Group gradations on Leavitt path algebras

TL;DR

This paper investigates the gradation properties of Leavitt path algebras over arbitrary groups, establishing conditions for various gradation strengths and providing new proofs and criteria related to their algebraic structure.

Contribution

It introduces the concept of nearly epsilon-strongly G-graded Leavitt path algebras and characterizes when these algebras are epsilon-strongly or strongly G-graded, including new proofs and criteria.

Findings

L_R(E) is always nearly epsilon-strongly G-graded.

If E is finite, then L_R(E) is epsilon-strongly G-graded.

L_R(E) is strongly Z-graded iff E has no sink when E is row-finite with no source.

Abstract

Given a directed graph $E$ and an associative unital ring $R$ one may define the Leavitt path algebra with coefficients in $R$, denoted by $L_R(E)$. For an arbitrary group $G$, $L_R(E)$ can be viewed as a $G$-graded ring. In this article, we show that $L_R(E)$ is always nearly epsilon-strongly $G$-graded. We also show that if $E$ is finite, then $L_R(E)$ is epsilon-strongly $G$-graded. We present a new proof of Hazrat's characterization of strongly $\mathbb{Z}$-graded Leavitt path algebras, when $E$ is finite. Moreover, if $E$ is row-finite and has no source, then we show that $L_R(E)$ is strongly $\mathbb{Z}$-graded if and only if $E$ has no sink. We also use a result concerning Frobenius epsilon-strongly $G$-graded rings, where $G$ is finite, to obtain criteria which ensure that $L_R(E)$ is Frobenius over its identity component.

Equations

$$A:$$

$$M$$

$$\{{ϵ}_g(s)\mid s\in S_g\}\cup \{{ϵ}_{g^{-1}}^{\prime }(s)\mid s\in S_g\}.$$

$$0\ne {\alpha }^{\ast }\beta =f_m^{\ast }\cdots f_1^{\ast }{f}_1^{\prime }\cdots f_n^{\prime }={\delta }_{f_1,f_1^{\prime }}\cdots {\delta }_{f_m,f_m^{\prime }}{f}_{m+1}^{\prime }\cdots f_n^{\prime }.$$

$$0\ne {\alpha }^{\ast }\beta =f_m^{\ast }\cdots f_1^{\ast }{f}_1^{\prime }\cdots f_n^{\prime }={\delta }_{f_1,f_1^{\prime }}\cdots {\delta }_{f_n,f_n^{\prime }}{f}_m^{\ast }\cdots f_{n+1}^{\ast }.$$

$${ϵ}_g=\sum _{i=1}^{n_g}{\mathcal{N}}_g(m_i)$$

$${\bullet }_{v_1}$$

$$S_0={\mathrm{span}}_R\{v_1,v_2,v_3,v_4,v_5,f_1{f}_1^{\ast },f_2{f}_3^{\ast },f_2{f}_2^{\ast },f_3{f}_2^{\ast }\},$$

$$S_1={\mathrm{span}}_R\{f_1,f_2,f_3,f_4,f_4{f}_3{f}_2^{\ast }\},S_{-1}={\mathrm{span}}_R\{f_1^{\ast },f_2^{\ast },f_3^{\ast },f_4^{\ast },f_2{f}_3^{\ast }{f}_4^{\ast }\},$$

$$S_2={\mathrm{span}}_R\{f_4{f}_3\},S_{-2}={\mathrm{span}}_R\{f_3^{\ast }{f}_4^{\ast }\},\text{and}{S}_n=\{0\},\text{ for }|n|>2.$$

$$0\ne v_2=f_k^{\ast }{v}_1{f}_k=\sum _{i\in F}{r}_i{f}_k^{\ast }{f}_{p_i}{f}_{q_i}^{\ast }{f}_k$$

$$\displaystyle\{\mbox{unital strongly $\mathbb{Z}$-graded Leavitt path algebras}\}$$

$$\displaystyle\subsetneq\{\mbox{epsilon-strongly $\mathbb{Z}$-graded Leavitt path algebras}\}$$

$$\displaystyle\subsetneq\{\mbox{nearly epsilon-strongly $\mathbb{Z}$-graded Leavitt path algebras}\}.$$

$${ϵ}_1=\sum _{\alpha \in E^1}{\mathcal{N}}_1(\alpha )=\sum _{\alpha \in E^1}\alpha {\alpha }^{\ast }=\sum _{v\in E^0\setminus \{\mathrm{sinks}\}}\sum _{\stackrel{\alpha \in E^1}{s(\alpha )=v}}\alpha {\alpha }^{\ast }=\sum _{v\in E^0\setminus \{\mathrm{sinks}\}}v.$$

$$v$$

$$=\sum _{i=1}^{n(v)}{\alpha }_{i,v}{\beta }_i^{\ast }{\beta }_i{\alpha }_{i,v}^{\ast }=\sum _{i=1}^{n(v)}{m}_{i,v}{m}_{i,v}^{\ast }=\sum _{i=1}^{n(v)}{\mathcal{N}}_{-1}(m_{i,v})\in S_{-1}{S}_1.$$

Full text

1. Introduction

A Leavitt path algebra associates with a directed graph $E$ and an associative unital ring $R$, an $R$-algebra $L_{R}(E)$. These algebras are algebraic analogues of graph $C^{*}$-algebras and are also natural generalizations of Leavitt algebras of type $(1,n)$ constructed in [8]. Leavitt path algebras were introduced, in the case when $R$ is a field, by Ara, Moreno and Pardo [3], and almost simultaneously by Abrams and Aranda Pino [2], using a different approach. In the case when $R$ is a commutative ring these structures were defined by Tomforde in [13]. Throughout this article we will follow Hazrat [6] and consider Leavitt path algebras with coefficients in an arbitrary associative unital ring $R$ (see Definition 18). Each Leavitt path algebra may, in a canonical way, be viewed as a $\mathbb{Z}$-graded ring (see Section 4). The characterization of Leavitt path algebras (e.g. simple, purely infinite simple, semisimple, prime, finite dimensional, locally finite, exchange, artinian, noetherian) in terms of properties of the underlying graph has been the subject of many studies, see Abrams’ extensive survey [1], and the references therein. Many of these investigations have, however, been carried out without taking into account the $\mathbb{Z}$-graded structure of $L_{R}(E)$. Although there are previous studies (see e.g. Ara, Moreno and Pardo [3], Tomforde [12]) where the $\mathbb{Z}$-graded structure of $L_{R}(E)$ has been taken into account, it seems to the authors of the present article that Hazrat [6] was first to utilize this philosophy systematically, and to e.g. study properties of the $\mathbb{Z}$-gradation itself. In loc. cit. he proves, among many other things, the following result.

Theorem 1** (Hazrat [6]).**

If $E$ is a finite directed graph and $R$ is an associative unital ring, then the Leavitt path algebra $L_{R}(E)$ is strongly $\mathbb{Z}$-graded if and only if $E$ has no sink.

To motivate the approach taken in this article, consider the following directed graphs.

[TABLE]

According to Theorem 1, the graph $A$ produces a strongly $\mathbb{Z}$-graded Leavitt path algebra, but the graphs $B$ and $C$ do not. In an attempt to single out a class of well-behaved group graded rings which naturally includes all crossed products by unital twisted partial actions, Nystedt, Öinert and Pinedo [10] recently introduced a generalization of unital strongly graded rings called epsilon-strongly graded rings (see Definition 6). In this article, we establish the following result which shows that $L_{R}(B)$ is epsilon-strongly $\mathbb{Z}$-graded.

Theorem 2**.**

If $E$ is a finite directed graph and $R$ is an associative unital ring, then the Leavitt path algebra $L_{R}(E)$ is epsilon-strongly $\mathbb{Z}$-graded.

It turns out that $L_{R}(C)$ is not epsilon-strongly $\mathbb{Z}$-graded (see Example 32). To capture Leavitt path algebras corresponding to the graph $C$ we introduce the class of nearly epsilon-strongly graded rings (see Definition 10), which is a further weakening of the notion of a strongly graded ring, and show the following result.

Theorem 3**.**

If $E$ is a (possibly non-finite) directed graph and $R$ is an associative unital ring, then the Leavitt path algebra $L_{R}(E)$ is nearly epsilon-strongly $\mathbb{Z}$-graded.

Here is a detailed outline of this article.

In Section 2, we recall some notions concerning unital and $s$-unital modules that we need in the sequel.

In Section 3, we provide the necessary background on group graded rings for use in subsequent sections. We define the following notions: symmetrically graded ring, epsilon-strongly graded ring, nearly epsilon-strongly graded ring, non-degenerately graded ring and strongly non-degenerately graded ring (see Definitions 5, 6, 10, 12 resp. 13) and we show a result connecting these notions (see Proposition 14).

In Section 4, we accomodate the Leavitt path algebra framework that we need in the sequel; in particular standard $G$-gradations on Leavitt path algebras, for an arbitrary group $G$. In the same section, we prove $G$-graded versions of Theorem 2 and Theorem 3 (see Theorem 28 and Theorem 30) through a series of propositions. At the end of the section, we apply a result, by Pinedo and the authors of the present article, concerning epsilon-strongly $G$-graded rings, to obtain criteria which ensure that a Leavitt path algebra is Frobenius over its identity component, when the group $G$ is finite (see Theorem 38).

In Section 5, we provide a useful characterization of strongly $\mathbb{Z}$-graded rings (see Proposition 39) and several corollaries.

In Section 6, we obtain a new proof of Hazrat’s Theorem 1 (see Proposition 45) by using our results from Section 4 and Section 5. We also characterize strongly $\mathbb{Z}$-graded Leavitt path algebras over row-finite graphs without sources (see Theorem 47).

2. s-unital modules

In this section, we recall some notions concerning unital and $s$-unital modules that we need in the sequel. Throughout this article all rings are associative but not necessarily unital. For the rest of this section $T$ and $T^{\prime}$ denote rings. Recall the following definitions.

If $M$ is a left $T$-module (right $T^{\prime}$-module), then $M$ is called left (right) unital if there is $t\in T$ ($t^{\prime}\in T^{\prime}$) such that, for all $m\in M$, the relation $tm=m$ ($mt^{\prime}=m$) holds. In that case $t$ ($t^{\prime}$) is said to be a left (right) identity for $M$. If $M$ is a $T$–$T^{\prime}$-bimodule, then $M$ is called unital if it is unital both as a left $T$-module and as a right $T^{\prime}$-module. The ring $T$ is said to be left (right) unital if it is left (right) unital as a left (right) module over itself. The ring $T$ is called unital if it is unital as a bimodule over itself.

In this article, we will use the notion of s-unitality, introduced in [14]. Namely, if $M$ is a left $T$-module (right $T^{\prime}$-module), then $M$ is said to be $s$-unital if for each $m\in M$ the relation $m\in Tm$ ($m\in mT^{\prime}$) holds. If $M$ is a $T$–$T^{\prime}$-bimodule, then $M$ is said to be $s$-unital if it is $s$-unital both as a left $T$-module and as a right $T^{\prime}$-module. The ring $T$ is said to be left (right) $s$-unital if it is left (right) $s$-unital as a left (right) module over itself. The ring $T$ is said to be $s$-unital if it is $s$-unital as a bimodule over itself.

Remark 4**.**

From [14, Theorem 1] it follows that if $M$ is a left $T$-module (right $T^{\prime}$-module), then $M$ is s-unital if and only if for all $n\in\mathbb{N}$ and all $m_{1},\ldots,m_{n}\in M$ there exists $t\in T$ ($t^{\prime}\in T$) such that, for each $i\in\{1,\ldots,n\}$, the equality $tm_{i}=m_{i}$ ($m_{i}t^{\prime}=m_{i}$) holds.

Let $M$ be a left $T$-module (right $T^{\prime}$-module). Recall that $M$ is called torsion-free if for all non-zero $m\in M$ the relation $Tm\neq\{0\}$ ($mT^{\prime}\neq\{0\}$) holds. If $M$ is a $T$–$T^{\prime}$-bimodule which is torsion-free both as a left $T$-module and as a right $T^{\prime}$-module, then $M$ is said to be torsion-free. It is clear that the following chain of implications hold:

[TABLE]

3. Graded rings

In this section, we provide the necessary background on group graded rings. In particular, we define the notion of a symmetrically graded ring, an epsilon-strongly graded ring, a nearly epsilon-strongly graded ring, a non-degenerately graded ring resp. a strongly non-degenerately graded ring (see Definitions 5, 6, 10, 12 resp. 13) and show a result connecting these notions (see Proposition 14).

Throughout this section, $G$ denotes a group with identity element $e$ and $S$ denotes an associative ring which is $G$-graded (or graded by $G$). Recall that this means that for each $g\in G$, there is an additive subgroup $S_{g}$ of $S$ such that $S=\oplus_{g\in G}S_{g}$ and for all $g,h\in G$ the inclusion $S_{g}S_{h}\subseteq S_{gh}$ holds. If, in addition, for all $g,h\in G$ the equality $S_{g}S_{h}=S_{gh}$ holds, then $S$ is said to be strongly $G$-graded (or strongly graded by $G$). For an overview of the theory of group graded rings, we refer the reader to [9].

Following [4, Definition 4.5] we make the following definition.

Definition 5**.**

The ring $S$ is said to be symmetrically $G$-graded (or symmetrically graded by $G$) if for each $g\in G$ the relation $S_{g}S_{g^{-1}}S_{g}=S_{g}$ holds.

We now recall from [10] the following weakening of the notion of a unital strongly graded ring.

Definition 6**.**

The ring $S$ is said to be epsilon-strongly $G$-graded (or epsilon-strongly graded by $G$) if for each $g\in G$ the $S_{g}S_{g^{-1}}$–$S_{g^{-1}}S_{g}$-bimodule $S_{g}$ is unital.

The following characterization of epsilon-strongly graded rings more or less follows from [10, Proposition 7]. For the convenience of the reader we include a proof.

Proposition 7**.**

The following assertions are equivalent:

• (i)

The ring $S$ is epsilon-strongly $G$-graded;

• (ii)

The ring $S$ is symmetrically $G$-graded and for each $g\in G$ the ring $S_{g}S_{g^{-1}}$ is unital;

• (iii)

For each $g\in G$ there exists $\epsilon_{g}\in S_{g}S_{g^{-1}}$ such that for all $s\in S_{g}$ the equalities $\epsilon_{g}s=s\epsilon_{g^{-1}}=s$ hold.

Proof.

(i)$\Rightarrow$(ii): Suppose that $S$ is epsilon-strongly $G$-graded. Take $g\in G$. First we show that $S$ is symmetrically $G$-graded. It is clear that $S_{g}S_{g^{-1}}S_{g}\subseteq S_{g}$. Now we show the reversed inclusion. Take $s\in S_{g}$. From Definition 6 it follows that there exists $\epsilon_{g}\in S_{g}S_{g^{-1}}$ such that $\epsilon_{g}s=s$. In particular, it follows that $s\in\epsilon_{g}S_{g}\subseteq S_{g}S_{g^{-1}}S_{g}$. Now we show that the ring $S_{g}S_{g^{-1}}$ is unital. By Definition 6, $S_{g}$ is a unital left $S_{g}S_{g^{-1}}$-module and $S_{g^{-1}}$ is a unital right $S_{g}S_{g^{-1}}$-module. Thus, $S_{g}S_{g^{-1}}$ is unital as an $S_{g}S_{g^{-1}}$-bimodule.

(ii)$\Rightarrow$(iii): Suppose that $S$ is symmetrically $G$-graded and for each $g\in G$ the ring $S_{g}S_{g^{-1}}$ is unital. Take $g\in G$. Let $\epsilon_{g}$ denote the multiplicative identity element of $S_{g}S_{g^{-1}}$. Take $s\in S_{g}$. Using that $S$ is symmetrically $G$-graded, there exist $n\in\mathbb{N}$, $a_{i},c_{i}\in S_{g}$ and $b_{i}\in S_{g^{-1}}$, for $i\in\{1,\ldots,n\}$, such that $s=\sum_{i=1}^{n}a_{i}b_{i}c_{i}$. Take $i\in\{1,\ldots,n\}$. Since $a_{i}b_{i}\in S_{g}S_{g^{-1}}$ we get that $\epsilon_{g}a_{i}b_{i}=a_{i}b_{i}$ and thus $\epsilon_{g}s=\sum_{i=1}^{n}\epsilon_{g}a_{i}b_{i}c_{i}=\sum_{i=1}^{n}a_{i}b_{i}c_{i}=s$. Similarly, $s\epsilon_{g^{-1}}=s$.

(iii)$\Rightarrow$(i): This is immediate. ∎

Remark 8**.**

If $S$ is epsilon-strongly $G$-graded, then the element $\epsilon_{g}$ from Proposition 7(iii) is a multiplicative identity element of the $S_{e}$-ideal $S_{g}S_{g^{-1}}$ and $\epsilon_{g}\in Z(S_{e})$. Moreover, $\epsilon_{e}$ is a multiplicative identity element of both $S_{e}$ and $S$, making $S$ a unital ring.

Proposition 9**.**

If $S$ is epsilon-strongly $G$-graded, then the following assertions are equivalent:

• (i)

$S$* is strongly $G$-graded;*

• (ii)

for each $g\in G$ the equality $S_{g}S_{g^{-1}}=S_{e}$ holds;

• (iii)

for each $g\in G$ the equality $\epsilon_{g}=1$ holds.

Proof.

The equivalence (i)$\Leftrightarrow$(ii) is [9, Proposition 1.1.1(3)] and the equivalence (ii)$\Leftrightarrow$(iii) follows from Remark 8. ∎

Now we introduce the following weakening of epsilon-strongly graded rings.

Definition 10**.**

The ring $S$ is said to be nearly epsilon-strongly $G$-graded (or nearly epsilon-strongly graded by $G$) if for each $g\in G$ the $S_{g}S_{g^{-1}}$–$S_{g^{-1}}S_{g}$-bimodule $S_{g}$ is s-unital.

Now we show an ”s-unital version” of Proposition 7.

Proposition 11**.**

The following assertions are equivalent:

• (i)

The ring $S$ is nearly epsilon-strongly $G$-graded;

• (ii)

The ring $S$ is symmetrically $G$-graded and for each $g\in G$ the ring $S_{g}S_{g^{-1}}$ is s-unital;

• (iii)

For each $g\in G$ and each $s\in S_{g}$ there exist $\epsilon_{g}(s)\in S_{g}S_{g^{-1}}$ and $\epsilon_{g}^{\prime}(s)\in S_{g^{-1}}S_{g}$ such that the equalities $\epsilon_{g}(s)s=s\epsilon_{g}^{\prime}(s)=s$ hold.

If (i), (ii) and (iii) are satisfied, then for all $g\in G$, the $S_{e}$-ideal $S_{g}S_{g^{-1}}$ is generated by

[TABLE]

Proof.

(i)$\Rightarrow$(ii): Suppose that $S$ is nearly epsilon-strongly $G$-graded. The symmetrical part is shown in the same way as for Proposition 7. Take $g\in G$. By Definition 10, $S_{g}$ is an s-unital left $S_{g}S_{g^{-1}}$-module and $S_{g^{-1}}$ is an s-unital right $S_{g}S_{g^{-1}}$-module. Thus, the ring $S_{g}S_{g^{-1}}$ is s-unital.

(ii)$\Rightarrow$(iii): Suppose that the ring $S$ is symmetrically $G$-graded and for each $g\in G$ the ring $S_{g}S_{g^{-1}}$ is s-unital. Take $g\in G$ and $s\in S_{g}$. Using that $S$ is symmetrically $G$-graded, there exist $n\in\mathbb{N}$, $a_{i},c_{i}\in S_{g}$ and $b_{i}\in S_{g^{-1}}$, for $i\in\{1,\ldots,n\}$, such that $s=\sum_{i=1}^{n}a_{i}b_{i}c_{i}$. Since $a_{i}b_{i}\in S_{g}S_{g^{-1}}$ and $b_{i}c_{i}\in S_{g^{-1}}S_{g}$, for $i\in\{1,\ldots,n\}$, it follows from s-unitality and Remark 4 that there exist $\epsilon_{g}(s)\in S_{g}S_{g^{-1}}$ and $\epsilon_{g}^{\prime}(s)\in S_{g^{-1}}S_{g}$ such that the equalities $\epsilon_{g}(s)a_{i}b_{i}=a_{i}b_{i}$ and $b_{i}c_{i}\epsilon_{g}^{\prime}(s)=b_{i}c_{i}$ hold, for $i\in\{1,\ldots,n\}$. Thus, $\epsilon_{g}(s)s=s\epsilon_{g}^{\prime}(s)=s$.

(iii)$\Rightarrow$(i): This is immediate. ∎

Following Cohen and Montgomery [5], and Lundström and Öinert [11], we make the following definition.

Definition 12**.**

The gradation on $S$ is said to be right non-degenerate (resp. left non-degenerate) if for each $g\in G$ and each non-zero $s\in S_{g}$, the relation $sS_{g^{-1}}\neq\{0\}$ (resp. $S_{g^{-1}}s\neq\{0\})$ holds. If the gradation on $S$ is right (resp. left) non-degenerate, then $S$ is said to be a right (resp. left) non-degenerately $G$-graded ring.

Now we define a strengthening of non-degenerate gradations.

Definition 13**.**

The gradation on $S$ is said to be strongly right non-degenerate (resp. strongly left non-degenerate) if for each $g\in G$ the left $S_{g}S_{g^{-1}}$-module (right $S_{g^{-1}}S_{g}$-module) $S_{g}$ is torsion-free. If the gradation on $S$ is strongly right (resp. left) non-degenerate, then $S$ is said to be a strongly right (resp. left) non-degenerately $G$-graded ring.

Proposition 14**.**

If $S$ is nearly epsilon-strongly $G$-graded, then $S$ is symmetrically $G$-graded, strongly right non-degenerately $G$-graded and strongly left non-degenerately $G$-graded.

Proof.

This follows immediately from (1) and Proposition 11(ii). ∎

Definition 15**.**

An involution $(\cdot)^{*}:S\to S$ is said to be anti-graded if for all $g\in G$ the equality $(S_{g})^{*}=S_{g^{-1}}$ holds.

Proposition 16**.**

If $S$ is equipped with an anti-graded involution, then the following assertions are equivalent:

• (i)

$S$* is epsilon-strongly $G$-graded;*

• (ii)

for each $g\in G$, there exists $\epsilon_{g}\in S_{g}S_{g^{-1}}$ such that $\epsilon_{g}^{*}=\epsilon_{g}$ and for all $s\in S_{g}$ the equality $\epsilon_{g}s=s$ holds;

• (iii)

for each $g\in G$, there exists $\epsilon_{g}\in S_{g}S_{g^{-1}}$ such that $\epsilon_{g}^{*}=\epsilon_{g}$ and for all $s\in S_{g}$ the equality $s\epsilon_{g^{-1}}=s$ holds.

Proof.

(i)$\Rightarrow$(ii): Suppose that $S$ is epsilon-strongly $G$-graded. Take $g\in G$ and an element $\epsilon_{g}\in S_{g}S_{g^{-1}}$ satisfying the conditions in Proposition 7(iii). Then $\epsilon_{g}^{*}\in(S_{g}S_{g^{-1}})^{*}=(S_{g^{-1}})^{*}S_{g}^{*}=S_{g}S_{g^{-1}}$ and thus, from Remark 8, it follows that $\epsilon_{g}^{*}=\epsilon_{g}\epsilon_{g}^{*}=(\epsilon_{g}^{*})^{*}\epsilon_{g}^{*}=(\epsilon_{g}\epsilon_{g}^{*})^{*}=(\epsilon_{g}^{*})^{*}=\epsilon_{g}$. Thus, (ii) holds.

(ii)$\Rightarrow$(i): Suppose that (ii) holds for some elements $\epsilon_{g}\in S_{g}S_{g^{-1}}$, for $g\in G$. Take $g\in G$ and $s\in S_{g}$. From the fact that $s^{*}\in S_{g^{-1}}$ it follows that $s\epsilon_{g^{-1}}=(\epsilon_{g^{-1}}^{*}s^{*})^{*}=(\epsilon_{g^{-1}}s^{*})^{*}=(s^{*})^{*}=s$.

The proof of the equivalence (i)$\Leftrightarrow$(iii) is analogous and is therefore left to the reader. ∎

Proposition 17**.**

Suppose that $S$ is equipped with an anti-graded involution.

• (a)

If for each $g\in G$ and all $s\in S_{g}$, there exists $\epsilon_{g}(s)\in S_{g}S_{g^{-1}}$ such that $\epsilon_{g}(s)^{*}=\epsilon_{g}(s)$ and the equality $\epsilon_{g}(s)s=s$ holds, then $S$ is nearly epsilon-strongly $G$-graded. In that case, for each $g\in G$, the $S_{e}$-ideal $S_{g}S_{g^{-1}}$ is generated by $\{\epsilon_{g}(s)\mid s\in S_{g}\}.$

• (b)

If for each $g\in G$ and all $s\in S_{g}$, there exists $\epsilon_{g}^{\prime}(s)\in S_{g^{-1}}S_{g}$ such that $\epsilon_{g}^{\prime}(s)^{*}=\epsilon_{g}^{\prime}(s)$ and the equality $s\epsilon_{g}^{\prime}(s)=s$ holds, then $S$ is nearly epsilon-strongly $G$-graded. In that case, for each $g\in G$, the $S_{e}$-ideal $S_{g}S_{g^{-1}}$ is generated by $\{\epsilon_{g^{-1}}^{\prime}(s)\mid s\in S_{g}\}.$

Proof.

Analogous to the proofs of (ii)$\Rightarrow$(i) resp. (iii)$\Rightarrow$(i) in Proposition 16. ∎

4. Leavitt path algebras

In this section, we accomodate the Leavitt path algebra framework that we need; in particular standard $G$-gradations on Leavitt path algebras, for an arbitrary group $G$. We prove $G$-graded versions of Theorem 2 and Theorem 3 (see Theorem 28 and Theorem 30). At the end of this section we establish criteria which ensure that a Leavitt path algebra is Frobenius over its identity component when the group $G$ is finite (see Theorem 38).

Let $R$ be an associative unital ring and let $E=(E^{0},E^{1},r,s)$ be a directed graph. Recall that $r$ (range) and $s$ (source) are maps $E^{1}\to E^{0}$. The elements of $E^{0}$ are called vertices and the elements of $E^{1}$ are called edges. A vertex $v$ for which $s^{-1}(v)$ is empty is called a sink. A vertex $v$ for which $r^{-1}(v)$ is empty is called a source. If $s^{-1}(v)$ is a finite set for every $v\in E^{0}$, then $E$ is called row-finite. If both $E^{0}$ and $E^{1}$ are finite sets, then we say that $E$ is finite. A path $\mu$ in $E$ is a sequence of edges $\mu=\mu_{1}\ldots\mu_{n}$ such that $r(\mu_{i})=s(\mu_{i+1})$ for $i\in\{1,\ldots,n-1\}$. In such a case, $s(\mu):=s(\mu_{1})$ is the source of $\mu$, $r(\mu):=r(\mu_{n})$ is the range of $\mu$, and $l(\mu):=n$ is the length of $\mu$. For any vertex $v\in E^{0}$ we put $s(v):=v$ and $r(v):=v$. The elements of $E^{1}$ are called real edges, while for $f\in E^{1}$ we call $f^{*}$ a ghost edge. The set $\{f^{*}\mid f\in E^{1}\}$ will be denoted by $(E^{1})^{*}$. We let $r(f^{*})$ denote $s(f)$, and we let $s(f^{*})$ denote $r(f)$. Following Hazrat [6] we make the following definition.

Definition 18**.**

The Leavitt path algebra of $E$ with coefficients in $R$, denoted by $L_{R}(E)$, is the algebra generated by the sets $\{v\mid v\in E^{0}\}$, $\{f\mid f\in E^{1}\}$ and $\{f^{*}\mid f\in E^{1}\}$ with the coefficients in $R$, subject to the relations:

1. (1)

$uv=\delta_{u,v}v$ for all $u,v\in E^{0}$; 2. (2)

$s(f)f=fr(f)=f$ and $r(f)f^{*}=f^{*}s(f)=f^{*}$, for all $f\in E^{1}$; 3. (3)

$f^{*}f^{\prime}=\delta_{f,f^{\prime}}r(f)$, for all $f,f^{\prime}\in E^{1}$; 4. (4)

$\sum_{f\in E^{1},s(f)=v}ff^{*}=v$, for every $v\in E^{0}$ for which $s^{-1}(v)$ is non-empty and finite.

Here the ring $R$ commutes with the generators.

There are many ways to assign a group gradation to $L_{R}(E)$. Indeed, let $G$ be an arbitrary group. Put $\operatorname{deg}(v)=e$, for each $v\in E^{0}$. For each $f\in E^{1}$, choose some $g\in G$ and put $\deg(f)=g$ and $\deg(f^{*})=g^{-1}$. By assigning degrees to the generators of the free algebra $F_{R}(E)=R\langle v,f,f^{*}\mid v\in E^{0},f\in E^{1}\rangle$ in this way, we are ensured that the ideal coming from relations (1)–(4) in Definition 18 is graded, i.e. homogeneous. Using this it is easy to see that the natural $G$-gradation on $F_{R}(E)$ carries over to a $G$-gradation on the quotient algebra $L_{R}(E)$. We will refer to such a gradation as a standard $G$-gradation on $L_{R}(E)$. If $\mu=\mu_{1}\ldots\mu_{n}$ is a path, then $\mu$ can be viewed as an element in $L_{R}(E)$, and we denote by $\mu^{*}$ the element $\mu_{n}^{*}\cdots\mu_{1}^{*}$ of $L_{R}(E)$. This gives rise to an anti-graded involution on $L_{R}(E)$. For $n\geq 1$ we define $E^{n}$ to be the set of paths of length $n$, and let $E^{*}=\cup_{n\geq 0}E^{n}$ be the set of all paths (and vertices). It is clear that the map $\operatorname{deg}:E^{0}\cup E^{1}\cup(E^{1})^{*}\to G$, described above, can be extended to a map $\operatorname{deg}:E^{*}\cup\{\mu^{*}\mid\mu\in E^{*}\}\to G$. One important instance of such a gradation is coming from the case $G=\mathbb{Z}$, when we put $\deg(v)=0$, for $v\in E^{0}$, $\deg(f)=1$ and $\deg(f^{*})=-1$, for $f\in E^{1}$. We will refer to this gradation as the canonical $\mathbb{Z}$-gradation on $L_{R}(E)$.

Assumption: Unless otherwise stated, throughout the rest of this section $E$ will denote an arbitrary directed graph, $R$ will denote an arbitrary associative unital ring, $G$ will denote an arbitrary group and we will assume that $L_{R}(E)$ is equipped with an arbitrary standard $G$-gradation, $S=L_{R}(E)=\oplus_{g\in G}S_{g}$.

We wish to introduce a partial order related to Leavitt path algebras. To do that we need some concepts and a well-known result which we include for the convenience of the reader.

Definition 19**.**

Let $\leq$ be a relation on a set $X$. Recall that $\leq$ is called a preorder if it is reflexive and transitive. If $\leq$ is a preorder, then it is called a partial order if it is antisymmetric.

Proposition 20**.**

Suppose that $\leq$ is a preorder on a set $X$. If we, for any $x,y\in X$, put $x\sim y$, whenever $x\leq y$ and $y\leq x$, then $\sim$ is an equivalence relation on $X$ which makes the quotient relation $\preceq$ on $X/\!\sim$, induced by $\leq$, a partial order.

Proof.

This is well-known and straightforward to check. ∎

Definition 21**.**

Let $X$ denote the set of all formal expressions of the form $\alpha\beta^{*}$ where $\alpha,\beta\in E^{*}$ and $r(\alpha)=r(\beta)$. Take $g\in G$ and put $X_{g}=\{x\in X\mid\operatorname{deg}(x)=g\}.$ Suppose that $\alpha,\beta,\gamma,\delta\in E^{*}$ satisfy $\alpha\beta^{*},\gamma\delta^{*}\in X_{g}$. We put $\alpha\beta^{*}\leq\gamma\delta^{*}$ if $\alpha$ is an initial subpath of $\gamma$. We put $\alpha\beta^{*}\sim\gamma\delta^{*}$ whenever $\alpha\beta^{*}\leq\gamma\delta^{*}$ and $\gamma\delta^{*}\leq\alpha\beta^{*}$, that is if $\alpha=\gamma$.

Remark 22**.**

(a) In the above definition of $X$, it is fully possible to let $\alpha$ or $\beta$ be a vertex.

(b) For each $g\in G$, the homogeneous component $S_{g}$ in the $G$-gradation of $L_{R}(E)$ is the $R$-linear span of $X_{g}$.

Proposition 23**.**

Let $g\in G$ be arbitrary. The following assertions hold:

• (a)

The relation $\leq$ is a preorder on $X_{g}$.

• (b)

The relation $\sim$ is an equivalence relation on $X_{g}$.

• (c)

The quotient relation $\preceq$ on $X_{g}/\!\sim$ induced by $\leq$ is a partial order.

Proof.

(a) Take $\alpha_{i},\beta_{i}\in E^{*}$, such that $\alpha_{i}\beta_{i}^{*}\in X_{g}$, for $i\in\{1,2,3\}$. It is clear that $\alpha_{1}\beta_{1}^{*}\leq\alpha_{1}\beta_{1}^{*}$, since $\alpha_{1}$ is an initial subpath of itself. Thus, $\leq$ is reflexive. Now we show that $\leq$ is transitive. Suppose that $\alpha_{1}\beta_{1}^{*}\leq\alpha_{2}\beta_{2}^{*}$ and $\alpha_{2}\beta_{2}^{*}\leq\alpha_{3}\beta_{3}^{*}$. Then $\alpha_{1}$ must be an initial subpath of $\alpha_{3}$. Thus, $\alpha_{1}\beta_{1}^{*}\leq\alpha_{3}\beta_{3}^{*}$.

(b) This is clear.

(c) This follows from Proposition 20. ∎

Proposition 24**.**

Let $g\in G$ be arbitrary. The map $\mathcal{N}_{g}:X_{g}/\!\sim\,\ni[x]\mapsto\mathcal{N}_{g}(x):=xx^{*}\in L_{R}(E)_{e}$ is well-defined.

Proof.

Suppose that $[x]=[y]$. Then $x\sim y$ which implies that $x\leq y$ and $y\leq x$. Then $x=\alpha\beta^{*}$ and $y=\alpha\gamma^{*}$ for some $\alpha,\beta,\gamma\in E^{*}$. Thus, $\mathcal{N}_{g}(x)=xx^{*}=\alpha\beta^{*}\beta\alpha^{*}=\alpha\alpha^{*}=\alpha\gamma^{*}\gamma\alpha^{*}=yy^{*}=\mathcal{N}_{g}(y)$. ∎

Lemma 25**.**

If $\alpha,\beta\in E^{*}$ are chosen so that $\alpha^{*}\beta\neq 0$ in $L_{R}(E)$, then $\alpha$ is an initial subpath of $\beta$ or vice versa.

Proof.

Put $m=l(\alpha)$ and $n=l(\beta)$. Suppose that $\alpha=f_{1}\ldots f_{m}$ and $\beta=f_{1}^{\prime}\ldots f_{n}^{\prime}$.

Case 1: $m\leq n$. Then

[TABLE]

This implies that $f_{i}=f_{i}^{\prime}$ for $i\in\{1,\ldots,m\}$. Thus, $\alpha$ is an initial subpath of $\beta$.

Case 2: $m>n$. Then

[TABLE]

Hence, $f_{i}=f_{i}^{\prime}$ for $i\in\{1,\ldots,n\}$. Thus, $\beta$ is an initial subpath of $\alpha$. ∎

Proposition 26**.**

Let $g\in G$ and $x,y\in X_{g}$ be arbitrary. The following assertions hold:

• (a)

If $[x]\preceq[y]$, then $\mathcal{N}_{g}(x)y=y$.

• (b)

If $[x]\npreceq[y]$ and $[y]\npreceq[x]$, then $\mathcal{N}_{g}(x)y=0$.

Proof.

Put $x=\alpha\beta^{*}$ and $y=\gamma\delta^{*}$.

(a) Suppose that $[x]\preceq[y]$. Then $\alpha$ is an initial subpath of $\gamma$. Hence, $\gamma=\alpha\alpha^{\prime}$ for some $\alpha^{\prime}\in E^{*}$ such that $r(\alpha)=s(\alpha^{\prime})$. Therefore, $\mathcal{N}_{g}(x)y=\alpha\beta^{*}\beta\alpha^{*}\gamma\delta^{*}=\alpha\alpha^{*}\alpha\alpha^{\prime}\delta^{*}=\alpha\alpha^{\prime}\delta^{*}=\gamma\delta^{*}=y$.

(b) From Lemma 25, we get that $\mathcal{N}_{g}(x)y=\alpha\beta^{*}\beta\alpha^{*}\gamma\delta^{*}=\alpha\alpha^{*}\gamma\delta^{*}=\alpha 0\delta^{*}=0$. ∎

Definition 27**.**

Suppose that $E$ is a finite directed graph and let $g\in G$. Since $E$ is finite, we can, on account of Proposition 23, choose $n_{g}\in\mathbb{N}$ and $m_{1},\ldots,m_{n_{g}}\in X_{g}$ such that $\{[m_{1}],\ldots,[m_{n_{g}}]\}$ equals the set of minimal elements of $X_{g}/\!\sim$, with respect to $\preceq$. We define

[TABLE]

in $L_{R}(E)$. Notice that we will always get $\epsilon_{e}=\sum_{v\in E^{0}}v$.

The following result allows us establish Theorem 2, in particular.

Theorem 28**.**

If $E$ is a finite directed graph and $R$ is an associative unital ring, then $L_{R}(E)$ is epsilon-strongly $G$-graded.

Proof.

Let $g\in G$ be arbitrary and put $\epsilon_{g}=\sum_{i=1}^{n_{g}}\mathcal{N}_{g}(m_{i})$ in $L_{R}(E)$. Clearly, $\epsilon_{g}\in S_{g}S_{g^{-1}}$. Take $\gamma\delta^{*}\in X_{g}$. There is a unique $j\in\{1,\ldots,n_{g}\}$ such that $[m_{j}]\preceq[\gamma\delta^{*}]$. From Proposition 26 it follows that $\mathcal{N}_{g}(m_{j})\gamma\delta^{*}=\gamma\delta^{*}$ and if $i\neq j$, then $\mathcal{N}_{g}(m_{i})\gamma\delta^{*}=0$. Thus, $\epsilon_{g}\gamma\delta^{*}=\gamma\delta^{*}$. The claim now follows from Proposition 16 and Remark 22(b). ∎

Remark 29**.**

Consider the directed graph $B$ from the introduction and suppose that $L_{R}(B)$ is equipped with a standard $G$-gradation. From Theorem 28 it follows that $L_{R}(B)$ is epsilon-strongly $G$-graded.

The following result allows us establish Theorem 3, in particular.

Theorem 30**.**

If $E$ is a directed graph and $R$ is an associative unital ring, then $L_{R}(E)$ is nearly epsilon-strongly $G$-graded.

Proof.

Take $g\in G$ and $s\in S_{g}$. There is a finite subset $I$ of $\mathbb{N}$ such that $s=\sum_{i\in I}r_{i}\alpha_{i}\beta_{i}^{*}$ for suitable $r_{i}\in R$ and $\alpha_{i}\beta_{i}^{*}\in X_{g}$. Take a subset $J$ of $I$ such that $\{[\alpha_{j}\beta_{j}^{*}]\}_{j\in J}$ equals the set of minimal elements in $\{[\alpha_{i}\beta_{i}^{*}]\}_{i\in I}$. Put $\epsilon_{g}(s)=\sum_{j\in J}\mathcal{N}_{g}(\alpha_{j}\beta_{j}^{*})\in S_{g}S_{g^{-1}}$. For each $i\in I$, there is a unique $j(i)\in J$ such that $[\alpha_{j(i)}\beta_{j(i)}^{*}]\preceq[\alpha_{i}\beta_{i}^{*}]$. From Proposition 26 it follows that $\mathcal{N}_{g}(\alpha_{j(i)}\beta_{j(i)}^{*})\alpha_{i}\beta_{i}^{*}=\alpha_{i}\beta_{i}^{*}$ and if $j\neq j(i)$, then $\mathcal{N}_{g}(\alpha_{j}\beta_{j}^{*})\alpha_{i}\beta_{i}^{*}=0$. Thus, $\epsilon_{g}(s)s=\sum_{i\in I}\sum_{j\in J}r_{i}\mathcal{N}_{g}(\alpha_{j}\beta_{j}^{*})\alpha_{i}\beta_{i}^{*}=\sum_{i\in I}r_{i}\mathcal{N}_{g}(\alpha_{j(i)}\beta_{j(i)}^{*})\alpha_{i}\beta_{i}^{*}=\sum_{i\in I}r_{i}\alpha_{i}\beta_{i}^{*}=s$. The claim now follows from Proposition 17. ∎

Example 31**.**

Consider the following finite directed graph $E$:

[TABLE]

Equip $S=L_{R}(E)$ with its canonical $\mathbb{Z}$-gradation, and view it as a $\mathbb{Z}$-graded ring. Notice that $v_{1}$ and $v_{3}$ are sinks. Hence, by Theorem 1, $S$ is not strongly $\mathbb{Z}$-graded. However, $S$ is epsilon-strongly $\mathbb{Z}$-graded, by Theorem 28. We shall now, for each $n\in\mathbb{Z}$, explicitly describe the homogeneous component $S_{n}$ and the element $\epsilon_{n}$. A quick calculation yields the following:

[TABLE]

Following the proof of Theorem 28 we shall now choose:

• •

$\epsilon_{0}=v_{1}+v_{2}+v_{3}+v_{4}+v_{5}\in S_{0}$;

• •

$\epsilon_{1}=v_{2}+v_{4}+v_{5}=(f_{2}f_{2}^{*}+f_{1}f_{1}^{*})+f_{3}f_{3}^{*}+f_{4}f_{4}^{*}\in S_{1}S_{-1}$;

• •

$\epsilon_{-1}=f_{2}f_{2}^{*}+v_{1}+v_{3}+v_{4}=f_{2}f_{3}^{*}f_{4}^{*}f_{4}f_{3}f_{2}^{*}+f_{1}^{*}f_{1}+f_{3}^{*}f_{3}+f_{4}^{*}f_{4}\in S_{-1}S_{1}$;

• •

$\epsilon_{2}=v_{5}=f_{4}f_{3}f_{3}^{*}f_{4}^{*}\in S_{2}S_{-2}$;

• •

$\epsilon_{-2}=v_{3}=f_{3}^{*}f_{4}^{*}f_{4}f_{3}\in S_{-2}S_{2}$;

• •

$\epsilon_{n}=0$, if $|n|>2$.

It is not difficult to see that, for any $n\in\mathbb{Z}$, $\epsilon_{n}$ satisfies the conditions in Proposition 7(iii).

The next example shows that Theorem 28 can not be generalized to arbitrary (possibly non-finite) directed graphs $E$.

Example 32**.**

Consider the graph from the introduction C:\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 7.75769pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\\}}}\ignorespaces{\hbox{\kern-7.75769pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet_{v_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 11.2299pt\raise 6.5pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{\scriptstyle{(\infty)}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 31.75769pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 31.75769pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{\bullet_{v_{2}}.}}}}}}}}\ignorespaces}}}}\ignorespaces For this graph $C^{1}=\{f_{1},f_{2},f_{3},\ldots\}$ is infinite. Is $L_{R}(C)$ epsilon-strongly $\mathbb{Z}$-graded? We begin by writing out some of the homogeneous components of the canonical $\mathbb{Z}$-gradation on $S=L_{R}(C)$.

• •

$S_{0}=\operatorname{span}_{R}\{v_{1},v_{2},f_{1}f_{1}^{*},f_{1}f_{2}^{*},\ldots,f_{i}f_{j}^{*},\ldots\}$

• •

$S_{1}=\operatorname{span}_{R}\{f_{1},f_{2},f_{3},\ldots\}$

• •

$S_{-1}=\operatorname{span}_{R}\{f_{1}^{*},f_{2}^{*},f_{3}^{*},\ldots\}$

If $S_{1}$ had only contained a finite number of elements, then we could have put $\epsilon_{1}=\sum_{f\in C^{1}}ff^{*}$. However, in the current situation that is not possible. The only option left for us is to put $\epsilon_{1}=v_{1}$. We need to check whether the conditions in Definition 6 are satisfied, and in particular whether $v_{1}\in S_{1}S_{-1}$. Seeking a contradiction, suppose that $v_{1}=\sum_{i\in F}r_{i}f_{p_{i}}f_{q_{i}}^{*}$, where $p_{i},q_{i}\in\mathbb{N}^{+}$, for some finite set $F$. For any $k\in\mathbb{N}^{+}$ we get

[TABLE]

which in particular yields that there is some $i\in F$ such that $k=p_{i}=q_{i}$. Thus, $F$ is not finite, which is a contradiction. This shows that $\epsilon_{1}\notin S_{1}S_{-1}$ and hence $S=L_{R}(C)$ is not epsilon-strongly $\mathbb{Z}$-graded.

Remark 33**.**

By Example 31 and Example 32 we have established the following chain of strict inclusions:

[TABLE]

Proposition 34**.**

If $E$ is a directed graph and $R$ is an associative unital ring, then the Leavitt path algebra $L_{R}(E)$ is symmetrically $G$-graded, strongly right non-degenerately $G$-graded and strongly left non-degenerately $G$-graded.

Proof.

This follows immediately from Proposition 14 and Theorem 30. ∎

Remark 35**.**

The ”symmetric” part of Proposition 34 can be shown directly without the use of Theorem 30. Indeed, take $g\in G$. If $S_{g}=\{0\}$, then clearly $S_{g}S_{g^{-1}}S_{g}=S_{g}$ holds. Now, suppose that $S_{g}\neq\{0\}$. It follows immediately from the $G$-gradation that $S_{g}S_{g^{-1}}S_{g}\subseteq S_{g}$ holds. To show the reversed inclusion, take an arbitrary non-zero monomial $\alpha\beta^{*}\in S_{g}$. Then $r(\alpha)=r(\beta)$ and $\beta\alpha^{*}\in S_{g^{-1}}$. We get $\alpha\beta^{*}=\alpha r(\beta)r(\alpha)\beta^{*}=\alpha(\beta^{*}\beta)(\alpha^{*}\alpha)\beta^{*}=(\alpha\beta^{*})(\beta\alpha^{*})(\alpha\beta^{*})\in S_{g}S_{g^{-1}}S_{g}.$ This shows that $S_{g}\subseteq S_{g}S_{g^{-1}}S_{g}$.

We end this section with an application to Frobenius extensions. To this end, recall the following.

Definition 36**.**

Suppose that $S/T$ is a ring extension. By this we mean that $T\subseteq S$ and both $S$ and $T$ are unital with a common multiplicative identity. The ring extension $S/T$ is called Frobenius if there is a finite set $J$, $x_{j},y_{j}\in S$, for $j\in J$, and a $T$-bimodule map $E:S\rightarrow T$ such that, for every $s\in S$, the equalities $s=\sum_{j\in J}x_{j}E(y_{j}s)=\sum_{j\in J}E(sx_{j})y_{j}$ hold. In that case, $(E,x_{j},y_{j})$ is called a Frobenius system.

In [10, Theorem 25] Nystedt, Öinert and Pinedo have shown the following result.

Theorem 37**.**

Suppose that $S$ is a ring which is epsilon-strongly graded by a finite group $G$. If we put $T=S_{e}$, then $S/T$ is a Frobenius extension.

As a direct consequence of the above result, in combination with Theorem 28, we get the following result.

Theorem 38**.**

Let $E$ be a finite directed graph and $R$ an associative unital ring. If $G$ is a finite group and we equip the Leavitt path algebra $L_{R}(E)$ with a standard $G$-gradation, then the ring extension $L_{R}(E)/L_{R}(E)_{e}$ is Frobenius.

5. Strongly $\mathbb{Z}$-graded rings

Throughout this section $S$ denotes a, not necessarily unital, $\mathbb{Z}$-graded ring.

Proposition 39**.**

The ring $S$ is strongly $\mathbb{Z}$-graded if and only if for every $n\in\mathbb{Z}$ the $S_{0}$-bimodule $S_{n}$ is unital, and the equalities $S_{1}S_{-1}=S_{-1}S_{1}=S_{0}$ hold.

Proof.

The ”only if” statement is immediate. Now we show the ”if” statement. Take positive integers $m$ and $n$. First we show by induction that $S_{m}=(S_{1})^{m}$. The base case $m=1$ is clear. Next suppose that $S_{m}=(S_{1})^{m}$. Then we get that $S_{m+1}=S_{0}S_{m+1}=S_{1}S_{-1}S_{m+1}\subseteq S_{1}S_{-1+m+1}=S_{1}S_{m}=S_{1}(S_{1})^{m}=(S_{1})^{m+1}\subseteq S_{m+1}$. Next we show by induction that $S_{-n}=(S_{-1})^{n}$. The base case $n=1$ is clear. Next suppose that $S_{-n}=(S_{-1})^{n}$. Then we get that $S_{-n-1}=S_{-n-1}S_{0}=S_{-n-1}S_{1}S_{-1}\subseteq S_{-n-1+1}S_{-1}=S_{-n}S_{-1}=(S_{-1})^{n}S_{-1}=(S_{-1})^{n+1}\subseteq S_{-n-1}$.

Case 1: $S_{m}S_{n}=(S_{1})^{m}(S_{1})^{n}=(S_{1})^{m+n}=S_{m+n}$.

Case 2: $S_{-m}S_{-n}=(S_{-1})^{m}(S_{-1})^{n}=(S_{-1})^{m+n}=S_{-m-n}$.

Case 3: Now we show that $S_{m}S_{-n}=S_{m-n}$. We get that $S_{m}S_{-n}=(S_{1})^{m}(S_{-1})^{n}$. By repeated application of the equality $S_{1}S_{-1}=S_{0}$, we get that $(S_{1})^{m}(S_{-1})^{n}=(S_{1})^{m-n}=S_{m-n}$, if $m\geq n$, or $(S_{1})^{m}(S_{-1})^{n}=(S_{-1})^{n-m}=S_{n-m}$, otherwise.

Case 4: $S_{-m}S_{n}=S_{n-m}$. This is shown in a similar fashion to Case 3, using the equality $S_{-1}S_{1}=S_{0}$, and is therefore left to the reader. ∎

Corollary 40**.**

If for every $n\in\mathbb{Z}$, the $S_{0}$-bimodule $S_{n}$ is s-unital, then $S$ is strongly $\mathbb{Z}$-graded if and only if the equalities $S_{1}S_{-1}=S_{-1}S_{1}=S_{0}$ hold.

Corollary 41**.**

If $S$ is symmetrically $\mathbb{Z}$-graded, then $S$ is strongly $\mathbb{Z}$-graded if and only if the equalities $S_{1}S_{-1}=S_{-1}S_{1}=S_{0}$ hold.

Corollary 42**.**

If $S$ is epsilon-strongly $\mathbb{Z}$-graded, then $S$ is strongly $\mathbb{Z}$-graded if and only if $\epsilon_{1}=\epsilon_{-1}=1$.

6. Strongly $\mathbb{Z}$-graded Leavitt path algebras

Throughout this section, $E$ denotes a directed graph, $R$ denotes an associative unital ring and we equip the associated Leavitt path algebra $S=L_{R}(E)$ with its canonical $\mathbb{Z}$-gradation. Recall that if $E$ is finite, then $L_{R}(E)$ is epsilon-strongly $\mathbb{Z}$-graded by Theorem 28. In that case, we will use the notation established in Section 4 without further mention.

Lemma 43**.**

If $E$ is a finite directed graph, then $\epsilon_{1}=\sum_{v\in E^{0}\setminus\{{\rm sinks}\}}v$.

Proof.

From the definition of the relation $\preceq$ it follows that $\{[\alpha]\mid\alpha\in E^{1}\}$ is a set of minimal elements of $X_{1}/\!\sim$. Thus

[TABLE]

∎

Lemma 44**.**

If $E$ is a finite directed graph which has no sink, then for every $v\in E^{0}$ there exist $n(v)\in\mathbb{N}$ and $m_{1,v},\ldots,m_{n(v),v}\in X_{-1}$ such that no pair of elements from $[m_{1,v}],\ldots,[m_{n(v),v}]$ has a common lesser element in $X_{-1}/\!\sim$, with respect to $\preceq$, and $v=\sum_{i=1}^{n(v)}\mathcal{N}_{-1}(m_{i,v})\in S_{-1}S_{1}$.

Proof.

Take $v\in E^{0}$. Since $E$ is finite and has no sink, it is clear that we can make repeated use of property (4) in Definition 18 to find $n(v)\in\mathbb{N}$ and $\alpha_{i,v}\in E^{*}$, for $i\in\{1,\ldots,n(v)\}$, such that $r(\alpha_{i,v})$ is a vertex belonging to a cycle $C_{i}$ in $E$, and $v=\sum_{i=1}^{n(v)}\alpha_{i,v}\alpha_{i,v}^{*}$. For each $i\in\{1,\ldots,n(v)\}$, choose a path $\beta_{i}$ in $C_{i}$ such that $r(\beta_{i})=r(\alpha_{i,v})$ and $\alpha_{i,v}\beta_{i}^{*}\in X_{-1}$. Then $\beta_{i}\alpha_{i,v}^{*}\in X_{1}$. For each $i\in\{1,\ldots,n(v)\}$, put $m_{i}=\alpha_{i,v}\beta_{i}^{*}$. Then

[TABLE]

Notice that the paths $\alpha_{1,v},\ldots,\alpha_{i,n(v)}$ are all distinct. Thus, no pair of elements from $[m_{1,v}],\ldots,[m_{n(v),v}]$ has a common lesser element in $X_{-1}/\!\sim$. ∎

We now prove Theorem 1 using an approach different from Hazrat’s [6].

Proposition 45**.**

If $E$ is a finite directed graph, then $L_{R}(E)$ is strongly $\mathbb{Z}$-graded if and only if $E$ has no sink.

Proof.

The ”only if” statement follows from Lemma 43, and also from Lemma 46. Now we show the ”if” statement. We claim that for all $v\in E^{0}$ the equality $\epsilon_{-1}v=v$ holds. If we assume that the claim holds, then we get that $\epsilon_{-1}=\epsilon_{-1}1=\sum_{v\in E^{0}}\epsilon_{-1}v=\sum_{v\in E^{0}}v=1$. From Lemma 43 we get that $\epsilon_{1}=1$. Thus, by Corollary 42, we get that $L_{R}(E)$ is strongly $\mathbb{Z}$-graded.

Now we show the claim. Take $v\in E^{0}$. By Lemma 44 we may write $v=\sum_{i=1}^{n(v)}\mathcal{N}_{-1}(m_{i,v})$ for suitable $n(v)\in\mathbb{N}$ and $m_{1,v},\ldots,m_{n(v),v}\in X_{-1}\subseteq S_{-1}$. Thus, $\epsilon_{-1}v=\epsilon_{-1}\sum_{i=1}^{n(v)}\mathcal{N}_{-1}(m_{i,v})=\sum_{i=1}^{n(v)}\epsilon_{-1}m_{i,v}m_{i,v}^{*}=\sum_{i=1}^{n(v)}m_{i,v}m_{i,v}^{*}=v$. ∎

Lemma 46**.**

Let $E$ be an arbitrary directed graph. The following assertions hold:

• (a)

If $L_{R}(E)$ is strongly $\mathbb{Z}$-graded, then $E$ has no sink.

• (b)

If $E$ is row-finite and has no sink, then $S_{1}S_{-1}=S_{0}$.

• (c)

If $E$ has no source, then $S_{-1}S_{1}=S_{0}$.

Proof.

(a) Suppose that $S=L_{R}(E)$ is strongly $\mathbb{Z}$-graded. Seeking a contradiction, suppose that there is a sink $v$ in $E$. Then $v\in S_{0}=S_{1}S_{-1}$. Using that $v$ is a sink, we get that $v=v^{2}\in vS_{1}S_{-1}=\{0\}$. This is a contradiction.

(b) It suffices to show that $E^{0}\subseteq S_{1}S_{-1}$. Take $v\in E^{0}$. Then $v=\sum_{s(f)=v}ff^{*}\in S_{1}S_{-1}$.

(c) It suffices to show that $E^{0}\subseteq S_{-1}S_{1}$. Take $v\in E^{0}$. Choose some $f\in E^{1}$ such that $v=r(f)=f^{*}f\in S_{-1}S_{1}$. ∎

The following results generalizes [7, Theorem 1.6.15].

Theorem 47**.**

Let $E$ be a row-finite directed graph which has no source. Then $L_{R}(E)$ is strongly $\mathbb{Z}$-graded if and only if $E$ has no sink.

Proof.

The ”only if” statement follows from Lemma 46(a). The ”if” statement follows from Lemma 46(b)-(c), Proposition 34 and Corollary 41. ∎