Subsets of vertices give Morita equivalences of Leavitt path algebras

TL;DR

This paper demonstrates that subsets of vertices in a directed graph induce Morita equivalences between subalgebras and ideals of associated Leavitt path algebras, and shows how graph contractions preserve Morita equivalence.

Contribution

It introduces a method to obtain Morita equivalences via vertex subsets and extends graph contraction techniques to preserve algebraic properties.

Findings

Subsets of vertices induce Morita equivalences in Leavitt path algebras.

Graph contractions can produce Morita equivalent Leavitt path algebras.

Examples include desingularisation and delaying of graphs.

Abstract

We show that every subset of vertices of a directed graph E gives a Morita equivalence between a subalgebra and an ideal of the associated Leavitt path algebra. We use this observation to prove an algebraic version of a theorem of Crisp and Gow: certain subgraphs of E can be contracted to a new graph G such that the Leavitt path algebras of E and G are Morita equivalent. We provide examples to illustrate how desingularising a graph, and in- or out-delaying of a graph, all fit into this setting.

Equations

$$p{C}^{\ast }(E)=\overline{\mathrm{span}}\{s_{\mu }{s}_{{\nu }^{\ast }}:\text{μ,ν are paths in E and μ has range in V}\}$$

$$M={\mathrm{span}}_R\{s_{\mu }{s}_{{\nu }^{\ast }}:\text{μ,ν are paths in E and μ has range in V}\},$$

$$L_R(E)={\mathrm{span}}_R\{s_{\mu }{s}_{\nu }^{\ast }:\mu ,\nu \in E^{\ast }\}.$$

$$M:={\mathrm{span}}_R\{s_{\mu }{s}_{{\nu }^{\ast }}:\mu ,\nu \in E^{\ast },r(\mu )\in V\}\text{ and }{M}^{\ast }:={\mathrm{span}}_R\{s_{\mu }{s}_{{\nu }^{\ast }}:\mu ,\nu \in E^{\ast },r(\nu )\in V\}.$$

$$\Psi :M{\otimes }_{M^{\ast }M}{M}^{\ast }\to M{M}^{\ast }\text{and}\Phi :M^{\ast }{\otimes }_{M{M}^{\ast }}M\to M^{\ast }M$$

$$M{M}^{\ast }={\mathrm{span}}_R\{p_v{s}_{\mu }{s}_{{\nu }^{\ast }}{s}_{\alpha }{s}_{\beta }^{\ast }{p}_w:v,w\in V,\alpha ,\beta ,\mu ,\nu \in E^{\ast }\}.$$

$$M{M}^{\ast }={\mathrm{span}}_R\{p_v{s}_{\mu }{s}_{{\nu }^{\ast }}{p}_w:v,w\in V,\mu ,\nu \in E^{\ast }\}=\mathrm{span}\{s_{\mu }{s}_{{\nu }^{\ast }}:\mu ,\nu \in E^{\ast },r(\mu ),r(\nu )\in V\}.$$

$$m\Phi (n\otimes m^{\prime })=mn{m}^{\prime }=\Psi (m\otimes n)m^{\prime }\text{and}n\Psi (m\otimes n^{\prime })=nm{n}^{\prime }=\Phi (n\otimes m)n^{\prime }.$$

$$v\in E^0,0<|r^{-1}(v)|<\infty \text{ and }s(r^{-1}(v))\subset H⟹v\in H.$$

$$p_v=\sum _{(\alpha ,\beta )\in F_1,(\mu ,\nu )\in F_2}{r}_{\alpha ,\beta ,\mu ,\nu }{s}_{\alpha }{s}_{{\beta }^{\ast }}{s}_{\mu }{s}_{{\nu }^{\ast }}$$

$$s_{\alpha }{s}_{{\beta }^{\ast }}{s}_{\mu }{s}_{{\nu }^{\ast }}=s_{\alpha }{p}_{s(\alpha )}{s}_{{\beta }^{\ast }}{s}_{\mu }{p}_{s(\mu )}{s}_{{\nu }^{\ast }}\in I_{\Sigma H(V)}.$$

$$T^0:=E^0\setminus G^0\text{and}{T}^1:=\{e\in E^1:s(e),r(e)\in T^0\}$$

$$B_v=\{\beta \in E^{\ast }\setminus E^0:r(\beta )=v,s(\beta )\in G^0\text{ and }s({\beta }_i)\in T^0\text{ for }1\le i<|\beta |\}.$$

$$M={\mathrm{span}}_R\{s_{\mu }{s}_{{\nu }^{\ast }}:r(\mu )\in G^0\}.$$

$$0=\varphi (t_{{\delta }^{\ast }}x{t}_{\gamma })=\varphi (r{q}_v)=r\varphi (q_v)=r{p}_{v^{\prime }}.$$

$$0=\phi(t_{\mu^{*}}xt_{\nu})=\phi\Big{(}\sum_{i=m}^{n}r_{i}t_{\alpha}^{i}\Big{)}=\sum_{i=m}^{n}r_{i}\phi(t_{\alpha})^{i}=\sum_{i=m}^{n}r_{i}s_{\alpha^{\prime}}^{i}.$$

$$P_{s(\mu )}=\sum _{\beta \in B_{s(\mu )}}{S}_{\beta }{S}_{{\beta }^{\ast }};$$

$$s_{\mu }{s}_{{\nu }^{\ast }}=s_{\mu }{p}_{s(\mu )}{s}_{{\nu }^{\ast }}=\sum _{\beta \in B_{s(\mu )}}{s}_{\mu }{s}_{\beta }{s}_{{\beta }^{\ast }}{s}_{{\nu }^{\ast }}=\sum _{\beta \in B_{s(\mu )}}{s}_{\mu \beta }{s}_{s(\beta {)}^{\ast }}{s}_{s(\beta )}{s}_{(\nu \beta {)}^{\ast }}\in M^{\ast }M.$$

$$s_{\mu }{s}_{{\nu }^{\ast }}=s_{\mu }{p}_{s(\mu )}{s}_{{\nu }^{\ast }}=s_{\mu }{s}_{{\alpha }^{\ast }}{s}_{\alpha }{s}_{{\nu }^{\ast }}\in M^{\ast }M.$$

$$Q_v=p_v,T_{e_{\beta }}=s_{\beta }\text{ and }{T}_{e_{\beta }^{\ast }}=s_{{\beta }^{\ast }}.$$

$$Q_v=p_v=\sum _{\beta \in B_v}{s}_{\beta }{s}_{{\beta }^{\ast }}=\sum _{e_{\beta }\in r_G^{-1}(v)}{T}_{e_{\beta }}{T}_{e_{\beta }^{\ast }}.$$

$$\varphi (q_v)=Q_v=p_v,\varphi (t_{e_{\beta }})=T_{e_{\beta }}=s_{\beta }\text{and}\varphi (t_{e_{\beta }^{\ast }})=T_{e_{\beta }^{\ast }}=s_{{\beta }^{\ast }}.$$

Full text

This research has been supported by a University of Otago Research Grant.

1. Introduction

Given a directed graph $E$, Crisp and Gow identified in [10, Theorem 3.1] a type of subgraph which can be “contracted” to give a new graph $G$ whose $C^{*}$-algebra $C^{*}(G)$ is Morita equivalent to $C^{*}(E)$. Crisp and Gow’s construction is widely applicable, as they point out in [10, §4]. It includes, for example, Morita equivalences of the $C^{*}$-algebras of graphs that are elementary-strong-shift-equivalent [4, 11], or are in- or out-delays of each other [5].

The $C^{*}$-algebra of a directed graph $E$ is the universal $C^{*}$-algebra generated by mutually orthogonal projections $p_{v}$ and partial isometries $s_{e}$ associated to the vertices $v$ and edges $e$ of $E$, respectively, subject to relations. In particular, the relations capture the connectivity of the graph. For any subset $V$ of vertices, $\sum_{v\in V}p_{v}$ converges to a projection $p$ in the multiplier algebra of $C^{*}(E)$. (If $V$ is finite, then $p$ is in $C^{*}(E)$.) Then the module $pC^{*}(E)$ implements a Morita equivalence between the corner $pC^{*}(E)p$ of $C^{*}(E)$ and the ideal $C^{*}(E)pC^{*}(E)$ of $C^{*}(E)$. The difficult part is to identify $pC^{*}(E)p$ and $C^{*}(E)pC^{*}(E)$ with known algebras. The corner $pC^{*}(E)p$ may not be another graph algebra, but sometimes it is (see, for example, [9]). The projection $p$ is called full when $C^{*}(E)pC^{*}(E)=C^{*}(E)$.

Now let $R$ be a commutative ring with identity. A purely algebraic analogue of the graph $C^{*}$-algebra is the Leavitt path algebra $L_{R}(E)$ over $R$. This paper is based on the very simple observation that every subset $V$ of the vertices of a directed graph $E$ gives an algebraic version of the Morita equivalence between $pC^{*}(E)p$ and $C^{*}(E)pC^{*}(E)$ for Leavitt path algebras (see Theorem 1). We show that this observation is widely applicable by proving an algebraic version of Crisp and Gow’s theorem (see Theorem 3). A special case of this result has been very successfully used in both [2, Section 3] and [13].

If $V$ is infinite, we cannot make sense of the projection $p$ in $L_{R}(E)$, but we can make sense of the algebraic analogues of the sets $pC^{*}(E)$, $pC^{*}(E)p$ and $C^{*}(E)pC^{*}(E)$. For example,

[TABLE]

has analogue

[TABLE]

where we also use $s_{e}$ and $p_{v}$ for universal generators of $L_{R}(E)$. Theorem 1 below gives a surjective Morita context $(M,M^{*},MM^{*},M^{*}M)$ between the $R$-subalgebra $MM^{*}$ and the ideal $M^{*}M$ of $L_{R}(E)$. The set $V$ is full, in the sense that $M^{*}M=L_{R}(E)$, if and only if the saturated hereditary closure of $V$ is the whole vertex set of $E$ (see Lemma 2).

Recently, the first author and Sims proved in [7, Theorem 5.1] that equivalent groupoids have Morita equivalent Steinberg $R$-algebras. They then proved that the graph groupoids of the graphs $G$ and $E$ appearing in Crisp and Gow’s theorem are equivalent groupoids [7, Proposition 6.2]. Since the Steinberg algebra of a graph groupoid is canonically isomorphic to the Leavitt path algebra of the graph, they deduced that the Leavitt path algebras of $L_{R}(G)$ and $L_{R}(E)$ are Morita equivalent.

In particular, we obtain a direct proof of [7, Proposition 6.2] using only elementary methods. There are two advantages to our elementary approach: it illustrates on the one hand where we have had to use different techniques from the $C^{*}$-algebraic analogue, and on the other hand where we can just use the $C^{*}$-algebraic results already established.

2. Preliminaries

A directed graph $E=(E^{0},E^{1},r,s)$ consists of countable sets $E^{0}$ and $E^{1}$, and range and source maps $r,s:E^{1}\to E^{0}$. We think of $E^{0}$ as the set of vertices, and of $E^{1}$ as the set of edges directed by $r$ and $s$. A vertex $v$ is called a infinite receiver if $|r^{-1}(v)|=\infty$ and is called a source if $|r^{-1}(v)|=0$. Sources and infinite receivers are called singular vertices.

We use the convention that a path is a sequence of edges $\mu=\mu_{1}\mu_{2}\cdots$ such that $s(\mu_{i})=r(\mu_{i+1})$. We denote the $i$th edge in a path $\mu$ by $\mu_{i}$. We say a path $\mu$ is finite if the sequence is finite and denote its length by $|\mu|$. Vertices are regarded as paths of length [math]. We denote the set of finite paths by $E^{*}$ and the set of infinite paths by $E^{\infty}$. We usually use the letters $x,y$ for infinite paths. We extend the range map $r$ to $\mu\in E^{*}\cup E^{\infty}$ by $r(\mu)=r(\mu_{1})$; for $\mu\in E^{*}$ we also extend the source map $s$ by $s(\mu)=s(\mu_{|\mu|})$.

Let $(E^{1})^{*}:=\{e^{*}:e\in E^{1}\}$ be a set of formal symbols called ghost edges. If $\mu\in E^{*}$, then we write $\mu^{*}$ for $\mu_{|\mu|}^{*}\dots\mu_{2}^{*}\mu_{1}^{*}$ and call it a ghost path. We extend $r$ and $s$ to the ghost paths by $r(\mu^{*})=s(\mu)$ and $s(\mu^{*})=r(\mu)$.

Let $R$ be a commutative ring with identity and let $A$ be an $R$-algebra. A Leavitt $E$-family in $A$ is a set $\{P_{v},S_{e},S_{e^{*}}:v\in E^{0},e\in E^{1}\}\subset A$ where $\{P_{v}:v\in E^{0}\}$ is a set of mutually orthogonal idempotents, and

1. (L1)

$P_{r(e)}S_{e}=S_{e}=S_{e}P_{s(e)}$ and $P_{s(e)}S_{e^{*}}=S_{e^{*}}=S_{e^{*}}P_{r(e)}$ for $e\in E^{1}$; 2. (L2)

$S_{e^{*}}S_{f}=\delta_{e,f}P_{s(e)}$ for $e,f\in E^{1}$; and 3. (L3)

for all non-singular $v\in E^{0}$, $P_{v}=\sum_{r(e)=v}S_{e}S_{e^{*}}$.

For a path $\mu\in E^{*}$ we set $S_{\mu}:=S_{\mu_{1}}\dots S_{\mu_{|\mu|}}$. The Leavitt path algebra $L_{R}(E)$ is the universal $R$-algebra generated by a universal Leavitt $E$-family $\{p_{v},s_{e},s_{e^{*}}\}$: that is, if $A$ is an $R$-algebra and $\{P_{v},S_{e},S_{e^{*}}\}$ is a Leavitt $E$-family in $A$, then there exists a unique $R$-algebra homomorphism $\pi:L_{R}(E)\to A$ such that $\pi(p_{v})=P_{v}$ and $\pi(s_{e})=S_{e}$ [15, §2-3]. It follows from (L2) that

[TABLE]

3. Subsets of vertices of a directed graph give Morita equivalences

Theorem 1**.**

Let $E$ be a directed graph, let $R$ be a commutative ring with identity and let $\{p_{v},s_{e},s_{e^{*}}\}$ be a universal generating Leavitt $E$-family in $L_{R}(E)$. Let $V\subset E^{0}$, and

[TABLE]

Then

1. (1)

$MM^{*}$* is an $R$-subalgebra of $L_{R}(E)$;* 2. (2)

$MM^{*}=\operatorname{span}\{s_{\mu}s_{\nu^{*}}:r(\mu),r(\nu)\in V\}$, and $M^{*}M$ is an ideal of $L_{R}(E)$ containing $MM^{*}$; 3. (3)

with actions given by multiplication in $L_{R}(E)$, $M$ is an $MM^{*}$–$M^{*}M$-bimodule and $M^{*}$ is an $M^{*}M$–$MM^{*}$-bimodule; 4. (4)

there are bimodule homomorphisms

[TABLE]

such that $(MM^{*},M^{*}M,M,M^{*},\Psi,\Phi)$ is a surjective Morita context.

Proof.

We have

[TABLE]

Products of the form $s_{\mu}s_{\nu^{*}}s_{\alpha}s_{\beta}^{*}$ are either zero or of the form $s_{\mu}s_{\gamma}s_{\delta^{*}}s_{\nu^{*}}=s_{\mu\gamma}s_{(\nu\delta)^{*}}$ for some $\gamma,\delta\in E^{*}$. Thus it is easy to see that $MM^{*}$ is a subalgebra of $L_{R}(E)$ and

[TABLE]

Similarly, $M^{*}M$ is an ideal.

To see that $MM^{*}\subset M^{*}M$, take a spanning element $s_{\mu}s_{\nu^{*}}$ of $MM^{*}$. Then $r(\mu)\in V$, $s_{\mu}s_{\nu^{*}}\in M$, and $s_{\mu}s_{\nu^{*}}=p_{r(\mu)}p_{r(\mu)^{*}}s_{\mu}s_{\nu^{*}}\in M^{*}M$. Thus $MM^{*}\subset M^{*}M$.

Since the module actions are given by multiplication in $L_{R}(E)$ it is easy to verify that $M$ is an $MM^{*}$–$M^{*}M$-bimodule and $M^{*}$ is an $M^{*}M$–$MM^{*}$-bimodule. The function $f\colon M\times M^{*}\to MM^{*}$ defined by $f(m,n)=mn$ is bilinear and $f(md,n)=f(m,dn)$ for all $d\in M^{*}M$. By the universal property of the balanced tensor product, there is a bimodule homomorphism $\Psi\colon M\otimes_{M^{*}M}M^{*}\to MM^{*}$ such that $\Psi(m\otimes n)=f(m,n)=mn$. Similarly, there is a bimodule homomorphism $\Phi\colon M^{*}\otimes_{MM^{*}}M\to M^{*}M$ such that $\Phi(n,m)=nm$. Both $\Psi$ and $\Phi$ are surjective. Since multiplication in $L_{R}(E)$ is associative, for $m,m^{\prime}\in M,n,n^{\prime}\in M^{*}$, we have

[TABLE]

Thus $(MM^{*},M^{*}M,M,M^{*},\Psi,\Phi)$ is a surjective Morita context. ∎

In the situation of Theorem 1, we say a subset $V$ of $E^{0}$ is full if the ideal $M^{*}M$ is all of $L_{R}(E)$. We want a graph-theoretic characterisation of fullness, and so we want an the algebraic version of [5, Lemma 2.2]. We need some definitions.

For $v,w\in E^{0}$ we write $v\leq w$ if there is a path $\mu\in E^{*}$ such that $s(\mu)=w$ and $r(\mu)=v$. We say a subset $H$ of $E^{0}$ is hereditary if $v\in H$ and $v\leq w$ implies $w\in H$. A hereditary subset $H$ of $E^{0}$ is saturated if

[TABLE]

We denote by $\Sigma H(V)$ the smallest saturated hereditary subset of $E^{0}$ containing $V$. For a saturated hereditary subset $H$ of $E^{0}$ we write $I_{H}$ for the ideal of $L_{R}(E)$ generated by $\{p_{v}:v\in H\}$.

Lemma 2**.**

Let $E$ be a directed graph, and let $V\subset E^{0}$. Then $V$ is full if and only if $\Sigma H(V)=E^{0}$.

Proof.

Let $R$ be a commutative ring with identity and let $\{p_{v},s_{e},s_{e^{*}}\}$ be a universal generating Leavitt $E$-family in $L_{R}(E)$. As in Theorem 1, let $M=\operatorname{span}_{R}\{s_{\mu}s_{\nu^{*}}\colon r(\mu)\in V\}$.

First suppose that $V$ is full, that is, that $M^{*}M=L_{R}(E)$. To see that $\Sigma H(V)=E^{0}$, fix $v\in E^{0}$. Then $p_{v}\in M^{*}M$, and we can write $p_{v}$ as a linear combination

[TABLE]

where $F_{1},F_{2}$ are finite subsets of $E^{*}\times E^{*}$, and each $r_{\alpha,\beta,\mu,\nu}\in R$ and $r(\beta)=r(\mu)\in V$.

Since $\Sigma H(V)$ is a hereditary subset containing $V$, we have $s(\beta)$, $s(\mu)\in\Sigma H(V)$, and hence $p_{s(\beta)},p_{s(\mu)}\in I_{\Sigma H(V)}$. Thus each summand

[TABLE]

It follows that $p_{v}\in I_{\Sigma H(V)}$. Thus $v\in\Sigma H(V)$, and hence $E^{0}\subset\Sigma H(V)$. The reverse set inclusion is trivial. Thus $\Sigma H(V)=E^{0}$.

Conversely, suppose that $\Sigma H(V)=E^{0}$. To see that $V$ is full, we need to show that the ideal $M^{*}M$ is all of $L_{R}(E)$. For this, suppose that $I$ is an ideal of $L_{R}(E)$ containing $MM^{*}$. It suffices to show that $L_{R}(E)=I$: by Theorem 1, $M^{*}M$ is an ideal of $L_{R}(E)$ containing $MM^{*}$, and taking $I=M^{*}M$ gives $L_{R}(E)=M^{*}M$, as needed.

By [15, Lemma 7.6], the subset $H_{I}:=\{v\in E^{0}\colon p_{v}\in I\}$ of $E^{0}$ is a saturated hereditary subset of $E^{0}$. Since $I$ contains $MM^{*}$, we have $p_{v}\in I$ for all $v\in V$. Thus $V\subset H_{I}$, and since $H_{I}$ is a saturated hereditary subset, we get $\Sigma H(V)\subset H_{I}$. By assumption, $\Sigma H(V)=E^{0}$, and now $L_{R}(E)=I_{E^{0}}=I_{\Sigma H(V)}\subset I_{H_{I}}\subset I\subset L_{R}(E)$. So $L_{R}(E)=I$ for any ideal $I$ containing $MM^{*}$. Thus $V$ is full. ∎

4. Contractible subgraphs of directed graphs

We start by stating the algebraic version of Crisp and Gow’s [10, Theorem 3.1]; for this we need a few more definitions.

Let $E$ be a directed graph. A finite path $\alpha=\alpha_{1}\alpha_{2}...\alpha_{|\alpha|}$ in $E$ with $|\alpha|\geq 1$ is a cycle if $s(\alpha)=r(\alpha)$ and $s(\alpha_{i})\neq s(\alpha_{j})$ when $i\neq j$. Then $E$ (respectively, a subgraph) is acyclic if it contains no cycles. An acyclic infinite path $x=x_{1}x_{2}....$ in $E$ is a head if each $r(x_{i})$ receives only $x_{i}$ and each $s(x_{i})$ emits only $x_{i}$.

If $E$ has a head, we can get a new graph $F$ by collapsing the head down to a source. This is an example of a desingularisation, and hence $L_{R}(F)$ and $L_{R}(E)$ are Morita equivalent by [1, Proposition 5.2]. Thus the “no heads” hypothesis in Theorem 3 below is not restrictive.

Theorem 3**.**

Let $R$ be a commutative ring with identity, let $E$ be a directed graph with no heads, and let $\{p_{v},s_{e},s_{e^{*}}\}$ be a universal generating Leavitt $E$-family in $L_{R}(E)$. Suppose that $G^{0}\subset E^{0}$ contains the singular vertices of $E$. Suppose also that the subgraph $T$ of $E$ defined by

[TABLE]

is acyclic. Suppose that

1. (T1)

each vertex in $G^{0}$ is the range of at most one infinite path $x\in E^{\infty}$ such that $s(x_{i})\in T^{0}$ for all $i\geq 1$.

Also suppose that for each $y\in T^{\infty}$,

1. (T2)

there is a path from $r(y)$ to a vertex in $G^{0}$; 2. (T3)

$|s^{-1}(r(y_{i}))|=1$* for all $i$; and* 3. (T4)

$e\in E^{1}$, $s(e)=r(y)\implies|r^{-1}(r(e))|<\infty$.

Let $G$ be the graph with vertex set $G^{0}$ and one edge $e_{\beta}$ for each $\beta\in E^{*}\setminus E^{0}$ with $s(\beta),r(\beta)\in G^{0}$ and $s(\beta_{i})\in T^{0}$ for $1\leq i<|\beta|$, such that $s(e_{\beta})=s(\beta)$ and $r(e_{\beta})=r(\beta)$. Then $L_{R}(G)$ is Morita equivalent to $L_{R}(E)$.

In words, the new graph $G$ of Theorem 3 is obtained by replacing each path $\beta\in E^{*}$ with $s(\beta),r(\beta)$ in $G^{0}$ of length at least $1$ which passes through $T$ by a single edge $e_{\beta}$, which has the same source and range as $\beta$. Note that the edges $e$ in $E$ with $r(e)$ and $s(e)$ in $G^{0}$ remain unchanged.

Let $v\in E^{0}$. As in [10] define

[TABLE]

Then $\bigcup_{w\in G^{0}}B_{w}$ of $E^{*}$ corresponds to the set of edges $G^{1}$ in $G$.

To prove Theorem 3, we apply Theorem 1 with $V=G^{0}$ so that

[TABLE]

Then $M^{*}M$ is an ideal of $L_{R}(E)$ containing the subalgebra $MM^{*}$, and $M^{*}M$ and $MM^{*}$ are Morita equivalent. We need to show that $M^{*}M=L_{R}(E)$ and that $MM^{*}$ is isomorphic to $L_{R}(G)$. Our proof uses quite a few of the arguments from Crisp and Gow’s proof of [10, Theorem 3.1]. In particular, Lemma 3.6 of [10] gives a Cuntz-Krieger $G$-family in $C^{*}(E)$, and since the proof is purely algebraic it also gives a Leavitt $G$-family in $L_{R}(E)$. The universal property of $L_{R}(G)$ then gives a unique homomorphism $\phi\colon L_{R}(G)\to L_{R}(E)$. Crisp and Gow used the gauge-invariant uniqueness theorem to show that their $C^{*}$-homomorphism is one-to-one. The analogue here would be the graded uniqueness theorem, however $\phi$ is not graded. Instead, to show $\phi$ is one-to-one, we adapt some clever arguments from the proof of [1, Proposition 5.1] in Lemma 5 below which uses a reduction theorem.

Theorem 4** (Reduction Theorem).**

Let $R$ be a commutative ring with identity, let $E$ be a directed graph, and let $\{p_{v},s_{e},s_{e^{*}}\}$ be a universal Leavitt $E$-family in $L_{R}(E)$. Suppose that $0\neq x\in L_{R}(E)$. There exist $\mu,\nu\in E^{*}$ such that either:

1. (1)

for some $v\in E^{0}$ and $0\neq r\in R$ we have $0\neq s_{\mu^{*}}xs_{\nu}=rp_{v}$, or 2. (2)

there exist $m,n\in\mathbb{Z}$ with $m\leq n$, $r_{i}\in R$, and a non-trivial cycle $\alpha\in E^{*}$ such that $0\neq s_{\mu^{*}}xs_{\nu}=\sum_{i=m}^{n}r_{i}s_{\alpha}^{i}$. (If $i$ is negative, then $s_{\alpha}^{i}:=s_{\alpha^{*}}^{|i|}$.)

Proof.

For Leavitt path algebras over a field this is proved in [3, Proposition 3.1]. We checked carefully that the same proof works over a commutative ring $R$ with identity. ∎

Lemma 5**.**

Let $R$ be a commutative ring with identity. Let $E$ and $G$ be directed graphs, and let $\phi\colon L_{R}(G)\to L_{R}(E)$ be an $R$-algebra $*$-homomorphism. Denote by $\{p_{v},s_{e},s_{e^{*}}\}$ and $\{q_{v},t_{e},t_{e^{*}}\}$ universal Leavitt $E$- and $G$-families in $L_{R}(E)$ and $L_{R}(G)$, respectively. Suppose that

1. (1)

for all $v\in G^{0}$, $\phi(q_{v})=p_{v^{\prime}}$ for some $v^{\prime}\in E^{0}$; and 2. (2)

for all $e\in G^{1}$, $\phi(t_{e})=s_{\beta}$ for some $\beta\in E^{*}$ with $|\beta|\geq 1$.

Then $\phi$ is injective.

Proof.

We follow an argument made in [1, Proposition 5.1]. Let $x\in\ker\phi$. Aiming for a contradiction, suppose that $x\neq 0$. By Theorem 4 there exist $\mu,\nu\in G^{*}$ such that either condition (1) or (2) of the theorem holds.

First suppose that (1) holds, that is, there exist $v\in G^{0}$ and $0\neq r\in R$ such that $0\neq t_{\mu^{*}}xt_{\nu}=rq_{v}$. Using assumption (1), there exists $v^{\prime}\in E^{0}$ such that $\phi(q_{v})=p_{v^{\prime}}$. Now

[TABLE]

But $rp_{v^{\prime}}\neq 0$ for all $0\neq r\in R$, giving a contradiction.

Second suppose that (2) holds, that is, there exist $m,n\in\mathbb{Z}$ with $m\leq n$, $r_{i}\in R$, and a non-trivial cycle $\alpha\in E^{*}$ such that $0\neq s_{\mu^{*}}xs_{\nu}=\sum_{i=m}^{n}r_{i}s_{\alpha}^{i}$. Since $\alpha$ is a non-trivial cycle, it has length at least $1$. By assumption (2), $\phi(t_{\alpha})=s_{\alpha^{\prime}}$ where $\alpha^{\prime}$ is a path in $E$ such that $|\alpha^{\prime}|_{E}\geq|\alpha|_{G}\geq 1$. Since $\phi$ is an $R$-algebra $*$-homomorphism we get

[TABLE]

Since $|\alpha^{\prime}|=k$ for some $k\geq 1$, $s_{\alpha^{\prime}}$ has grading $k$, and hence each $s_{\alpha^{\prime}}^{i}$ has grading $ik$. Thus each term in the sum $\sum_{i=m}^{n}r_{i}s_{\alpha^{\prime}}^{i}$ is in a distinct graded component. But since $s_{\alpha^{\prime}}\neq 0$, we must have $r_{i}=0$ for all $i$. Thus $\sum_{i=m}^{n}r_{i}t_{\alpha}^{i}=0$, which is a contradiction. In either case, we obtained a contradiction to the assumption that $x\neq 0$. Thus $x=0$ and $\phi$ is injective. ∎

Proof of Theorem 3.

Let $\{p_{v},s_{e},s_{e^{*}}\}$ be a universal Leavitt $E$-family in $L_{R}(E)$. We apply Theorem 1 with $V=G^{0}$ to get a surjective Morita context between $MM^{*}$ and $M^{*}M$.

Since $M$ and $M^{*}\subset L_{R}(E)$, we have $M^{*}M\subset L_{R}(E)$. To see that $L_{R}(E)\subset M^{*}M$, let $s_{\mu}s_{\nu^{*}}\in L_{R}(E)$. We may assume that $s(\mu)=s(\nu)$, for otherwise $s_{\mu}s_{\nu^{*}}=0$. If $s(\mu)\in G^{0}$, then the Leavitt $E$-family relations give $s_{\mu}s_{\nu^{*}}=s_{\mu}s_{s(\mu)^{*}}s_{s(\mu)}s_{\nu^{*}}\in M^{*}M$ and we are done. So suppose $s(\mu)\in T^{0}$. Then the graph-theoretic [10, Lemma 3.4(c)] implies that $B_{s(\mu)}\neq\emptyset$. Suppose first that $B_{s(\mu)}$ is finite. It then follows from the first part of [10, Lemma 3.6] that $s(\mu)$ is a nonsingular vertex. The second part of [10, Lemma 3.6] implies that for any Cuntz-Krieger $E$-family $\{P_{v},S_{e},S_{e^{*}}\}$ in $C^{*}(E)$,

[TABLE]

the proof is purely algebraic and works for any Leavitt $E$-family in $L_{R}(E)$. Thus

[TABLE]

Next suppose that $B_{s(\mu)}$ is infinite. Since $s(\mu)\in T^{0}$ and $B_{s(\mu)}$ is infinite, the graph-theoretic [10, Lemma 3.4(d)] implies that there exists $x\in T^{\infty}$ such that $s(\mu)=r(x)$. By assumption (T2), there is a path $\alpha\in E^{*}$ with $r(\alpha)\in G^{0}$ such that $s(\alpha)=r(x)=s(\mu)$. Now

[TABLE]

Thus $L_{R}(E)=M^{*}M$. (We could have used Lemma 2 to prove that $L_{R}(E)=M^{*}M$, as Crisp and Gow do, but this seemed easier.)

Next we show that $L_{R}(G)$ and $M^{*}M$ are isomorphic. For $v\in G^{0}$ and $\beta\in\cup_{w\in G^{0}}B_{w}$ define

[TABLE]

Then $\{Q_{v},T_{e},T_{e^{*}}\}$ is a Leavitt $G$-family in $L_{R}(E)$; again this follows as in the proof of [10, Theorem 3.1]. To see what is involved, we briefly step through this. Relations (L1) follow immediately from the relations for $\{p_{v},s_{e},s_{e^{*}}\}$. To see that (L2) holds, let $\gamma,\beta\in\cup_{w\in G^{0}}B_{w}$. Then $T_{e_{\beta}^{*}}T_{e_{\gamma}}=s_{\beta^{*}}s_{\gamma}$. By the graph-theoretic [10, Lemma 3.4(a)] neither $\gamma$ nor $\beta$ can be a proper extension of the other. Thus $T_{e_{\beta}^{*}}T_{e_{\gamma}}=s_{\beta^{*}}s_{\gamma}=\delta_{\beta,\gamma}p_{s(\beta)}=\delta_{e_{\beta},e_{\gamma}}Q_{s(e_{\beta})}$, and (L2) holds.

To see that (L3) holds, let $v\in G^{0}$ be a non-singular vertex. Then $B_{v}$ is finite and non-empty because it is equinumerous with $r_{G}^{-1}(v)$. Using the algebraic analogue of [10, Lemma 3.6] again, we have

[TABLE]

Thus (L3) holds and $\{Q_{v},T_{e},T_{e^{*}}\}$ is a Leavitt $G$-family in $L_{R}(E)$.

Now let $\{q_{v},t_{e},t_{e^{*}}\}$ be a universal Leavitt $G$-family in $L_{R}(G)$. The universal property of $L_{R}(G)$ gives a unique homomorphism $\phi\colon L_{R}(G)\to L_{R}(E)$ such that for $v\in G^{0}$, $\beta\in\cup_{w\in G^{0}}B_{w}$,

[TABLE]

If $v\in G^{0}$, then $p_{v}=s_{v}s_{v^{*}}\in MM^{*}$; if $\beta\in B_{w}$ for some $w\in G^{0}$, then $r(\beta)\in G^{0}$, and $s_{\beta}=s_{\beta}s_{s(\beta)^{*}}$ and $s_{\beta^{*}}=s_{s(\beta)}s_{\beta^{*}}\in MM^{*}$. It follows that the range of $\phi$ is contained in $MM^{*}$. That $\phi$ is onto $MM^{*}$ again follows from work of Crisp and Gow. They take a non-zero spanning element $s_{\mu}s_{\nu^{*}}\in MM^{*}$ and use the graph-theoretic [10, Lemma 3.4(b)], the algebraic [10, Lemma 3.6] and assumptions (T1)-(T4) to show that $s_{\mu}s_{\nu^{*}}$ is in the range of $\phi$. Thus $\phi$ is onto.

Finally, $\phi$ satisfies the hypotheses of Lemma 5, and hence is one-to-one. Thus $\phi$ is an isomorphism of $L_{R}(G)$ onto $MM^{*}$. ∎

Remark 6*.*

A version of Theorem 1 should hold for the Kumjian-Pask algebras associated to locally convex or finitely aligned $k$-graphs [6, 8]. But the challenge would be to formulate an appropriate notion of contractible subgraph in that setting.

5. Examples

As mentioned in the introduction, the setting of Theorem 3 includes many known examples. We found it helpful to see how some concrete examples fit.

Example 7**.**

An infinite path $x=x_{1}x_{2}\dots$ in a directed graph is collapsible if $x$ has no exits except at $r(x)$, the set $r^{-1}(r(x_{i}))$ of edges is finite for every $i$, and $r^{-1}(r(x))=\{x_{1}\}$ (see [14, Chapter 5]). Consider the following row-finite directed graph $E$:

v_{0}$$v_{1}$$v_{2}$$v_{3}$$v_{4}……x_{1}$$x_{2}$$x_{3}$$x_{4}

The infinite path $x=x_{1}x_{2}\cdots$ is collapsible. When we collapse $x$ to the vertex $v_{0}$, as described in [14, Proposition 5.2], we get the following graph $F$ with infinite receiver $v_{0}$:

v_{0}$$v_{1}$$v_{2}$$v_{3}…

This fits the setting of Theorem 3: take $G^{0}=\{v_{i}\colon i\geq 0\}$, and then $T$ is the subgraph defined by $T^{0}=\{s(x_{i})\colon i\geq 1\}$ and $T^{1}=\{x_{i}\colon i\geq 2\}$. Then $T$ contains none of the singular vertices $\{v_{i}\colon i\geq 2\}$ of $E$, is acyclic, and satisfies the conditions (T1)–(T4). Thus $F$ is the graph $G$ described in the theorem.

Example 8**.**

Consider the directed graph $F$ with source $w$ and infinite receiver $v$:

v$$w$$\infty

An example of a Drinen-Tomforde desingularisation [12] of $F$ is the row-finite graph $E$ with no sources below on the left: a head has been added at the source $w$ of $F$ and each edge from $w$ to $v$ in $F$ has been replaced with paths as shown. (This desingularisation is also an example of an out-delay.) Then, since we are interested in Morita equivalence, we delete the head at $w$ to get the graph $E$ below on the right.

v$$w…⋮

v$$w⋮

Set $T^{0}=E^{0}\setminus\{v,w\}$ and $T^{1}=\{e\in E^{1}\colon s(e),r(e)\in T^{0}\}.$ Then the subgraph $T$ contains none of the singularities of $E$, is acyclic, and satisfies conditions (T1)–(T4) of Theorem 3. The graph $F$ we started with is the graph $G$ of Theorem 3.

Example 9**.**

Consider again the graph $F$ of Example 8. Label the infinitely many edges from $w$ to $v$ by $e_{i}$ for $i\geq 1$. This time we will consider the in-delayed graph $d_{s}(E)$ given by the Drinen source-vector $d_{s}\colon E^{0}\cup E^{1}\to\mathbb{N}\cup\{\infty\}$ (see [5, Section 4]) to be the function defined by $d_{s}(e_{i})=i-1$ for $i\geq 1$, $d_{s}(v)=0$ and $d_{s}(w)=\infty$. Then the in-delayed graph $d_{s}(E)$ given by $d_{s}$, as described in [5], is

v^{0}$$w^{0}$$w^{1}$$w^{2}$$e_{1}$$e_{2}$$e_{3}⋮

Now take $T^{0}=d_{s}(E)^{0}\setminus\{v^{0},w^{0}\}$. Then $T^{0}$ contains none of the singular vertices of $d_{s}(E)$, and the corresponding subgraph $T$ is acyclic. There are no infinite paths in $d_{s}(E)$, and hence conditions (T1)–(T4) of Theorem 3 hold trivially. The graph $G$ of the theorem is again the graph $F$ that we started out with.