic to the Exel-Laca algebra , where A_E is the edge matrix of E. However, the class of graph C^*-algebras and Exel-Laca algebras are differ from each other.

To study both graph C^*-algebras and Exel-Laca algebras under one theory, Tomforde [15] introduced the notion of an ultragraph and its associated C^*-algebra. Briefly, an ultragraph is a directed graph which allows the range of each edge to be a nonempty set of vertices rather than a singleton vertex. We see in [15] that for each binary-valued matrix A there exists an ultragraph so that the C^*-algebra of is isomorphic to the Exel-Laca algebra of A. Furthermore, every graph C^*-algebra can be considered as an ultragraph C^*-algebra, whereas there is an ultragraph C^*-algebra which is not a graph C^*-algebra nor an Exel-Laca algebra.

Recently many authors have discussed the algebraic versions of matrix and graph C^*-algebras. For example, in [3] the algebraic analogue of the Cuntz-Krieger algebra , denoted by , was studied for finite matrix A, where K is a field. Also the Leavitt path algebra L_K(E) of directed graph E was introduced in [1, 2] as the algebraic version of graph C^*-algebra C^*(E). The class of Leavitt path algebras includes naturally the algebras of [3] as well as the well-known Leavitt algebras L(1, n) of [14]. More recently, Tomforde defined a new version of Leavitt path algebras with coefficients in a unital commutative ring [18]. In the case R = ℂ, the Leavitt path algebra L_ℂ(E) is isomorphic to a dense *-subalgebra of the graph C^*-algebra C^*(E) [17].

The purpose of this paper is to define the algebraic versions of ultragraphs C^*-algebras and Exel-Laca algebras. For an ultragraph and unital commutative ring R, we define the Leavitt path algebra . To study the ideal structure of , we use the notion of quotient ultragraphs from [13]. Given an admissible pair (H, S) in , 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 R. We continue by considering the definition of quotient ultragraphs of [13]. For any admissible pair (H, S) in an ultragraph , 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 R-algebras of finite graphs.

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_A(R), for every infinite matrix A with entries in {0, 1} and every unital commutative ring R. In the case R = ℂ, the Exel-Laca ℂ-algebra ℰℒ_A(ℂ) is isomorphic to dense *-subalgebra of . 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 R-algebra.

2. Leavitt Path Algebras

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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 ultragraph consists of a set of vertices G⁰, a set of edges , the source map and the range map , where ℘(G⁰) is the collection of all subsets of G⁰. Throughout this work, ultragraph will be assumed to be countable in the sense that both G⁰ and are countable.

For a set X, a subcollection of ℘(X) is said to be lattice if and it is closed under the set operations ∩ and ∪. An algebra is a lattice such that for all . If is an ultragraph, we write for the algebra in ℘(G⁰) generated by .

A path in ultragraph is a sequence α = e₁e₂ · · · e_n of edges with for 1 ≤ i ≤ n − 1 and we say that the path α has length |α| := n. We write for the set of all paths of length n and $Path (G) : = \cup_{n = 0}^{\infty} G^{n}$ for the set of finite paths. We may extend the maps and on by setting and for |α| ≥ 1 and for . For every edge e, We say that e^* is the ghost edge associated to e. The following definition is the algebraic version of [15, Definition 2.7].

Definition 2.1

Let be an ultragraph and let R be a unital commutative ring. A Leavitt-family in an R-algebra A is a set {p_A, s_e, s_e_^* : and } of elements in A such that

p_∅︀ = 0, p_Ap_B = p_A∩B and p_A∪B = p_A + p_B − p_A∩B for all ;
and for all ;
for all e, ;
for every vertex v with $0 < ∣ s_{G}^{- 1} (v) ∣ < \infty$ ,

where p_v denotes p_{_v_}. The R-algebra generated by the Leavitt -family {s, p} is denoted by L_R(s, p).

We say that the Leavitt -family {s, p} is universal, if B is an R-algebra and {S, P} is a Leavitt -family in B, then there exists an algebra homomorphism φ : L_R(s, p) → B such that φ(p_A) = P_A, φ(s_e) = S_e and φ(s_e_^*) = S_e_^* for every and every . The Leavitt path algebra ofwith coefficients in R, denoted by , is the (unique up to isomorphism) R-algebra generated by a universal Leavitt -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.

Definition 2.2. ([16, Definition 3.1])

Let be an ultragraph. A subcollection is called hereditary if satisfying the following conditions:

implies for all e ∈ .
A ∪ B ∈ H for all A, B ∈ H.
A ∈ H, and B ⊆ A, imply B ∈ H.

Also, is called saturated if for any v ∈ G⁰ with $0 < ∣ s_{G}^{- 1} (v) ∣ < \infty$ , we have

{r_{G} (e) : e \in G^{1} and s_{G} (e) = v} \subseteq H implies {v} \in H .

For a saturated hereditary subcollection , the breaking vertices of H is denoted by

B_{H} : = {v \in G^{0} : ∣ s_{G}^{- 1} (v) ∣ = \infty but 0 < ∣ S_{G}^{- 1} (v) \cap {e : r_{G} (e) \notin H} ∣ < \infty} .

An admissible pair in is a pair (H, S) of a saturated hereditary set and some S ⊆ B_H.

In order to define the quotient of ultragraphs we need to recall and introduce some notations from [13, Section 3]. Let (H, S) be an admissible pair in . For each , we denote Ā := A ∪ {w′ : w ∈ A ∩ (B_H S)}, where w′ denotes another copy of w. Consider the ultragraph , where , Ḡ⁰ := G⁰ ∪ {w′ : w ∈ B_H S} and the maps r̄, s̄ are defined by $\bar{r} (e) : = \bar{r_{G} (e)}$ and

\bar{s} (e) : = {\begin{array}{l} s_{G} {(e)}^{'} & if s_{G} (e) \in B_{H} S and r_{G} (e) \in H, \\ s_{G} (e) & otherwise, \end{array}

for every , respectively. We write for the algebra generated by the sets {v}, {w′} and r(e).

Lemma 2.3. ([13, Lemma 3.5])

Let (H, S) be an admissible pair in an ultragraphand let ~ be a relation ondefined by A ~ B if and only if there exists V ∈ H such that A ∪ V = B ∪ V. Then ~ is an equivalence relation onand the operations

[A] \cup [B] : = [A \cup B], [A] \cap [B] : = [A \cap B] and [A] [B] : = [A B]

are well-defined on the equivalence classes.

We usually denote [v] instead of [{v}] for every v ∈ Ḡ⁰. For , we write [A] ⊆ [B] whenever [A] ∩ [B] = [A]. The set ∪_A_∈_H A is denoted by ∪H.

Definition 2.4. ([13, Definition 3.6])

Let (H, S) be an admissible pair in . The quotient ultragraph ofby (H, S) is the quadruple , where

\begin{matrix} Φ (G^{0}) : = {[v] : v \in G^{0} ⋃ H} \cup {[w^{'}] : w \in B_{H} S}, \\ Φ (G^{1}) : = {e \in G^{1} : r_{G} (e) \notin H}, \end{matrix}

and and are the maps defined by and $r (e) : = [\bar{r_{G} (e)}]$ for every , respectively.

Lemma 2.5

Ifis a quotient ultragraph, then

Φ (G^{0}) = {⋃_{j = 1}^{k} ⋂_{i = 1}^{n_{j}} A_{i, j} B_{i, j} : A_{i, j}, B_{i, j} \in Φ (G^{0}) \cup {r (e) : e \in Φ (G^{1})}},

whereis the smallest algebra incontaining

{[v], [w^{'}] : v \in G^{0} ⋃ H, w \in B_{H} S} \cup {r (e) : e \in Φ (G^{1})} .

Proof

We denote by X the right hand side of the above equality. It is clear that , because is an algebra generated by the elements [v], [w′] and r(e). For 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], [w′] and r(e) and consequently .

Remark 2.6

If , then $\bar{A \cup B} = \bar{A} \cup \bar{B}, \bar{A \cap B} = \bar{A} \cap \bar{B}$ and $\bar{A B} = \bar{A} \bar{B}$ . Thus, by applying an analogous lemma of Lemma 2.5 for and , we deduce that for all . One can see that

{\bar{G}}^{0} = {\bar{A} \cup F_{1} \cup F_{2} : A \in G^{0}, F_{1} and F_{2} are finite subsets of G^{0} and {w^{'} : w \in B_{H}}, respectively} .

For example we have

\bar{A} {v} = \bar{A {v}} \cup (\bar{{v}} {v} \cap \bar{A} A),

and

\bar{A} {w^{'}} = \bar{A {w}} \cup (A \cap {w}) .

Furthermore, it follows from Lemma 2.3 that .

Remark 2.7

The hereditary property of H and Remark 2.6 imply that A ~ B if and only if both A B and B A belong to H.

Similar to ultragraphs, a path in is a sequence α := e₁e₂ · · · e_n of edges in such that s(e_i₊₁) ⊆ r(e_i) for 1 ≤ i ≤ n − 1. We say the path α has length |α| := n and we consider the elements in to be paths of length zero. We denote by , the union of paths with finite length. The maps r and s can be naturally extended on . Let be the set of ghost edges. We also define the ghost path $α^{*} : = e_{n}^{*} e_{n - 1}^{*} \dots e_{1}^{*}$ for every and [A]^* := [A] for every .

Using Theorem 3.4, we define the Leavitt path algebra of a quotient ultragraph which is corresponding to the quotient R-algebra . 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 R be a unital commutative ring. A Leavitt-family in an R-algebra A is a set and of elements in A such that

q_[∅︀] = 0, q_[_A_]q_[_B_] = q_[_A_]_∩_[_B_] and q_[_A_]_∪_[_B_] = q_[_A_] + q_[_B_] − q_[_A_]_∩_[_B_];
q_s₍_e₎t_e = t_eq_r₍_e₎ = t_e and q_r₍_e₎t_e_^* = t_e_^*q_s₍_e₎ = t_e_^*;
t_e_^*t_f = δ_{e, f} q_r₍_e₎;
q_[_v_] = ∑_s₍_e_)=[_v_]t_et_e_^* for every [v] ∈ Φ(G⁰) with 0 < |s⁻¹([v])| < ∞.

The R-algebra generated by the Leavitt -family {t, q} is denoted by L_R(t, q). The Leavitt path algebra ofwith coefficients in R, denoted by , is the (unique up to isomorphism) R-algebra generated by a universal Leavitt -family (the definition of universal Leavitt -family is similar to ultragraph case).

If (H, S) = (∅︀, ∅︀), then we have [A] = {A} for each . In this case, every Leavitt -family is a Leavitt -family and vice versa. So, we can consider the ultragraph Leavitt path algebra as .

Let R be a unital commutative ring. For a nonempty set X, we write w(X) for the set of words w := w₁w₂ · · ·w_n from the alphabet X. The free R-algebra generated by X is denoted by . For definition of the free R-algebra we refer the reader to [4, Section 2.3].

Now, we show that for every quotient ultragraph , there exists a universal Leavitt -family. Suppose that and I is the ideal of the free R-algebra 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 – δ_{e, 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 q_[_A_] := π(A), t_e := π(e) and t_e_^* := π(e^*) for every and , and we show that the Leavitt -family {t, q} has the universal property.

Assume that {T, Q} is a Leavitt -family in an R-algebra A. If we define f : X → A by f([A]) = P_[_A_], f(e) = T_e and f(e^*) = T_e_^*, then by [4, Proposition 2.6], there is an R-algebra homomorphism such that φ_f | _X = f. Since {T, Q} is a Leavitt -family, φ_f vanishes on the generator of I. Hence, we can define an R-algebra homomorphism by φ(π(x)) = φ_f (x) for every . Furthermore, we have φ(q_[_A_]) = Q_[_A_], φ(t_e) = T_e and φ(t_e_^*) = T_e_^*.

From now on we denote the universal Leavitt -family and -family by {s, p} and {t, q}, respectively. So, we suppose that {s, p} and {t, q} are the canonical generators of and , respectively.

Theorem 2.9

Letbe a quotient ultragraph and let R be a unital commutative ring. Thenis of the form

{span}_{R} {t_{α} q_{[A]} t_{β^{*}} \in L_{R} (G / H, S)) : α, β \in Path (G / (H, S)), r (α) \cap [A] \cap r (β) \neq [\emptyset]} .

Furthermore, is a ℤ-graded ring by the grading

L_{R} {(G / (H, S))}_{n} = {span}_{R} {t_{α} q_{[A]} t_{β^{*}} \in L_{R} (G / (H, S)) : ∣ α ∣ - ∣ β ∣ = n} (n \in ℤ) .

Proof

If t_αq_[_A_]t_β_^*, , then

(t_{α} q_{[A]} t_{β *}) (t_{μ} q_{[B]} t_{ν *}) = {\begin{array}{l} t_{α μ^{'}} q_{[B]} t_{ν *} & if μ = β μ^{'} and s (μ^{'}) \subseteq A \cap r (α), \\ t_{α} q_{[A] \cap r (β) \cap [B]} t_{ν *} & if μ = β, \\ t_{α} q_{[A]} t_{(ν β^{'}) *} & if β = μ β^{'} and s (β^{'}) \subseteq B \cap r (ν), \\ 0 & otherwise, \end{array}

which proves the first statement. Let and consider as defined before. If we define a degree map d : X → ℤ by d([A]) = 0, d(e) = 1 and d(e^*) = −1 for every and , then by [4, Proposition 2.7], is a ℤ-graded ring with the grading

L_{R} {(G / (H, S))}_{n} = {span}_{R} {t_{α} q_{[A]} t_{β^{*}} \in L_{R} (G / (H, S)) : ∣ α ∣ - ∣ β ∣ = n} .

Theorem 2.10

Letbe an ultragraph and let R be a unital commutative ring. Then for all nonempty setsand all r ∈ R {0}, the elements rp_A ofare nonzero. In particular, rs_e ≠ 0 and rs_e_^*≠ 0 for everyand every r ∈ R {0}.

Proof

By the universality, it suffices to generate a Leavitt -family {s̃, p̃} in which rp̃_A ≠ 0 for every nonempty set and every r ∈ R {0}. For each , we define a disjoint copy Z_e := ⊕ R, where the direct sum is taken over countably many copies of R and for each v ∈ G⁰, let

Z_{v} : = {\begin{matrix} \underset{s (e) = v}{\oplus} Z_{e} & if ∣ s_{G}^{- 1} (v) ∣ \neq 0, \\ \oplus R & if ∣ s_{G}^{- 1} (v) ∣ = 0, \end{matrix}

where ⊕ R is another disjoint copy for each v. For every , define P_A : ⊕_v_∈_A Z_v → ⊕_v_∈_A Z_v to be the identity map. Also, for each choose an isomorphism and let $S_{e *} : = S_{e}^{- 1} : Z_{e} \to \oplus_{v \in r_{G} (e)} Z_{v}$ . Now if Z := ⊕_v_∈_G_⁰Z_v, then we naturally extend all P_A, S_e, S_e_^* to homomorphisms p̃_A, s̃_e, s̃_e_^* ∈ Hom_R(Z, Z), respectively by setting the yet undefined components to zero. It is straightforward to verify that {s̃, p̃} is a Leavitt -family in Hom_R(Z, Z) such that rp̃_A ≠ 0 for every and every r ∈ R {0}.

Note that we cannot follow the argument of Theorem 2.10 to show that rq_[_A_] ≠ 0. For example, suppose that is the ultragraph

and let H be the collection of all finite subsets of {v₁, v₂, . . .}, which is a hereditary and saturated subcollection of . If we consider the quotient ultragraph , then {[∅︀] ≠ [v] : [v] ⊆ r(e)} = ∅︀. So we can not define the idempotent q_r₍_e₎ : ⊕ _[_v_]⊆_r₍_e₎Z_[_v_] → ⊕ _[_v_]⊆_r₍_e₎Z_[_v_] as in the proof of Theorem 2.10. In Section 3 we will solve this problem.

Remark 2.11

Every directed graph E = (E⁰, E¹, r, s) can be considered as an ultragraph , where G⁰ := E⁰, and the map is defined by for every . In this case, the algebra is the collection of all finite subsets of G⁰. The Leavitt path algebra L_R(E) is naturally isomorphic to (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

Letbe an ultragraph and let R be a unital commutative ring. Thenis unital if and only ifand in this case.

Proof

If , then the Relations of Definition 2.1 imply that p_G_⁰ is a unit for .

Conversely, suppose that is unital and write

1_{L_{R} (G)} = \sum_{k = 1}^{n} r_{k} s_{α_{k}} p_{A_{k}} s_{β_{k}^{*}},

where r_k ∈ R, and . Let $A : = \cup_{k = 1}^{n} s (α_{k}) \in G^{0}$ . If , then we can choose an element v ∈ G⁰ A and derive

p_{v} = p_{v} \cdot 1_{L_{R} (G)} = \sum_{k = 1}^{n} r_{k} p_{v} s_{α_{k}} p_{A_{k}} s_{β_{k}^{*}} = \sum_{k = 1}^{n} r_{k} p_{v} p_{s_{G} (α_{k})} s_{α_{k}} p_{A_{k}} s_{β_{k}^{*}} = 0,

this contradicts Theorem 2.10 and it follows , as desired.

We note that, for directed graph E, the algebra L_R(E) is unital if and only if E⁰ is finite. If E⁰ is infinite, then we can define an ultragraph associated to E such that is unital and L_R(E) is an embedding subalgebra of . More precisely, Consider the ultragraph where G⁰ = E⁰ ∪ {v}, and . Since , by Lemma 2.12, is unital. Define S_e = s_e and P_v = p_v for e ∈ E¹ and v ∈ E⁰, respectively. It is straightforward to see that {S, P} is a Leavitt E-family for directed graph E. The result now follows by [18, Theorem 5.3].

2.3. Uniqueness Theorems

Go to

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 [v] ∈ Φ(G⁰) is called a sink if s⁻¹([v]) = ∅︀ and [v] is called an infinite 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 Φ_sg(G⁰).

Let be a finite subset and write F⁰ := F ∩ Φ_sg(G⁰) and . Following [13], we construct a finite graph E_F as follows. First, for every ω = (ω₁, . . . , ω_n) ∈ {0, 1}ⁿ {0ⁿ}, we define r(ω) := ∩_ω_{_i=1}r(e_i) ∪_ω_{_j=0}r(e_j) and R(ω) := r(ω) ∪_[_v_]∈_F_⁰ [v] which belong to . We also set

\begin{array}{l} Γ_{0} : = {ω \in {0, 1}^{n} {0^{n}} : there are vertices [v_{1}], \dots, [v_{m}]} \\ such that R (ω) = \cup_{i = 1}^{m} [v_{i}] and \emptyset \neq s^{- 1} ([v_{i}]) \subseteq F^{1} for 1 \leq i \leq m}, \end{array}

and

Γ_{F} : = {ω \in {0, 1}^{n} {0^{n}} : R (ω) \neq [\emptyset] and ω \notin Γ_{0}} .

Now we define the finite graph $E_{F} = (E_{F}^{0}, E_{F}^{1}, r_{F}, s_{F})$ , where

\begin{array}{l} E_{F}^{0} : = F^{0} \cup F^{1} \cup Γ_{F}, \\ E_{F}^{1} : = {(e, f) \in F^{1} \times F^{1} : s (f) \subseteq r (e)} \\ \cup {(e, [v]) \in F^{1} \times F^{0} : [v] \subseteq r (e)} \\ \cup {(e, ω) \in F^{1} \times Γ_{F} : ω_{i} = 1 whenever e = e_{i}}, \end{array}

with s_F (e, f) = s_F (e, [v]) = s_F (e, ω) = e and r_F (e, f) = f, r_F (e, [v]) = [v], r_F (e, ω) = ω.

Lemma 2.13

Letbe a quotient ultragraph and let R be a unital commutative ring. Then we have the following assertion:

For every finite set, the elements
$\begin{array}{l} P_{e} : t_{e} t_{e *}, & P_{[v]} : = q_{[v]} (1 - \sum_{e \in F^{1}} t_{e} t_{e *}), & P_{ω} : = q_{R (ω)} (1 - \sum_{e \in F^{1}} t_{e} t_{e *}), \\ S_{(e, f)} : = t_{e} P_{f}, & S_{(e, [v])} : = t_{e} P_{[v]}, & S_{(e, ω)} : = t_{e} P_{ω}, \\ S_{(e, f) *} : = P_{f} t_{e *}, & S_{(e, [v]) *} : = P_{[v]} t_{e *}, & S_{(e, ω) *} : = P_{ω} t_{e *}, \end{array}$
form a Leavitt E_F -family which generates the subalgebra ofgenerated by {q_[_v_], t_e, t_e_^* : [v] ∈ F⁰, e ∈ F¹}.
For r ∈ R {0}, if rq_[_A_] ≠ 0 for all [A] ≠ [∅︀] in, then rP_z ≠ 0 for all $z \in E_{F}^{0}$ . In this case, we have
$L_{R} (E_{F}) ≅ L_{R} (S, P) = Alg {q_{[v]}, t_{e}, t_{e *} \in L_{R} (G / (H, S)) : [v] \in F^{0}, e \in F^{1}} .$

Proof

The statement (i) follows from the fact that {t, q} is a Leavitt -family, or see the similar [13, Proposition 4.2].

For (ii), fix r ∈ R {0}. If rq_[_A_] ≠ 0 for every , then rt_e ≠ 0 and rt_e_^* ≠ 0 for every edge e. Thus rP_e ≠ 0 for every edge e ∈ F¹. Let [v] ∈ F⁰. If [v] is a sink, then rP_[_v_] = rq_[_v_] ≠ 0. If [v] is an infinite emitter, then there is such that s(f) = [v]. In this case, we have rP_[_v_]t_f = rq_[_v_]t_f = t_f ≠ 0. Therefore rP_[_v_] ≠ 0 for all [v] ∈ F⁰. Moreover, for each ω ∈ Γ_F, there is a vertex [v] ⊆ R(ω) such that either [v] is a sink or there is an edge with s(f) = [v]. In the former case, we have q_[_v_](rP_ω) = rq_[_v_] ≠ 0 and in the later, t_f_^* (rP_ω) = rt_f_^* ≠ 0. Thus all rP_ω are nonzero. Consequently, rP_z ≠ 0 for every vertex $z \in E_{F}^{0}$ .

Now we show that L_R(E_F) ~= L_R(S, P). Note that for $z \in E_{F}^{0}$ and $g \in E_{F}^{1}$ , the degree of P_z, S_g and S_g_^* as elements in are 0, 1 and −1, respectively. So L_R(S, P) is a graded subalgebra of with the grading

L_{R} {(S, P)}_{n} : = L_{R} (S, P) \cap L_{R} {(G / (H, S))}_{n} .

Let {s̃, p̃} be the canonical generators of L_R(E_F). By the universal property, there is an R-algebra homomorphism π : L_R(E_F) → L_R(S, P) such that π(rp̃_z) = rP_z ≠ 0, π(s̃_g) = S_g and π(s̃_g_^*) = S_g_^* for $z \in E_{F}^{0}, g \in E_{F}^{1}$ and r ∈ R{0}. Since π preserves the degree of generators, the graded-uniqueness theorem for graphs [18, Theorem 5.3] implies that π is injective. As π is also surjective, we conclude that L_R(E_F) is isomorphic to L_R(S, P) as R-algebras.

Theorem 2.14. (The graded-uniqueness theorem)

Letbe a quotient ultragraph and let R be a unital commutative ring. If T is a ℤ-graded ring andis a graded ring homomorphism with π(rq_[_A_]) ≠ 0 for all [A] ≠ [∅︀] inand r ∈ R {0}, then π is injective.

Proof

Let {F_n} be an increasing sequence of finite subsets of such that $\cup_{n = 1}^{\infty} F_{n} = Φ_{sg} (G^{0}) \cup Φ (G^{1})$ . For each n, the graded subalgebra of generated by { $q_{[v]}, t_{e}, t_{e *} : [v] \in F_{n}^{0}$ and $e \in F_{n}^{1}$ }is denoted by X_n. Since π(rq_[_A_]) ≠ 0 for all and r ∈ R {0}, by Lemma 2.13, there is an graded isomorphism π_n : L_R(E_F_{_n}) → X_n. Thus π ∘ π_n : L_R(E_F_{_n}) → T is a graded homomorphism.

Fix r ∈ R {0}. We show that π ∘ π_n(rp̃_z) = π(rP_z) ≠ 0 for all $z \in E_{F_{n}}^{0}$ . Since π(rq_[_A_]) ≠ 0 for all , the elements π(rt_e) and π(rt_e_^*) are nonzero for every edge e. So π(rP_et_e) = π(rt_e) ≠ 0 and thus π(rP_e) ≠ 0. Let [v] ∈ F⁰. If [v] is a sink, then π(rP_[_v_]) = π(rq_[_v_]) ≠ 0. If [v] is an infinite emitter, then there is such that s(f) = [v]. In this case, we have π(rP_[_v_]t_f) = π(rq_[_v_]t_f) = π(t_f) ≠ 0, hence π(rP_[_v_]) ≠ 0. For every ω ∈ Γ_F, there is a vertex [v] ⊆ R(ω) such that either [v] is a sink or there is an edge with s(f) = [v]. In the former case, we have π(q_[_v_](rP_ω)) = π(rq_[_v_]) ≠ 0 and in the later, π(t_f_^* (rP_ω)) = π(rt_f_^*) ≠ 0. Thus π ∘ π_n(rp̃_z) ≠ 0 for every $z \in E_{F_{n}}^{0}$ and r ∈ R {0}. Hence, we may apply the graded-uniqueness theorem for graphs [18, Theorem 5.3] to obtain the injectivity of π ∘ π_n.

If [v] is a non-singular vertex, then we have q_[_v_] = ∑_s₍_e_)=[_v_]t_et_e_^*. Furthermore, q_[_A_]_[_B_] = q_[_A_] – q_[_A_]q_[_B_] for every . Thus, by Lemma 2.5, is an R-algebra generated by

{q_{[v]}, t_{e}, t_{e *} : [v] \in Φ_{sg} (G^{0}) and e \in Φ (G^{1})},

and so $\cup_{n = 1}^{\infty} X_{n} - L_{R} (G / (H, S))$ . It follows that π is injective on , as desired.

Corollary 2.15

Letbe an ultragraph, R a unital commutative ring and T a ℤ-graded ring. Ifis a graded ring homomorphism such that π(rp_A) ≠ 0 for alland r ∈ R {0}, then π is injective.

Definition 2.16

A loop in a quotient ultragraph is a path α with |α| ≥ 1 and s(α) ⊆ r(α). An exit for a loop α₁ · · · α_n is an edge with the property that s(f) ⊆ r(α_i) but f ≠ α_i₊₁ for some 1 ≤ i ≤ n, where α_n₊₁ := α₁. We say that satisfies Condition (L) if every loop α := α₁ · · · α_n in has an exit, or r(α_i) ≠ s(α_i₊₁) for some 1 ≤ i ≤ n.

Theorem 2.17. (The Cuntz-Krieger uniqueness theorem)

Letbe a quotient ultragraph satisfying Condition (L) and let R be a unital commutative ring. If T is a ring andis a ring homomorphism such that π(rq_[_A_]) ≠ 0 for everyand r ∈ R {0}, then π is injective.

Proof

Choose an increasing sequence {F_n} of finite subsets of such that $\cup_{n = 1}^{\infty} F_{n} = Φ_{sg} (G^{0}) \cup Φ (G^{1})$ . Let X_n be the subalgebras of as in Theorem 2.14. Since π(rq_[_A_]) ≠ 0 for all and r ∈ R {0}, by Lemma 2.13, there exists an isomorphism π_n : L_R(E_F_{_n}) → X_n for each n ∈ ℕ. Furthermore, π ∘ π_n(rp̃_z) ≠ 0 for every $z \in E_{F_{n}}^{0}$ and r ∈ R {0}. Since satisfies Condition (L), By [13, Lemma 4.8], all finite graphs E_F_{_n} satisfy Condition (L). So, the Cuntz-Krieger uniqueness theorem for graphs [18, Theorem 6.5] implies that π ∘ π_n is injective for n ≥ 1. Now by the fact $\cup_{n = 1}^{\infty} X_{n} = L_{R} (G / (H, S))$ , we conclude that π is an injective homomorphism.

Corollary 2.18

Letbe an ultragraph satisfying Condition (L), R a unital commutative ring and T a ring. Ifis a ring homomorphism such that π(rp_A) ≠ 0 for alland r ∈ R {0}, then π is injective.

3. Basic Graded Ideals

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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 (H, S) be an admissible pair in an ultragraph . For any w ∈ B_H, we define the gap idempotent

p_{w}^{H} : = p_{w} - \sum_{s (e) = w, r (e) \notin H} s_{e} s_{e *} .

Let I be an ideal in . We write , which is a saturated hereditary subcollection of . Also, we set $S_{I} : = {w \in B_{H_{I}} : p_{w}^{H_{I}} \in I}$ . We say that the ideal I is basic if the following conditions hold:

rp_A ∈ I implies p_A ∈ I for and r ∈ R {0}.
$r p_{w}^{H_{I}} \in I$ implies $p_{w}^{H_{I}} \in I$ for w ∈ B_H_{_I} and r ∈ R {0}.

For an admissible pair (H, S) in , the (two-sided) ideal of generated by the idempotents ${p_{A} : A \in H} \cup {p_{w}^{H} : w \in S}$ is denoted by I₍_{H, S}₎.

Lemma 3.1. (cf. [12, Lemma 3.9])

If (H, S) is an admissible pair in ultragraph, then

I_{(H, S)} : = {span}_{R} {s_{α} p_{A} s_{β *}, s_{μ} p_{w}^{H} s_{ν *} \in L_{R} (G) : A \in H and w \in S}

and I₍_{H, S}₎is a graded basic ideal of.

Proof

We denote the right-hand side of the above equality by J. The hereditary property of H implies that J is an ideal of being contained in I₍_{H, S}₎. On the other hand, all generators of I₍_{H, S}₎ belong to J and so we have I₍_{H, S}₎ = J.

Note that the elements s_αp_As_β_^* and $s_{μ} p_{w}^{H} s_{ν *}$ are homogeneous elements of degrees |α| – |β| and |μ| – |ν|, respectively. Thus I₍_{H, S}₎ is a graded ideal.

To show that I₍_{H, S}₎ is a basic ideal, suppose rp_A ∈ I₍_{H, S}₎ for some and r ∈ R {0} and write

r p_{A} = x : = \sum_{i = 1}^{n} r_{i} s_{α_{i}} p_{A_{i}} s_{β_{i}^{*}} + \sum_{j = 1}^{m} s_{j} s_{μ_{j}} p_{w_{j}}^{H} s_{ν_{j}^{*}},

where A_i ∈ H, w_j ∈ S and r_i, s_j ∈ R for all i, j. We first show assertion (1) of the definition of basic ideal in several steps.

Step I

If A = {v} ∉ H and v ∉ S, then $0 < ∣ s_{G}^{- 1} (v) ∣ < \infty$ .

Note that p_vx = x, so the assumption v ∉ ∪ H ∪ S and the hereditarity of H imply that we may assume that |α_i|, |μ_j| ≥ 1 and for every i, j (because a sum like ∑r_iA_i could be zero). Hence v is not a sink. Set α_i = α_i,₁α_i,₂ · · · α_i,_|_α_|. If $∣ s_{G}^{- 1} (v) ∣ = \infty$ , then there is an edge e ≠ α_1,1, . . . , α_n,₁, μ_1,1, . . . , μ_m,₁ with . So rs_e_^* = s_e_^* (rp_v) = s_e_^*x = 0, contradicting Theorem 2.3. Thus $0 < ∣ s_{G}^{- 1} (v) ∣ < \infty$ .

Step II

If A ∉ H, then there exists v ∈ A such that {v} ∉ H.

Let rp_A = x. Assume first that |μ_j | = 0 for some j. Then, since rp_A = p_Arp_A = p_Ax = x, w_j ∈ A. As w_j ∈ B_H we deduce that {w_j} ∉ H. So let |μ_j| ≥ 1 for all j. If |α_i| = 0 for some i then we let . In this case, we have , then rp_v = rp_vp_A = p_vx = 0, a contradiction). Thus . Suppose that {{v} : v ∈ A} ⊆ H. Since H is hereditary, A ∈ H, which is impossible. Hence there is a vertex v ∈ A such that {v} ∉ H.

Step III

If A = {v}, then {v} ∈ H.

We go toward a contradiction and assume {v} ∉ H. Set v₁ := v. If v₁ ∈ B_H, then there is an edge such that and . If v₁ ∉ B_H, we have $0 < ∣ s_{G}^{- 1} (v) ∣ < \infty$ by Step I. The saturation of H gives an edge e₁ with and . By Step II, there is a vertex such that {v₂} ∉ H. We may repeat the argument to choose a path γ = γ₁. . . γ_L for L = max_{i, j} {|β_i|, |ν_j |} + 1, such that and for all 1 ≤ k ≤ L. Note that and so for all i, k. Moreover, since r(γ_k) ∉ H we have $p_{w_{j}}^{H} s_{γ_{k}} = 0$ for all j, k. It follows that rs_γ = (rp_v)s_γ = xs_γ = 0, a contradiction. Therefore {v} ∈ H.

Step IV

If rp_A ∈ I₍_{H, S}₎, then A ∈ H.

If A ∉ H, then by Step II there is a vertex v ∈ A such that {v} ∉ H, which contradicts the Step III. Hence A ∈ H and consequently p_A ∈ I₍_{H, S}₎, as desired.

Now, we show that I₍_{H, S}₎ satisfies assertion (2) of the definition of basic ideal. Note that, by Step IV, we have H_I_{₍_{H, S}₎} = H. Let w ∈ B_H, r ∈ R {0} and $r p_{w}^{H} \in I_{(H, S)}$ . Using the first part of the lemma, write

r p_{w}^{H} = x : = \sum_{i = 1}^{n} r_{i} s_{α_{i}} p_{A_{i}} s_{β_{i}^{*}} + \sum_{j = 1}^{m} s_{j} s_{μ_{j}} p_{w_{j}}^{H} s_{ν_{j}^{*}},

where A_i ∈ H, w_j ∈ S and r_i, s_j ∈ R for all i, j. Since $r p_{w}^{H} = p_{w}^{H} r p_{w}^{H} = p_{w}^{H} x = x$ and $p_{w}^{H} A_{i} = 0$ , We may assume that |α_i| ≥ 1 for all i. As w ∈ B_H, we can choose an edge e ≠ α_1,1, . . . , α_n,₁, μ_1,1, . . . , μ_m,₁ such that and . If w ∉ S, then $r s_{e *} = s_{e *} (r p_{w}^{H}) = s_{e *} x = 0$ which is a contradiction. Therefore w ∈ S and $p_{w}^{H} \in I_{(H, S)}$

Remark 3.2

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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 (H, S) be an admissible pair in ultragraph and let r ∈ R{0}. The argument of Lemma 3.1 implies that

H = {A \in G^{0} : r p_{A} \in I_{(H, S)}} and S = {w \in B_{H} : r p_{w}^{H} \in I_{(H, S)}} .

Lemma 3.3. (cf. [13, Proposition 3.3])

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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

Letbe an ultragraph and let R be a unital commutative ring. If (H, S) is an admissible pair in, then.

Proof

Suppose that {S̃, P̃} is a universal Leavitt -family. If we define

(3.1)

\begin{array}{l} P_{A} : = {\tilde{P}}_{\bar{A}} & for A \in G^{0}, \\ S_{e} : = {\tilde{S}}_{e} & for e \in G^{1}, \\ S_{e *} : = {\tilde{S}}_{e *} & for e \in G^{1}, \end{array}

then it is straightforward to see that {S, P} is a Leavitt -family in . Note that L_R(S, P) inherits the graded structure of . Since rP_A ≠ 0 for all and r ∈ R {0}, by Corollary 2.15, .

We show that . For , P̃_A = P_Ā ∈ L_R(S, P). If v ∉ B_H S, then ${\tilde{P}}_{\bar{{v}}} = {\tilde{P}}_{v} = P_{v} \in L_{R} (S, P)$ . We note that

\bar{s} (e) = {\begin{array}{l} s_{G} {(e)}^{'} & if s_{G} (e) \in B_{H} S and r_{G} (e) \in H, \\ s_{G} (e) & otherwise, \end{array}

for every . Thus for w ∈ B_H S, we have 0 < |s̄⁻¹(w)| < ∞ and for e ∈ s̄⁻¹(w). Hence

{\tilde{P}}_{w} = \sum_{\bar{s} (e) = w} {\tilde{S}}_{e} {\tilde{S}}_{e *} = \sum_{s_{G} (e) = w, r_{G} (e) \notin H} S_{e} S_{e *} \in L_{R} (S, P) .

Also,

{\tilde{P}}_{w^{'}} = {\tilde{P}}_{{w, w^{'}}} - {\tilde{P}}_{w} = P_{w} - \sum_{s_{G} (e) = w, r_{G} (e) \notin H} S_{e} S_{e *} = P_{w}^{H} \in L_{R} (S, P) .

Thus, by Remark 2.6, P̃_A ∈ L_R(S, P) for all . Since S̃_e, S̃_e_^* ∈ L_R(S, P), we deduce that . Consequently, .

Theorem 3.4. (cf. [12, Theorem 3.10])

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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

Letbe an ultragraph and let R be a unital commutative ring. Then

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 ofto the set of all graded basic ideals of.

Proof

(1) Let {S̃, P̃} be a universal Leavitt -family and let , where {S, P} is the Leavitt -family as defined in 3.1. Define

(3.2)

\begin{array}{l} Q_{[A]} : = {\tilde{P}}_{A} + I_{(H, S)} & for A \in {\bar{G}}^{0} \\ T_{e} : = {\tilde{S}}_{e} + I_{(H, S)} & for e \in Φ (G^{1}), \\ T_{e *} : = {\tilde{S}}_{e *} + I_{(H, S)} & for e \in Φ (G^{1}) . \end{array}

It can be shown that the family is a Leavitt -family that generates . Furthermore, by Remark 3.2, rQ_[_A_] ≠ 0 for all and r ∈ R {0}.

Now we use the universal property of to get an R-homomorphism such that π(t_e) = T_e, π(t_e_^*) = T_e_^* and π(rq_[_A_]) = rQ_[_A_] ≠ 0 for and r ∈ R {0}. Since I₍_{H, S}₎ is a graded ideal, the quotient is graded. Moreover, the elements Q_[_A_], T_e and T_e_^* have degrees 0, 1 and −1 in , respectively and thus π is a graded homomorphism. It follows from the graded-uniqueness Theorem 2.14 that π is injective. Since is generated by , we deduce that π is also surjective. Hence .

(2) The injectivity of the map (H, S) ↦ I₍_{H, S}₎ is a consequence of Remark 3.2. To see that it is onto, let I be a graded basic ideal in . Then I₍_H_{_I}, _S_{_I)} ⊆ I. Consider the ultragraph with respect to admissible pair (H_I, S_I). Since I is a graded ideal, the quotient ring is graded. Let be the quotient map. For (H_I, S_I), consider {S̃, P̃} and {T, Q} as defined in Equations 3.1 and 3.2, respectively. Since I is basic, we have rp_A ∉ I and $r p_{w}^{H_{I}} \notin I$ for , w ∈ B_H_{_I} S_I and r ∈ R {0}.

We show that π(rq_[_A_]) = π(rQ_[_A_]) = r P̃_A +I ≠ 0 for all and r ∈ R {0}. Fix r ∈ R {0}. We know that . Let A = B for some . If [A] ≠ [∅︀], then B ∉ H_I and thus r P̃_A = rp_B ∉ I. Therefore π(rq_[_A_]) ≠ 0. If w ∈ B_H_{_I} S_I, then $r {\tilde{P}}_{w^{'}} = r p_{w}^{H_{I}} \notin I$ . Hence π(rq_[_w′_]) ≠ 0. In view of Remark 2.10, we deduce that π(rq_[_A_]) ≠ 0 for every . Furthermore, π is a graded homomorphism. It follows from Theorem 2.14 that the quotient map π is injective. Hence I = I₍_H_{_I}, _S_{_I)}.

As we have seen in the proof of Theorem 3.4, if (H, S) is an admissible pair in , then there is a Leavitt -family {T, Q} in such that rQ_[_A_] ≠ 0 for all and r ∈ R {0}. So we have the following proposition.

Proposition 3.5

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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 (H, S) be an admissible pair in ultragraphand let R be a unital commutative ring. If {t, q} is the universal Leavitt-family, then rq_[_A_] ≠ 0 for everyand every r ∈ R {0}.

4. Condition (K)

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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 v ∈ G⁰. A first-return path based at v in is a loop α = e₁e₂ · · · e_n such that and for i = 2, 3, . . . , n.

Definition 4.1. ([9, Section 7])

An ultragraph satisfies Condition (K) if every vertex in G⁰ is either the base of no first-return path or it is the base of at least two first-return paths.

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 hereditary if satisfying the following conditions:

s(e) ∈ K implies r(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 I_K in generated by {q_[_A_] : [A] ∈ K} is equal to

{span}_{R} {t_{α} q_{[A]} t_{β *} \in L_{R} (G / (H, S)) : α, β \in Path (G / (H, S)) and [A] \in K} .

Lemma 4.2

Letbe a quotient ultragraph and let R be a unital commutative ring. Ifdoes not satisfy Condition (L), thencontains a non-graded ideal I such that rq_[_A_] ∉ I for alland r ∈ R{0}.

Proof

Suppose that contains a closed path γ := e₁e₂ · · · e_n without exits and r(e_i) = s(e_i₊₁) for 1 ≤ i ≤ n where e_n₊₁ := e₁. Thus we can assume that s(e_i) ≠ s(e_j) for all i, j. Let s(e_i) = [v_i] for 1 ≤ i ≤ n and let X be the subalgebra of generated by {t_e_{_i}, $t_{e_{i}^{*}}$ , q_[_v_{_i]} : 1 ≤ i ≤ n}.

Claim 1

The subalgebra X is isomorphic to the Leavitt path algebra L_R(E), where E is the graph containing a single simple closed path of length n, that is,

\begin{matrix} E^{0} = {w_{1}, \dots, w_{n}}, E^{1} = {f_{1}, \dots, f_{n}}, \\ r (f_{i}) = s (f_{i + 1}) = w_{i + 1} for 1 \leq i \leq n where w_{n + 1} : = w_{1} . \end{matrix}

Proof of Claim 1

Set P_w_{_i} := q_[_v_{_i]}, S_f_{_i} := t_e_{_i} and $S_{f_{i}^{*}} : = t_{e_{i}^{*}}$ for all i. Then {S, P} is a Leavitt E-family in X such that L_R(S, P) = X. So there is a homomorphism π : L_R(E) → X such that π(s_f_{_i}) = t_e_{_i}, $π (s_{f_{i}^{*}}) = t_{e_{i}^{*}}$ and π(p_w_{_i}) = q_[_v_{_i]} for all i. Since π preserves the degree of generators, π is a graded homomorphism. By Proposition 3.5, π(rp_w_{_i}) = rq_[_v_{_i]} ≠ 0 for all r ∈ R {0} and all i. Now apply [18, Theorem 5.3] to obtain the injectivity of π. Hence, X ~= L_R(E) which proves the claim.

By [18, Lemma 7.16], L_R(E) contains an ideal J that is basic but not graded. Thus π(J) is a non-graded ideal of X.

Claim 2

rq_[_A_] ∉ π(J) for every r ∈ R {0} and .

Proof of Claim 2

We show that rp_w_{_i} ∉ J for all r ∈ R{0} and all i. Let rp_w_{_j} ∈ J for some r ∈ R {0} and some 1 ≤ j ≤ n. Since J is a basic ideal, p_w_{_j} ∈ J. For every 1 ≤ i ≤ n, there is a path α ∈ Path(E) such that s(α) = w_j and r(α) = w_i. Hence $p_{w_{i}} = s_{α_{i}^{*}} p_{w_{j}} s_{α_{i}} \in J$ for all i. Thus J = L_R(E), contradicting the hypothesis that J is non-graded.

Suppose that there exist r ∈ R{0} and such that rq_[_A_] ∈ π(J). If we identify X with the subalgebra

{span}_{R} {t_{α} t_{β *} \in L_{R} (G / (H, S)) : s (α), s (β) \in {[v_{1}], \dots, [v_{n}]}}

of , then $r q_{[A]} = r_{1} t_{α_{1}} t_{β_{1}^{*}} + \dots + r_{m} t_{α_{m}} t_{β_{m}^{*}}$ , where r_k ∈ R {0} and s(α_k), s(β_k) ∈ {[v₁], . . . , [v_n]}. Thus

r q_{[A]} = r q_{[A]} q_{[A]} = r_{1} q_{[A]} t_{α_{1}} t_{β_{1}^{*}} + \dots + r_{m} q_{[A]} t_{α_{m}} t_{β_{m}^{*}} \neq 0,

which implies that $[A] \cap (\cup_{i = 1}^{n} [v_{i}]) \neq [\emptyset]$ . So there is j such that [v_j ] ⊆ [A]. In this case we have rq_[_v_{_j]} = q_[_v_{_j]}rq_[_A_] ∈ π(J). Hence rp_w_{_j} = π⁻¹(rq_[_v_{_j]}) ∈ J, a contradiction.

Let

K : = {⋃_{k = 1}^{m} [v_{n_{k}}] : 1 \leq m \leq n and n_{k} \in {1, 2, \dots, n}}

and set $[A] : = \cup_{i = 1}^{n} [v_{i}]$ . Since γ is a closed path without exits and r(e_i) = s(e_i₊₁), K is a hereditary subset of .

Claim 3

If rq_[_v_] ∈ I_K for some and some r ∈ R{0}, then there exists such that s(α) = [v] and r(α) ∩ A ≠ [∅︀].

Proof of Claim 3

Suppose that

r q_{[v]} = x : = r_{1} t_{α_{1}} q_{[v_{n_{1}}]} t_{β_{1}^{*}} + \dots + r_{m} t_{α_{m}} q_{[v_{n_{m}}]} t_{β_{m}^{*}},

where r_i ∈ R {0} and . If there is no such path, then we can choose an edge e such that s(e) = [v] and r(e) ∩ [A] = [∅︀]. Since rq_r₍_e₎ ∈ I_K, there is a vertex [w] ⊆ r(e) such that rq_[_w_] ∈ I_K. By a continuing process, one can choose a path η = η₁ · · · η_L for L = max_i {|β_i|} + 1, such that s(η) = [v] and r(η_k) ∩ [A] = [∅︀] for all 1 ≤ k ≤ L. Thus we have rt_η = (rq_[_v_])t_η = xt_η = 0, a contradiction. Therefore, there is a path α such that s(α) = [v] and r(α)∩[A] ≠ [∅︀].

Claim 4

I := I_Kπ(J)I_K is a non-graded ideal of .

Proof of Claim 4

Suppose that I is graded. Thus I = ⊕ I_n, where . For x ∈ π(J) and we have q_[_A_]xq_[_A_] = x and q_[_A_]y, yq_[_A_] ∈ X. Thus π(J) ⊆ I. Let x ∈ π(J). Since I = ⊕ I_n we have x = r_n_₁y_n_₁x_n_₁z_n_₁ + . . . + r_n_{_m}y_n_{_m}x_n_{_m}z_n_{_m}, where x_n_{_k} ∈ π(J), y_n_{_k}, z_n_{_k} ∈ I_K and y_n_{_k}x_n_{_k}z_n_{_k} ∈ I_n_{_k}. Thus

x = q_{[A]} x q_{[A]} = r_{n_{1}} q_{[A]} y_{n_{1}} x_{n_{1}} z_{n_{1}} q_{[A]} + \dots + r_{n_{m}} q_{[A]} y_{n_{m}} x_{n_{m}} z_{n_{m}} q_{[A]} .

Since q_[_A_]y_n_{_k}, z_n_{_k}q_[_A_] ∈ X, we have r_n_{_k}q_[_A_]y_n_{_k}x_n_{_k}z_n_{_k}q_[_A_] ∈ π(J) ∩X_n, where . Hence π(J) is a graded ideal of X, a contradiction.

Finally, we show that rq_[_B_] ∉ I for all and r ∈ R{0}. Assume rq_[_B_] ∈ I for some r ∈ R{0} and some . Since rq_[_B_] ∈ I_K, there is a vertex [v] ⊆ [B] such that rq_[_v_] ∈ I ⊆ I_K. By Claim 2, there is a path α such that s(α) = [v] and [v_i] ⊆ r(α)∩[A] for some i ∈ {1, . . . , n}. Sorq_[_v_{_i]} = t_α_^*rq_[_v_]t_αq_[_v_{_i]} ∈ I. If we consider rq_[_v_{_i]} in terms of its representation in I_Kπ(J)I_K, then one can show that rq_[_v_{_i]} ∈ Xπ(J)X = π(J) which is a contradiction. Therefore, we have rq_[_B_] ∉ I for every r ∈ R {0} and every .

Theorem 4.3. (cf. [12, Theorem 3.18], [18, Theorem 7.17])

Go to

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

Letbe an ultragraph and let R be a unital commutative ring. Thensatisfies Condition (K) if and only if every basic ideal inis graded.

Proof

Suppose that satisfies Condition (K). If I is a basic ideal of , then by Theorem 3.4, we have the quotient map

π : L_{R} (G / (H_{I}, S_{I})) ≅ L_{R} (G) / I_{(H_{I}, S_{I})} \to L_{R} (G) / I,

such that π(rq_[_A_]) ≠ 0 for all and all r ∈ R {0}. By Proposition [13, Proposition 6.2] and the Cuntz-Krieger uniqueness Theorem 2.17, π is injective and I = I₍_H_{_I}, _S_{_I)}. It follows from Lemma 3.1 that I is a graded ideal.

Conversely, if does not satisfy Condition (K), then [13, Proposition 6.2] and Lemma 4.2 imply that there exist H and S such that contains a non-graded ideal J such that rq_[_A_] ∉ J for all and all r ∈ R {0}. If is the quotient map, then I := π⁻¹(J) is a non-graded ideal of . Similar to the last paragraph of the proof of Theorem 3.4, we have rp_A ∉ I for ∅︀ ≠ A ∉ H and r ∈ R {0}, that is, . Also, $r p_{w}^{H} \notin I$

for w ∈ B_H S and r ∈ R {0}. Consequently I is a basic ideal. Therefore satisfies Condition (K).

Corollary 4.4

Go to

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

Letbe an ultragraph and let R be a unital commutative ring. Ifsatisfies Condition (K), then the map (H, S) ↦ I₍_{H, S}₎is a bijection from the set of all admissible pairs ofinto the set of all basic ideals of.

5. Exel-Laca R-algebras

Go to

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 A with no identically zero rows [7, Definition 8.1]. In this section, we define an algebraic version of the Exel-Laca algebras.

Definition 5.1

Let I be a countable set and let A be an I ×I matrix with entries in {0, 1}. The ultragraph $G_{A} : = (G_{A}^{0}, G_{A}^{1}, r, s)$ is defined by $G_{A}^{0} : = {v_{i} : i \in I}, G_{A}^{1} : = I$ , s(i) := v_i and r(i) := {v_j : A(i, j) = 1} for all i ∈ I.

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 R be a unital commutative ring, let I be a countable set and let A be a {0, 1}-valued matrix over I with no identically zero rows. The Exel-Laca R-algebra associated to A, denoted by ℰℒ_A(R), is defined by .

Example 5.3

Let R be a unital commutative ring and let A = (A(i, j)) be an n × n matrix, with with no identically zero rows and A(i, j) ∈ {0_R, 1_R} for all i, j. The Cuntz-Krieger R-algebra associated to A, as defined in [3, Example 2.5], is the free R-algebra generated by a set {x₁, y₁, . . . , x_n, y_n}, modulo the ideal generated by the following relations:

x_iy_ix_i = x_i and y_ix_iy_i = y_i for all i,
yix_j = 0 for all i ≠ j,
$y_{i} x_{i} = \sum_{j = 1}^{n} A (i, j) x_{j} y_{j}$ for all i,
$\sum_{j = 1}^{n} x_{j} y_{j} = 1$ .

The Cuntz-Krieger R-algebra associated to A is denoted by . We will show that . Note that is a finite ultragraph. If we define P_B := ∑_v_{_i∈}_B x_iy_i, S_i := x_i and S_i_^* := y_i for all i and all $B \in G_{A}^{0}$ , then it is easy to verify that { $P_{B}, S_{i}, S_{i *} : B \in G_{A}^{0}, 1 \leq i \leq n$ }is a Leavitt -family.

On the other hand, if {s, p} is the universal Leavitt -family for , then the elements X_i := s_i and Y_i := s_i_^*, 1 ≤ i ≤ n, satisfy relations (1)–(4). Now the universality of and conclude that .

Note that for any {0, 1}-valued matrix A with no identically zero rows, the ultragraph 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 v ∈ G⁰ and e ∈ s⁻¹(v), we introduce a vertex v_e and set Ã := {v_e : v ∈ A and e ∈ s⁻¹(v)} for every . Next we define the ultragraph , where G̃⁰ := {v_e : v ∈ G⁰ and e ∈ s⁻¹(v)}, , the source map s̃(e) := s(e)_e and the range map $\tilde{r} (e) : = \tilde{r (e)}$ , where $\tilde{r (e)} = {v_{e} : v \in r (e) and e \in s^{- 1} (v)}$ .

Remark 5.4

We notice that each vertex v_e in emits exactly one edge e. Moreover, Lemma 2.5 implies that

{\tilde{G}}^{0} = {\tilde{A} \cup F : A \in G^{0} and F is a finite subset of {\tilde{G}}^{0}} .

Lemma 5.5

Letbe a row-finite ultragraph with no singular vertices and let R be a unital commutative ring. Thenis isomorphic toas algebras.

Proof

Let {t, q} be the universal Leavitt -family for . Since is row-finite, for every v ∈ G⁰ we have . Define

\begin{array}{l} P_{A} : = q_{\tilde{A}} & for A \in G^{0}, \\ S_{e} : = t_{e} & for e \in G^{1}, \\ S_{e *} : = t_{e *} & for e \in G^{1} . \end{array}

By Remark 5.4, for each , the idempotent P_A satisfies the condition (1) of Definition 2.1. It is straightforward to check that {S, P} is a Leavitt -family in . For example, to verify condition (2) of Definition 2.1 suppose that . Then S_eP_r₍_e₎ = t_eq_r̃₍_e₎ = t_e = S_e and

P_{s (e)} S_{e} = (\sum_{f \in {\tilde{G}}^{1}} q_{s {(e)}_{f}}) q_{\tilde{s} (e)} t_{e} = (\sum_{f \in {\tilde{G}}^{1}} q_{s {(e)}_{f}}) q_{s {(e)}_{e}} t_{e} = q_{s {(e)}_{e}} t_{e} = t_{e} = S_{e} .

Thus there is an R-algebra homomorphism such that φ(s_e) = t_e, φ(s_e_^*) = t_e_^* and for v ∈ G⁰, and r ∈ R {0}. So rφ(p_A) ≠ 0 for all . As φ is a graded homomorphism, Corollary 2.15 implies that φ is injective. Moreover, for any v_e ∈ G̃⁰, we have (s̃)⁻¹(v_e) = {e} and hence S_eS_e_^* = t_et_e_^* = q_v_{_e}. Therefore, φ is an isomorphism and we get the result.

Definition 5.6. ([15, Definition 2.4])

Let be an ultragraph. The edge matrix of an ultragraph is the matrix given by

A_{G} (e, f) : = {\begin{array}{l} 1 & if s (f) \in r (e), \\ 0 & otherwise . \end{array}

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

Let R be a unital commutative ring and letbe an ultragraph with no singular vertices. Thenis isomorphic to.

Example 5.8

Suppose that E is a graph with one vertex E⁰ = {v} and n edges E¹ = {e₁, . . . .e_n}. We know that L_R(E) is isomorphic to the Leavitt algebra L_R(1, n). If all entries of matrix A_n_×_n is 1, then A = A_E. Hence, Theorem 5.7 shows that ℰℒ_A(R) ~= L_R(1, n).

Remark 5.9

Theorem 5.7 shows that the Leavitt path algebras of ultragraphs with no singular vertices are precisely the Exel-Laca R-algebras.

Remark 5.10

We know that for every directed graph E, L_ℂ(E) is isomorphic to a dense *-subalgebra of the C^*-algebra C^*(E) (see [17, Section 7]). Using a similar argument to [17, Theorem 7.3], one can show that is isomorphic to a dense *-subalgebra of the C^*-algebra introduced in [15]. In particular, for any countable {0, 1}-valued matrix A, the Exel-Laca algebra ℰℒ_A(ℂ) is isomorphic to a dense *-subalgebra of [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 R = ℤ₂ and let be the ultragraph

with one ultraedge e such that s(e) = v₀ and r(e) = {v₁, v₂, . . .}. Note that 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 E. Let {t, q} be the canonical generators for L_ℤ₂ (E). If E has a loop α, then for t_α ∈ L_ℤ₂ (E) we have $t_{α}^{n} \neq t_{α}^{m}$ for all m ≠ n, but one can show that does not include such member. Thus L_ℤ₂ (E) does not have any loop and consequently E satisfies Condition (K) for directed graphs. Hence by [12, Theorem 3.18] any ideal of L_ℤ₂ (E) is graded. Since L_ℤ₂ (E) is unital, E⁰ must be finite and hence L_ℤ₂ (E) contains finitely many (graded) ideals by Theorem 3.4(2), which contradicts the fact that has infinitely many pairwise orthogonal ideals I_{_v_{_n}} for n ≥ 1.

Now assume for some matrix A and let be an isomorphism. We may consider the ideal I_{_v_₁} in . Since v₁ is a sink, we have

I_{{v_{1}}} = {span}_{ℤ_{2}} {p_{v_{1}}, s_{e} p_{v_{1}}, p_{v_{1}} s_{e *}, s_{e} p_{v_{1}} s_{e *}},

and so |I_{_v_₁}| < ∞. On the other hand, we know that has no sinks. Hence the graded ideal φ(I_{_v_₁}) in includes infinitely many elements, which is a contradiction.

Acknowledgements

Go to

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

Go to

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

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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.
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M. Tomforde. A unified approach to Exel-Laca algebras and C^*-algebras associated to graphs. J Operator Theory., 50(2003), 345-368.
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