Exel-Pardo Algebras of Self-Similar k-Graphs

Abstract

We introduce the Exel-Pardo $*$-algebra ${\textrm{EP}}_R(G,\Lambda )$ associated to a self-similar k-graph $(G,\Lambda ,\varphi )$. We also prove the ${\mathbb {Z}}^k$-graded and Cuntz–Krieger uniqueness theorems for such algebras and investigate their ideal structure. In particular, we modify the graded uniqueness theorem for self-similar 1-graphs and then apply it to present ${\textrm{EP}}_R(G,\Lambda )$ as a Steinberg algebra and to study the ideal structure.

1 Introduction

To give a unified framework like graph $C^*$-algebras for the Katsura’s [11] and Nekrashevyche’s algebras [17, 18], Exel and Pardo introduced self-similar graphs and their $C^*$-algebras in [7]. They then associated an inverse semigroup and groupoid model to this class of $C^*$-algebras and studied structural features by underlying self-similar graphs. Note that although only finite graphs are considered in [7], many of arguments and results may be easily generalized for countable row-finite graphs with no sources (see [8, 10] for example). Inspired from [7], Li and Yang in [15, 16] introduced self-similar action of a discrete countable group G on a row-finite k-graph $\Lambda$. They then associated a universal $C^*$-algebra ${\mathcal {O}}_{G,\Lambda }$ to $(G,\Lambda )$ satisfying specific relations.

The algebraic analogues of Exel-Pardo $C^*$-algebras, denoted by ${\mathcal {O}}_{(G,E)}^{\textrm{alg}}$ in [6] and by $L_R(G,E)$ in [9], were introduced and studied in [6, 9]. In particular, Hazrat et al. proved a ${\mathbb {Z}}$-graded uniqueness theorem and gave a model of Steinberg algebras for $L_R(G,E)$ [9]. The initial aims to write the present paper are to give a much easier proof for [9, Theorem B] (a groupoid model for $L_R(G,E)$) using the ${\mathbb {Z}}$-graded uniqueness theorem and then to study the ideal structure. However, we do these here, among others, for a more general class of algebras associated to self-similar higher rank graphs $(G,\Lambda )$, which is introduced in Sect. 2.

This article is organized as follows. Let R be a unital commutative $*$-ring. In Sect. 2, we introduce a universal $*$-algebra ${\textrm{EP}}_R(G,\Lambda )$ of a self-similar k-graph $(G,\Lambda )$ satisfying specific properties, which is called the Exel-Pardo algebra of $(G,\Lambda )$. Our algebras are the higher rank generalization of those in [9, Theorem 1.6] and the algebraic analogue of ${\mathcal {O}}_{G,\Lambda }$ [15, 16]. Moreover, this class of algebras includes many important known algebras such as the algebraic Katsura algebras [9], Kumjian-Pask algebras [2], and the quotient boundary algebras ${\mathcal {Q}}_R^{\textrm{alg}}(\Lambda \bowtie G)$ of a Zappa-Sz$\acute{\textrm{e}}$p product $\Lambda \bowtie G$ introduced in Sect. 3. In Sect. 3, we give a specific example of Exel-Pardo algebras using boundary quotient algebras of semigroups. Indeed, for a single-vertex self-similar k-graph $(G,\Lambda )$, the Zappa-Szép product $\Lambda \bowtie G$ is a cancellative semigroup. We prove that the quotient boundary algebra ${\mathcal {Q}}_R^{\textrm{alg}}(\Lambda \bowtie G)$ (defined in Definition 3.1) is isomorphic to ${\textrm{EP}}_R(G,\Lambda )$. Section 4 is devoted to proving a graded uniqueness theorem for the Exel-Pardo algebras. Note that using the description in Proposition 2.7, there is a natural ${\mathbb {Z}}^k$-grading on ${\textrm{EP}}_R(G,\Lambda )$. Then, in Theorem 4.2, a ${\mathbb {Z}}^k$-graded uniqueness theorem is proved for ${\textrm{EP}}_R(G,\Lambda )$ which generalizes and modifies [9, Theorem A]. In particular, we will see in Sects. 5 and 6 that this modification makes it more applicable.

In Sects. 5 and 6, we assume that our self-similar k-graphs are pseudo-free (Definition 5.1). In Sect. 5, we prove that every Exel-Pardo algebra ${\textrm{EP}}_R(G,\Lambda )$ is isomorphic to the Steinberg algebra $A_R({\mathcal {G}}_{G,\Lambda })$, where ${\mathcal {G}}_{G,\Lambda }$ is the groupoid introduced in [15]. We should note that the proof of this result is completely different from that of [9, Theorem B]. Indeed, the main difference between the proof of Theorem 5.5 and that of [9, Theorem B] is due to showing the injectivity of defined correspondence. In fact, in [9, Theorem B], the authors try to define a representation for ${\mathcal {S}}_{(G,E)}$ in $\textrm{EP}_R(G,E)$ while we apply our graded uniqueness theorem, Theorem 4.2. This gives us an easier proof for Theorem 5.5, even in the 1-graph case. Finally, in Sect. 6, we investigate the ideal structure of ${\textrm{EP}}_R(G,\Lambda )$. Using the Steinberg algebras, we can define a conditional expectation ${\mathcal {E}}$ on ${\textrm{EP}}_R(G,\Lambda )$ and then characterize basic, ${\mathbb {Z}}^k$-graded, diagonal-invariant ideals of ${\textrm{EP}}_R(G,\Lambda )$ by G-saturated G-hereditary subsets of $\Lambda ^0$. These ideals are exactly basic, $Q({\mathbb {N}}^k,G)\rtimes _{{\mathcal {T}}}{\mathbb {Z}}^k$-graded ideals of ${\textrm{EP}}_R(G,\Lambda )$.

1.1 Notation and Terminology

Let ${\mathbb {N}}=\{0,1,2,\ldots \}$. For $k\ge 1$, we regard ${\mathbb {N}}^k$ as an additive semigroup with the generators $e_1,\ldots ,e_k$. We use $\le$ for the partial order on ${\mathbb {N}}^k$ given by $m\le n$ if and only if $m_i\le n_i$ for $1\le i\le k$. We also write $m\vee n$ and $m\wedge n$ for the coordinate-wise maximum and minimum, respectively.

A k-graph is a countable small category $\Lambda =(\Lambda ^0,\Lambda ,r,s)$ equipped with a degree functor $d:\Lambda \rightarrow {\mathbb {N}}^k$ satisfying the unique factorization property: for $\mu \in \Lambda$ and $m,n\in {\mathbb {N}}^k$ with $d(\mu )=m+n$, then there exist unique $\alpha ,\beta \in \Lambda$ such that $d(\alpha )=m$, $d(\beta )=n$, and $\mu =\alpha \beta$. We usually denote $\mu (0,m):=\alpha$ and $\mu (m,d(\mu )):=\beta$. We refer to $\Lambda ^0$ as the vertex set and define $\Lambda ^n:=\{\mu \in \Lambda : d(\mu )=n\}$ for every $n\in {\mathbb {N}}^k$. For $A,B\subseteq \Lambda$, define $AB=\{\mu \nu :\mu \in A,\nu \in B, ~\textrm{and}~ s(\mu )=r(\nu )\}$. Also, for $\mu ,\nu \in \Lambda$, define $\Lambda ^{min}(\mu ,\nu )=\{(\alpha ,\beta )\in \Lambda \times \Lambda : \mu \alpha =\nu \beta , d(\mu \alpha )=d(\mu )\vee d(\nu )\}$.

We say that $\Lambda$ is row-finite if $v\Lambda ^n$ is finite for all $n\in {\mathbb {N}}^k$ and $v\in \Lambda ^0$. A source in $\Lambda$ is a vertex $v\in \Lambda ^0$ such that $v\Lambda ^{e_i}=\emptyset$ for some $1\le i\le k$.

Standing assumption. Throughout the article, we work only with row-finite k-graphs without sources.

Let $\Omega _k:=\{(m,n)\in {\mathbb {N}}^k\times {\mathbb {N}}^k:m\le n\}$. By defining $(m,n).(n,l):=(m,l)$, $r(m,n):=(m,m)$, and $s(m,n):=(n,n)$, then $\Omega _k$ is a row-finite k-graph without sources. A graph homomorphism $x:\Omega _k\rightarrow \Lambda$ is called an infinite path of $\Lambda$ with the range $r(x)=x(0,0)$, and we write $\Lambda ^\infty$ for the set of all infinite paths of $\Lambda$.

2 Exel-Pardo Algebras of Self-Similar k-Graphs

In this section, we associate a $*$-algebra to a self-similar k-graph as the algebraic analogue of [16, Definition 3.9]. Let us first review some definitions and notations.

Following [9], we consider $*$-algebras over $*$-rings. Let R be a unital commutative $*$-ring. Recall that a $*$-algebra over R is an algebra A equipped with an involution such that $(a^*)^*=a$, $(ab)^*=b^* a^*$, and $(ra+b)^*=r^* a^*+b^*$ for all $a,b\in A$ and $r\in R$. Then $p\in A$ is called a projection if $p^2=p=p^*$, and $s\in A$ a partial isometry if $s=ss^*s$.

Definition 2.1

([2, Definition 3.1]) Let $\Lambda$ be a row-finite k-graph without sources. A Kumjian-Pask $\Lambda$-family is a collection $\{s_{\mu }:\mu \in \Lambda \}$ of partial isometries in a $*$-algebra A such that

• (KP1) $\{s_v:v\in \Lambda ^0\}$ is a family of pairwise orthogonal projections;

• (KP2) $s_{\mu \nu }=s_\mu s_\nu$ for all $\mu ,\nu \in \Lambda$ with $s(\mu )=r(\nu )$;

• (KP3) $s_\mu ^* s_\mu =s_{s(\mu )}$ for all $\mu \in \Lambda$; and

• (KP4) $s_v=\sum _{\mu \in v\Lambda ^n}s_\mu s_\mu ^*$ for all $v\in \Lambda ^0$ and $n\in {\mathbb {N}}^k$.

2.1 Self-Similar k-Graphs and Their Algebras

Let $\Lambda$ be a row-finite k-graph without sources. An automorphism of $\Lambda$ is a bijection $\psi :\Lambda \rightarrow \Lambda$ such that $\psi (\Lambda ^n)\subseteq \Lambda ^n$ for all $n\in {\mathbb {N}}^k$ with the properties $s\circ \psi =\psi \circ s$ and $r\circ \psi =\psi \circ r$. We denote by $\textrm{Aut}(\Lambda )$ the group of automorphisms on $\Lambda$. Furthermore, if G is a countable discrete group, an action of G on $\Lambda$ is a group homomorphism $g\mapsto \psi _g$ from G into $\textrm{Aut}(\Lambda )$.

Definition 2.2

([15]) Let $\Lambda$ be a row-finite k-graph without sources and G a discrete group with identity $e_G$. We say that a triple $(G,\Lambda ,\varphi )$ is a self-similar k-graph whenever the following properties hold:

• (1) G acts on $\Lambda$ by a group homomorphism $g\mapsto \psi _g$. We prefer to write $g\cdot \mu$ for $\psi _g(\mu )$ to ease the notation.

• (2) $\varphi :G\times \Lambda \rightarrow G$ is a 1-cocycle for the action $G\curvearrowright \Lambda$ such that for every $g\in G$, $\mu ,\nu \in \Lambda$ and $v\in \Lambda ^0$ we have

  • (a) $\varphi (gh,\mu )=\varphi (g,h \cdot \mu )\varphi (h,\mu )$ (the 1-cocycle property),

  • (b) $g\cdot (\mu \nu )=(g\cdot \mu )(\varphi (g,\mu ) \cdot \nu )$ (the self-similar equation),

  • (c) $\varphi (g,\mu \nu )=\varphi (\varphi (g,\mu ),\nu )$, and

  • (d) $\varphi (g,v)=g$.

For convenience, we usually write $(G,\Lambda )$ instead of $(G,\Lambda ,\varphi )$.

Remark 2.3

In [15], the authors used the notation $g|_\mu$ for $\varphi (g,\mu )$. However, we would prefer to follow [7,8,9] for writing $\varphi (g,\mu )$.

Remark 2.4

If in equation (2)(a) of Definition 2.2, we set $g=h=e_G$, then we get $\varphi (e_G,\mu )=e_G$ for every $\mu \in \Lambda$. Moreover, [15, Lemma 3.5(ii)] shows that $\varphi (g,\mu )\cdot v=g\cdot v$ for all $g\in G$, $v\in \Lambda ^0$, and $\mu \in \Lambda$.

Now we generalize the definition of Exel-Pardo $*$-algebras [9] to the k-graph case.

Definition 2.5

Let $(G,\Lambda )$ be a self-similar k-graph. An Exel-Pardo $(G,\Lambda )$-family (or briefly $(G,\Lambda )$-family) is a set

$$\begin{aligned} \{s_\mu :\mu \in \Lambda \}\cup \{u_{v,g}:v\in \Lambda ^0,g\in G\} \end{aligned}$$

in a $*$-algebra satisfying

• (1) $\{s_\mu :\mu \in \Lambda \}$ is a Kumjian-Pask $\Lambda$-family,

• (2) $u_{v,e_G}=s_v$ for all $v\in \Lambda ^0$,

• (3) $u_{v,g}^*=u_{g^{-1} \cdot v,g^{-1}}$ for all $v\in \Lambda ^0$ and $g\in G$,

• (4) $u_{v,g}s_\mu =\delta _{v,g\cdot r(\mu )}s_{g\cdot \mu }u_{g\cdot s(\mu ),\varphi (g,\mu )}$ for all $v\in \Lambda ^0$, $\mu \in \Lambda$, and $g\in G$,

• (5) $u_{v,g}u_{w,h}=\delta _{v,g\cdot w}u_{v,gh}$ for all $v,w\in \Lambda ^0$ and $g,h\in G$.

Then the Exel-Pardo algebra ${\textrm{EP}}_R(G,\Lambda )$ is the universal $*$-algebra over R generated by a $(G,\Lambda )$-family $\{s_\mu ,u_{v,g}\}$.

Recall that the universality of ${\textrm{EP}}_R(G,\Lambda )$ means that for every $(G,\Lambda )$-family $\{S_\mu ,U_{v,g}\}$ in a $*$-algebra A, there exists a $*$-homomorphism $\phi :{\textrm{EP}}_R(G,\Lambda )\rightarrow A$ such that $\phi (s_\mu )=S_\mu$ and $\phi (u_{v,g})=U_{v,g}$ for all $v\in \Lambda ^0$, $\mu \in \Lambda$, and $g\in G$. (See Sect. 2.2 for the construction of ${\textrm{EP}}_R(G,\Lambda )$.) Throughout the paper, we will denote by $\{s_\mu ,u_{v,g}\}$ the $(G,\Lambda )$-family generating ${\textrm{EP}}_R(G,\Lambda )$.

2.2 The Construction of ${\textrm{EP}}_R(G,\Lambda )$

Let $(G,\Lambda )$ be a self-similar k-graph as in Definition 2.2. The following is a standard construction of a universal algebra ${\textrm{EP}}_R(G,\Lambda )$ subject to desired relations. Consider the set of formal symbols

$$\begin{aligned} S=\left\{ S_\mu ,S_\mu ^*,U_{v,g},U_{v,g}^*:\mu \in \Lambda , v\in \Lambda ^0, g\in G\right\} . \end{aligned}$$

Let $X=w(S)$ be the collection of finite words in S. We equip the free R-module

$$\begin{aligned} {\mathbb {F}}_R(X):=\bigg \{\sum _{i=1}^l r_ix_i: l\ge 1, r_i\in R, x_i\in X\bigg \} \end{aligned}$$

with the multiplication

$$\begin{aligned} \bigg (\sum _{i=1}^l r_i x_i\bigg )\bigg (\sum _{j=1}^{l'} s_j y_j\bigg ):=\sum _{i,j} r_is_j x_iy_j, \end{aligned}$$

and the involution

$$\begin{aligned} \bigg (\sum r_i x_i\bigg )^*:=\bigg (\sum r_i^* x_i^*\bigg ) \end{aligned}$$

where $x^*=s_l^*\ldots s_1^*$ for each $x=s_1\ldots s_l$. Then ${\mathbb {F}}_R(X)$ is a $*$-algebra over R. If I is the (two-sided and self-adjoint) ideal of ${\mathbb {F}}_R(X)$ containing the roots of relations (1)-(5) in Definition 2.5, then the quotient ${\mathbb {F}}_R(X)/I$ is the Exel-Pardo algebra ${\textrm{EP}}_R(G,\Lambda )$ with the desired universal property. Let us define $s_\mu :=S_\mu +I$ and $u_{v,g}:=U_{v,g}+I$ for every $\mu \in \Lambda$, $v\in \Lambda ^0$, and $g\in G$. In case $(G,\Lambda )$ is pseudo-free (Definition 5.1), Theorem 4.2 insures that all generators $\{s_\mu ,u_{v,g}\}$ of ${\textrm{EP}}_R(G,\Lambda )$ are nonzero.

Proposition 2.7 describes the elements of ${\textrm{EP}}_R(G,\Lambda )$. First, see a simple lemma.

Lemma 2.6

Let $(G,\Lambda )$ be a self-similar graph (as in Definition 2.2) and $\{S,U\}$ a $(G,\Lambda )$-family. If $S_\mu U_{v,g} S_\nu ^*\ne 0$ where $\mu ,\nu \in \Lambda$, $v\in \Lambda ^0$ and $g\in G$, then $s(\mu )=v=g\cdot s(\nu )$.

Proof

If $a=S_\mu U_{v,g} S_\nu ^*$ is nonzero, then by Definition 2.5 we can write

$$\begin{aligned} S_\mu U_{v,g}S_\nu ^*&= S_\mu (S_{s(\mu )}U_{v,g} ) S_\nu ^*\\&= S_\mu (U_{s(\mu ),e_G} U_{v,g}) S_\nu ^*\\&=S_\mu (\delta _{s(\mu ),e_G\cdot v} U_{s(\mu ),g}) S_\nu ^*. \end{aligned}$$

Now, the hypothesis $a\ne 0$ forces $s(\mu )=v$. On the other hand, a similar computation gives

$$\begin{aligned} a&= S_\mu (U_{v,g} S_{s(\nu )}) S_\nu ^*\\&= S_\mu ( U_{v,g} U_{s(\nu ),e_G}) S_\nu ^*\\&=S_\mu (\delta _{v, g\cdot s(\nu )} U_{v,g}) S_\nu ^*, \end{aligned}$$

and thus $v=g\cdot s(\nu )$. $\square$

Proposition 2.7

Let $(G,\Lambda )$ be a self-similar graph. Then

$$\begin{aligned} {\textrm{EP}}_R(G,\Lambda )=\textrm{span}_R\{s_\mu u_{s(\mu ),g} s_\nu ^*: g\in G,~ \mu ,\nu \in \Lambda ,~ \textrm{and}~ s(\mu )=g\cdot s(\nu )\}. \end{aligned}$$

(2.1)

Proof

Define $M:=\textrm{span}_R\{s_\mu u_{s(\mu ),g} s_\nu ^*: g\in G,~ \mu ,\nu \in \Lambda \}$. For every $g,h\in G$ and $\mu ,\nu ,\alpha ,\beta \in \Lambda$ with $\alpha =\nu \alpha '$ for some $\alpha '\in \Lambda$, the relations of Definition 2.5 imply that

$$\begin{aligned} \bigg (s_\mu u_{s(\mu ),g} s_\nu ^* \bigg ) \bigg (s_\alpha u_{s(\alpha ),h} s_\beta ^* \bigg )&= s_\mu u_{s(\mu ),g}(s_\nu ^* s_\alpha ) u_{s(\alpha ),h} s_\beta ^*\\&= s_\mu u_{s(\mu ),g}(s_{\alpha '}) u_{s(\alpha ),h} s_\beta ^* \\&= s_\mu \bigg ( \delta _{s(\mu ),g\cdot r(\alpha ')} s_{g\cdot \alpha '} u_{g\cdot s(\alpha '), \varphi (g,\alpha ')} u_{s(\alpha ),h}\bigg ) s_\beta ^* \\&= \delta _{s(\mu ),g\cdot s(\nu )} s_{\mu (g\cdot \alpha ')}\\&\quad \bigg (\delta _{g\cdot s(\alpha '),\varphi (g,\alpha ') \cdot s(\alpha )} u_{g\cdot s(\alpha '), \varphi (g,\alpha ') h} \bigg ) s_\beta ^*\\&\qquad (\textrm{as}~~ r(\alpha ')=s(\nu )). \end{aligned}$$

In the case $\nu =\alpha \nu '$ for some $\nu '\in \Lambda$, the above multiplication may be computed similarly, and otherwise is zero. Hence, M is closed under multiplication. Also, we have

$$\begin{aligned} \bigg (s_\mu u_{s(\mu ),g}s_\nu ^* \bigg )^*= s_\nu u_{g^{-1} \cdot s(\mu ), g^{-1}} s_\mu ^*, \end{aligned}$$

so $M^*\subseteq M$. Since

$$\begin{aligned} s_\mu =s_\mu u_{s(\mu ),e_G} s_{s(\mu )}^* \textrm{ and } u_{v,g}=s_v u_{v,g} s_{g\cdot v}^* \end{aligned}$$

for all $g\in G$, $v\in \Lambda ^0$, and $\mu \in \Lambda$, it follows that M is a $*$-subalgebra of ${\textrm{EP}}_R(G,\Lambda )$ containing the generators of ${\textrm{EP}}_R(G,\Lambda )$. In light of Lemma 2.6, this concludes the identification (2.1). $\square$

2.3 The Unital Case

In case $\Lambda$ is a k-graph with finite $\Lambda ^0$, we may give a better description for Definition 2.5. Note that this case covers all unital Exel-Pardo algebras ${\textrm{EP}}_R(G,\Lambda )$.

Lemma 2.8

Let $(G,\Lambda )$ be a self-similar k-graph and let $s_v$ be nonzero in ${\textrm{EP}}_R(G,\Lambda )$ for every $v\in \Lambda ^0$. Then ${\textrm{EP}}_R(G,\Lambda )$ is a unital algebra if and only if the vertex set $\Lambda ^0$ is finite.

Proof

If $\Lambda ^0=\{v_1,\ldots , v_l\}$ is finite, then using identification (2.1), $P=\sum _{i=1}^l s_{v_i}$ is the unit of ${\textrm{EP}}_R(G,\Lambda )$. Conversely, if $\Lambda ^0$ is infinite, then the set $\{s_v:v\in \Lambda ^0\}\subseteq {\textrm{EP}}_R(G,\Lambda )$ contains infinitely many mutually orthogonal projections. Now again by (2.1), there is no element of ${\textrm{EP}}_R(G,\Lambda )$ which acts as an identity on each element of $\{s_v:v\in \Lambda ^0\}$. $\square$

Note that if $\{s,u\}$ is a $(G,\Lambda )$-family in a $*$-algebra A, then for each $g\in G$ we may define $u_g:=\sum _{v\in \Lambda ^0}u_{v,g}$ as an element of the multiplier algebra ${\mathcal {M}}(A)$ with the property $s_v u_g=u_{v,g}$ for all $v\in \Lambda ^0$ (relations (2) and (5) of Definition 2.5 yield $s_vu_{w,g}=\delta _{v,w} u_{v,g}$). (See [1] for the definition of multiplier algebras.) Thus relations (3) and (5) of Definition 2.5 imply that $u:G\rightarrow {\mathcal {M}}(A)$, defined by $g \mapsto u_g$, is a unitary $*$-representation of G on ${\mathcal {M}}(A)$. In particular, in case $\Lambda ^0$ is finite, $u_g$’s lie all in A, and we may describe Definition 2.5 as the following:

Proposition 2.9

Let $(G,\Lambda )$ be a self-similar k-graph. Suppose also that $\Lambda ^0$ is finite. Then ${\textrm{EP}}_R(G,\Lambda )$ is the universal $*$-algebra generated by families $\{s_\mu :\mu \in \Lambda \}$ of partial isometries and $\{u_g:g\in G\}$ of unitaries satisfying

• (1) $\{s_\mu :\mu \in \Lambda \}$ is a Kumjian-Pask $\Lambda$-family;

• (2) $u:G\rightarrow {\textrm{EP}}_R(G,\Lambda )$, by $g\mapsto u_g$, is a unitary $*$-representation of G on ${\textrm{EP}}_R(G,\Lambda )$, in the sense that

  • (a) $u_g u_h=u_{gh}$ for all $g,h\in G$, and

  • (b) $u^*_g= u^{-1}_g= u_{g^{-1}}$ for all $g\in G$;

• (3) $u_g s_\mu =s_{g\cdot \mu } u_{\varphi (g,\mu )}$ for all $g\in G$ and $\mu \in \Lambda$.

3 An example: The Zappa-Szép Product $\Lambda \bowtie G$ and its $*$-Algebra

Let $(G,\Lambda )$ be a self-similar k-graph such that $|\Lambda ^0|=1$. The $C^*$-algebra and quotient boundary $C^*$-algebra associated to the Zappa-Szép product $\Lambda \bowtie G$ as a semigroup were studied in [4, 14]. In this section, we first define ${\mathcal {Q}}_R^{\textrm{alg}}(S)$ as the algebraic analogue of the quotient boundary $C^*$-algebra ${\mathcal {Q}}(S)$ of a cancellative semigroup S. Then we show that ${\mathcal {Q}}_R^{\textrm{alg}}(\Lambda \bowtie G)$ is isomorphic to the Exel-Pardo algebra ${\textrm{EP}}_R(G,\Lambda )$.

Let us recall some terminology from [4, 13]. Let S be a left-cancellative semigroup with an identity. Given $X\subseteq S$ and $s\in S$, define $sX:=\{sx:x\in X\}$ and $s^{-1}X:=\{r\in S:sr\in X\}$. Also, the set of constructible right ideals in S is defined as

$$\begin{aligned} {\mathcal {J}}(S):=\{s^{-1}_1r_1\ldots s_l^{-1} r_l S: l\ge 1, ~ s_i,r_i\in S\}\cup \{\emptyset \}. \end{aligned}$$

Then, a foundation set in ${\mathcal {J}}(S)$ is a finite subset $F\subseteq {\mathcal {J}}(S)$ such that for each $Y\in {\mathcal {J}}(S)$, there exists $X\in F$ with $X\cap Y\ne \emptyset$.

The following is the algebraic analogue of [13, Definition 2.2].

Definition 3.1

Let S be a left-cancellative semigroup and R be a unital commutative $*$-ring. The boundary quotient $*$-algebra of S is the universal unital $*$-algebra ${\mathcal {Q}}_R^{\textrm{alg}}(S)$ over R generated by a set of isometries $\{t_s:s\in S\}$ and a set of projections $\{q_X:X\in {\mathcal {J}}(S)\}$ satisfying

• (1) $t_s t_r=t_{sr}$,

• (2) $t_s q_X t_s^*=q_{sX}$,

• (3) $q_S=1$ and $q_\emptyset =0$,

• (4) $q_X q_Y=q_{X\cap Y}$, and moreover

• (5) $\prod _{X\in F} (1-q_X)=0$

for all $s,r\in S$, $X,Y\in {\mathcal {J}}(S)$, and foundation sets $F\subseteq {\mathcal {J}}(S)$.

Let $\Lambda$ be a k-graph such that $\Lambda ^0=\{v\}$. Then $\mu \nu$ is composable for all $\mu ,\nu \in \Lambda$, and hence $\Lambda$ may be considered as a semigroup with the identity v. Also, the unique factorization property implies that $\Lambda$ is cancellative.

Definition 3.2

([4, Definition 3.1]) Let $(G,\Lambda )$ is a single-vertex self-similar k-graph. If we consider $\Lambda$ as a semigroup, then the Zappa-Szép product $\Lambda \bowtie G$ is the semigroup $\Lambda \times G$ with the multiplication

$$\begin{aligned} (\mu ,g)(\nu ,h):=(\mu (g\cdot \nu ), \varphi (g,\nu ) \cdot h) \qquad (\mu ,\nu \in \Lambda ~ \textrm{and}~ g,h\in G). \end{aligned}$$

Remark 3.3

If $\Lambda$ is a single-vertex k-graph, then [14, Lemma 3.2 (iv)] follows that

$$\begin{aligned} {\mathcal {J}}(\Lambda )=\bigg \{\bigcup _{i=1}^l\mu _i \Lambda : l\ge 1, \mu _i\in \Lambda , d(\mu _1)=\cdots =d(\mu _l)\bigg \}. \end{aligned}$$

In order to prove Theorem 3.6, the following lemmas are useful.

Lemma 3.4

Let $(G,\Lambda )$ be a self-similar k-graph with $\Lambda ^0=\{v\}$. Suppose that for each $\mu \in \Lambda$, the map $g\mapsto \varphi (g,\mu )$ is surjective. Then

• (1) ${\mathcal {J}}(\Lambda \bowtie G)={\mathcal {J}}(\Lambda )\times \{G\}$, where $\emptyset \times G:=\emptyset$.

• (2) A finite subset $F\subseteq {\mathcal {J}}(\Lambda )$ is a foundation set if and only if $F'=F\times \{G\}$ is a foundation set in ${\mathcal {J}}(\Lambda \bowtie G)$.

Proof

Statement (1) is just [14, Lemma 2.13]. For (2), suppose that $F\subseteq {\mathcal {J}}(\Lambda )$ is a foundation set, and let $Y\times G \in {\mathcal {J}}(\Lambda \bowtie G)$. Then there exists $X\in F$ such that $X\cap Y\ne \emptyset$. Thus $(X\times G)\cap (Y\times G)\ne \emptyset$, from which we conclude that $F\times \{G\}$ is a foundation set in ${\mathcal {J}}(\Lambda \bowtie G)$. The converse may be shown analogously. $\square$

In the following, for $\mu \in \Lambda$ and $E\subseteq \Lambda$ we define

$$\begin{aligned} \textrm{Ext}(\mu ;E):=\{\alpha : (\alpha ,\beta )\in \Lambda ^{\textrm{min}}(\mu ,\nu ) \mathrm {~ for ~ some ~} \nu \in E\}. \end{aligned}$$

Lemma 3.5

Let $(G,\Lambda )$ be a self-similar k-graph with $\Lambda ^0=\{v\}$. For every $X=\cup _{i=1}^l\mu _i \Lambda$ and $Y=\cup _{j=1}^{l'} \nu _j \Lambda$ in ${\mathcal {J}}(\Lambda )$, we have

$$\begin{aligned} X\cap Y=\cup \{\mu _i\alpha \Lambda : 1\le i\le l, \alpha \in \textrm{Ext}(\mu _i;\{\nu _i,\ldots , \nu _{l'}\})\}. \end{aligned}$$

Proof

For any $\lambda \in X\cap Y$, there are $\alpha ,\beta \in \Lambda$, $1\le i\le l$, and $1\le j \le l'$ such that $\lambda =\mu _i\alpha =\nu _j\beta$. Define

$$\begin{aligned} \alpha ':=\alpha (0,d(\mu _i)\vee d(\nu _j)-d(\mu _i)) ~~~ \textrm{and} ~~~ \beta ':=\beta (0,d(\mu _i)\vee d(\nu _j)-d(\nu _j)). \end{aligned}$$

Then the factorization property implies that $\lambda =\mu _i\alpha '\lambda '=\nu _j\beta '\lambda '$ where $d(\mu _i\alpha ')=d(\nu _j \beta ')=d(\mu _i)\vee d(\nu _j)$ and $\lambda '=\lambda (d(\mu _i)\vee d(\nu _j),d(\lambda ))$. It follows that $\lambda \in \mu _i\alpha '\Lambda$ with $\alpha '\in \textrm{Ext}(\mu _i;\{\nu _j\})$ as desired. The reverse containment is trivial. $\square$

The following result is inspired by [14, Theorem 3.3].

Theorem 3.6

Let $(G,\Lambda )$ be a self-similar k-graph with $\Lambda ^0=\{v\}$ and let $\{s_\mu ,u_g\}$ be the $(G,\Lambda )$-family generating ${\textrm{EP}}_R(G,\Lambda )$ as in Proposition 2.9. Suppose that for every $\mu \in \Lambda$ the map $g\mapsto \varphi (g,\mu )$ is surjective. If the family $\{t_{(\mu ,g)}, q_X: (\mu ,g)\in \Lambda \bowtie G, X\in {\mathcal {J}}(\Lambda \bowtie G)\}$ generates ${\mathcal {Q}}_R^{\textrm{alg}}(\Lambda \bowtie G)$, then there exists an R-algebra $*$-isomorphism $\pi :{\textrm{EP}}_R(G,\Lambda )\rightarrow {\mathcal {Q}}_R^{\textrm{alg}}(\Lambda \bowtie G)$ such that $\pi (s_\mu )=t_{(\mu ,e_G)}$ and $\pi (u_g)=q_{(v,g)}$ for all $\mu \in \Lambda$ and $g\in G$.

Proof

For every $\mu \in \Lambda$ and $g\in G$, define

$$\begin{aligned} S_\mu :=t_{(\mu ,e_G)} ~~ \textrm{and} ~~ U_g:=t_{(v,g)}. \end{aligned}$$

We will show that $\{S, U\}$ is a $(G,\Lambda )$-family in ${\mathcal {Q}}_R^{\textrm{alg}}(\Lambda \bowtie G)$, which is described in Proposition 2.9. First, for each $g\in G$ we have

$$\begin{aligned} U_g U_g^*=t_{(v,g)} ~ q_{\Lambda \bowtie G} ~ t_{(v,g)}^* = q_{(v,g)\Lambda \bowtie G}= q_{\Lambda \bowtie G}=1_{{\mathcal {Q}}_R^{\textrm{alg}}(\Lambda \bowtie G)}. \end{aligned}$$

So, $g\mapsto U_g$ is a unitary $*$-representation of G into ${\mathcal {Q}}_R^{\textrm{alg}}(\Lambda \bowtie G)$. Moreover, (KP1)-(KP3) can be easily checked, so we verify (KP4) for $\{s_\mu :\mu \in \Lambda \}$. Fix some $n\in {\mathbb {N}}^k$. Then $\{\mu \Lambda :\mu \in \Lambda ^n\}$ is a foundation set in ${\mathcal {J}}(\Lambda )$, and thus so is $F=\{\mu \Lambda \times G:\mu \in \Lambda ^n\}$ in ${\mathcal {J}}(\Lambda \bowtie G)$ by Lemma 3.4. Hence we have

$$\begin{aligned} 1-\sum _{\mu \in \Lambda ^n} S_\mu S_\mu ^*&=\prod _{\mu \in \Lambda ^n}(1-S_\mu S_\mu ^*) \qquad (\mathrm {because ~} S_\mu S_\mu ^*\mathrm {s ~ are ~ pairwise ~ orthogonal})\\&=\prod _{\mu \in \Lambda ^n}(1-t_{(\mu ,e_G)} ~ q_{\Lambda \bowtie G} ~ t_{(\mu ,e_G)}^*)\\&=\prod _{\mu \in \Lambda ^n}(1-q_{\mu \Lambda \times G}) \qquad (\mathrm {by ~ eq.~ (2) ~ of ~ Definition ~ 3.1}) \\&=\prod _{X\in F} (1-q_X)=0 \qquad (\mathrm {by ~ eq.~ (5) ~ of ~ Definition ~ 3.1}). \end{aligned}$$

Because $S_v=U_{e_G}=1$, (KP4) is verified, and therefore $\{S_\mu :\mu \in \Lambda \}$ is a Kumjian-Pask $\Lambda$-family. Since for each $\mu \in \Lambda$ and $g\in G$,

$$\begin{aligned} U_g S_\mu =t_{(v,g)}t_{(\mu ,e_G)} = t_{(v,g)(\mu ,e_G)}=t_{(g\cdot \mu ,\varphi (g,\mu ))}= S_{g\cdot \mu } U_{\varphi (g,\mu )}, \end{aligned}$$

and so we have shown that $\{S,U\}$ is a $(G,\Lambda )$-family in ${\mathcal {Q}}_R^{\textrm{alg}}(\Lambda \bowtie G)$. Now the universality implies that the desired $*$-homomorphism $\pi :{\textrm{EP}}_R(G,\Lambda )\rightarrow {\mathcal {Q}}_R^{\textrm{alg}}(\Lambda \bowtie G)$ exists.

Now we prove that $\pi$ is an isomorphism. In order to do this, it suffices to find a homomorphism $\rho :{\mathcal {Q}}_R^{\textrm{alg}}(\Lambda \bowtie G)\rightarrow {\textrm{EP}}_R(G,\Lambda )$ such that $\rho \circ \pi =\textrm{id}_{{\textrm{EP}}_R(G,\Lambda )}$ and $\pi \circ \rho =\textrm{id}_{{\mathcal {Q}}_R^{\textrm{alg}}(\Lambda \bowtie G)}$. For any $(\mu ,g)\in \Lambda \bowtie G$ and $X=(\cup _{i=1}^l \mu _i\Lambda ) \times G\in {\mathcal {J}}(\Lambda \bowtie G)$, we define

$$\begin{aligned} T_{(\mu ,g)}:=s_\mu u_g \textrm{and} Q_X:=\sum _{i=1}^l s_{\mu _i}s_{\mu _i}^*. \end{aligned}$$

We will show that the family $\{T,Q\}$ satisfies the properties of Definitions 3.1. Relations (1)-(3) easily hold by the $(G,\Lambda )$-relations for $\{s,u\}$. Also, for every $X=(\cup _{i=1}^l \mu _i\Lambda ) \times G$ and $Y=(\cup _{j=1}^{l'}\nu _j\Lambda ) \times G$ in ${\mathcal {J}}(\Lambda \bowtie G)$, we have

$$\begin{aligned} Q_X Q_Y&=\bigg (\sum _{i=1}^l s_{\mu _i}s_{\mu _i}^* \bigg )\bigg (\sum _{j=1}^{l'} s_{\nu _j}s_{\nu _j}^* \bigg )\\&=\sum _{i,j} s_{\mu _i}(s_{\mu _i}^* s_{\nu _j})s_{\nu _j}^*\\&=\sum _{i,j} s_{\mu _i}\bigg (\sum _{(\alpha ,\beta )\in \Lambda ^{\textrm{min}}(\mu _i,\nu _j)}s_\alpha s_\beta ^* \bigg ) s_{\nu _j}^* \qquad (\mathrm {by ~ [1, Lemma~ 3.3]})\\&=\sum _{i,j} \sum _{\begin{array}{c} \mu _i\alpha =\nu _j \beta \\ d(\mu _i\alpha )=d(\mu _i)\vee d(\nu _j) \end{array}} s_{\mu _i\alpha } s_{\nu _j \beta }^*\\&=Q_{X\cap Y} \qquad (\mathrm {by ~ Lemma} ~ 3.5). \end{aligned}$$

For eq. (5) of Definition 3.1, let $F=\{X_i\times G:=\cup _{j=1}^{t_i}\mu _{ij}\Lambda \times G: 1\le i\le l\}$ be a foundation set in ${\mathcal {J}}(\Lambda \bowtie G)$. Then $F'=\{X_i=\cup _{j=1}^{t_i}\mu _{ij}\Lambda \}_{i=1}^l$ is a foundation set in ${\mathcal {J}}(\Lambda )$ by Lemma 3.4(2). Defining $n:=\bigvee _{i,j}d(\mu _{ij})$, we claim that the set $M=\{\mu _{ij}\alpha : \alpha \in \Lambda ^{n-d(\mu _{ij})}, 1\le i\le l, 1\le j\le t_i\}$ coincides with $\Lambda ^n$. Indeed, if on the contrary there exists some $\lambda \in \Lambda ^n \setminus M$, then $\Lambda ^{\textrm{min}}(\lambda ,\mu _{ij})=\emptyset$, and hence $\lambda \Lambda \cap \mu _{ij}\Lambda =\emptyset$ for all i and j. This yields that $\lambda \Lambda \cap X_i=\emptyset$ for every $X_i\in F'$, contradicting that $F'$ is a foundation set in ${\mathcal {J}}(\Lambda )$.

Now one may compute

$$\begin{aligned} \prod _{X_i\times G\in F} (1-Q_{X_i\times G})&= \prod _{i=1}^l (1-\sum _{j=1}^{t_i} s_{\mu _{ij}} s_{\mu _{ij}}^*)\\&= \prod _{i=1}^l \bigg (1- \sum _{j=1}^{t_i} s_{\mu _{ij}} (\sum _{\alpha \in \Lambda ^{n-d(\mu _{ij})}} s_\alpha s_\alpha ^*) s_{\mu _{ij}}^* \bigg )\\&=\prod _{i=1}^l \bigg (1-\sum _{j=1}^{t_i} \sum _{\alpha \in \Lambda ^{n-d(\mu _{ij})}} s_{\mu _{ij}\alpha } s_{\mu _{ij}\alpha }^* \bigg ) \qquad (\star ). \end{aligned}$$

Observe that the projections $s_{\mu _{ij}\alpha }s_{\mu _{ij}\alpha }^*$ are pairwise orthogonal because $d(\mu _{ij}\alpha )=n$ for all i, j (see [2, Remark 3.2(c)]). Hence, using the above claim, expression $(\star )$ equals to

$$\begin{aligned} (\star )=1-\sum _{\lambda \in \Lambda ^n}s_\lambda s_\lambda ^*=0 (\mathrm {by ~ (KP4)}). \end{aligned}$$

Therefore, the family $\{T,Q\}$ satisfies the relations of Definition 3.1, and by the universality there exists an algebra $*$-homomorphism $\rho :{\mathcal {Q}}_R^{\textrm{alg}}(\Lambda \bowtie G)\rightarrow {\textrm{EP}}_R(G,\Lambda )$ such that $\rho (t_{(\mu ,g)})=T_{(\mu ,g)}$ and $\rho (q_X)=Q_X$ for $(\mu ,g)\in \Lambda \bowtie G$ and $X\in {\mathcal {J}}(\Lambda \bowtie G)$. It is clear that $\rho \circ \pi =\textrm{id}_{{\textrm{EP}}_R(G,\Lambda )}$ and $\pi \circ \rho =\textrm{id}_{{\mathcal {Q}}_R^{\textrm{alg}}(\Lambda \bowtie G)}$ because they fix the generators of ${\textrm{EP}}_R(G,\Lambda )$ and ${\mathcal {Q}}_R^{\textrm{alg}}(\Lambda \bowtie G)$, respectively. Consequently, $\pi$ is an isomorphism, completing the proof. $\square$

4 A Graded Uniqueness Theorem

In this section, we prove a graded uniqueness theorem for ${\textrm{EP}}_R(G,\Lambda )$ which generalizes and modifies [9, Theorem A] for self-similar k-graphs. This modification, in particular, helps us to prove Theorems 5.5 and 6.8.

Let us first recall some definitions. Let $\Gamma$ be a group and A be an algebra over a ring R. A is called $\Gamma$-graded (or briefly, graded whenever the group is clear) if there is a family of R-submodules $\{A_\gamma :\gamma \in \Gamma \}$ of A such that $A=\bigoplus _{\gamma \in \Gamma } A_\gamma$ and $A_\gamma A_{\gamma '}\subseteq A_{\gamma \gamma '}$ for all $\gamma ,\gamma '\in \Gamma$. Then each set $A_\gamma$ is called a $\gamma$-homogeneous component of A. In this case, we say an ideal I of A is $\Gamma$-graded if $I=\bigoplus _{\gamma \in \Gamma }(I\cap A_\gamma )$. Note that an ideal I of A is $\Gamma$-graded if and only if it is generated by a subset of $\bigcup _{\gamma \in \Gamma } A_\gamma$, the homogeneous elements of A.

Furthermore, if A and B are two $\Gamma$-graded algebras over R, a homomorphism $\phi :A\rightarrow B$ is said to be a graded homomorphism if $\phi (A_\gamma )\subseteq B_\gamma$ for all $\gamma \in \Gamma$. Hence the kernel of a graded homomorphism is always a graded ideal. Also, if I is a graded ideal of A, then there is a natural $\Gamma$-grading $(A_\gamma +I)_{\gamma \in \Gamma }$ on the quotient algebra A/I, and thus the quotient map $A\rightarrow A/I$ is a graded homomorphism.

Lemma 4.1

Let $(G,\Lambda )$ be a self-similar k-graph. If for every $n\in {\mathbb {Z}}^k$, we define

$$\begin{aligned} {\textrm{EP}}_R(G,\Lambda )_n:=\textrm{span}_R\bigg \{s_\mu u_{s(\mu ),g} s_\nu ^*: g\in G,~ \mu ,\nu \in \Lambda ,~ \textrm{and}~ d(\mu )-d(\nu )=n\bigg \}, \end{aligned}$$

then $({\textrm{EP}}_R(G,\Lambda )_n)_{n\in {\mathbb {Z}}^k}$ is a ${\mathbb {Z}}^k$-grading on ${\textrm{EP}}_R(G,\Lambda )$.

Proof

Consider the free $*$-algebra ${\mathbb {F}}_R(X)$ and its ideal I as in Sect. 2.2. If we define

$$\begin{aligned} \theta (S_\mu ):=d(\mu ),~ \theta (S_\mu ^*):=-d(\mu ), ~\textrm{and} ~ \theta (U_{v,g}):=0 \end{aligned}$$

for all $g\in G$, $v\in \Lambda ^0$ and $\mu \in \Lambda$, then $\theta$ induces a ${\mathbb {Z}}^k$-grading on ${\mathbb {F}}_R(X)$. Also, since the generators of I are all homogenous, I is a graded ideal. Therefore, ${\textrm{EP}}_R(G,\Lambda )\cong {\mathbb {F}}_R(X)/I$ is a ${\mathbb {Z}}^k$-graded algebra, and Proposition 2.7 concludes the result. $\square$

Theorem 4.2

(Graded Uniqueness) Let $(G,\Lambda )$ be a self-similar k-graph. Let $\phi :{\textrm{EP}}_R(G,\Lambda )\rightarrow B$ be a ${\mathbb {Z}}^k$-graded R-algebra $*$-homomorphism into a ${\mathbb {Z}}^k$-graded $*$-algebra B. If $\phi (a)\ne 0$ for every nonzero element of the form $a=\sum _{i=1}^l r_i u_{v,g_i}$ with $v\in \Lambda ^0$ and $g_i^{-1} \cdot v =g_j^{-1} \cdot v$ for $1\le i,j\le l$, then $\phi$ is injective.

Proof

For convenience, we write $A={\textrm{EP}}_R(G,\Lambda )$. Since $A=\bigoplus _{n\in {\mathbb {Z}}^k}A_n$ and $\phi$ preserves the grading, it suffices to show that $\phi$ is injective on each $A_n$. So, fix some $b\in A_n$, and assume $\phi (b)=0$. By equation (2.1), we can write

$$\begin{aligned} b=\sum _{i=1}^lr_i s_{\mu _i}u_{w_i,g_i}s_{\nu _i}^* \end{aligned}$$

(4.1)

where $w_i=s(\mu _i)=g_i \cdot s(\nu _i)$ and $d(\mu _i)-d(\nu _i)=n$ for $1\le i\le l$. Define $n'=\vee _{1\le i\le l}d(\mu _i)$. Then, for each $i\in \{1\ldots l\}$, (KP4) says that

$$\begin{aligned} s_{w_i}=\sum _{\lambda \in w_i\Lambda ^{n'-d(\mu _i)}}s_\lambda s_\lambda ^*, \end{aligned}$$

and we can write

$$\begin{aligned} s_{\mu _i}u_{w_i,g_i}s_{\nu _i}^*&=s_{\mu _i}(s_{s(\mu _i)})u_{w_i,g_i}s_{\nu _i}^*\\&=\sum s_{\mu _i}(s_\lambda s_\lambda ^*)u_{w_i,g_i}s_{\nu _i}^*\\&=\sum s_{\mu _i \lambda }\bigg (s_{\nu _i} u_{w_i,g_i}^*s_{\lambda }\bigg )^*\\&=\sum s_{\mu _i \lambda }\bigg (s_{\nu _i}u_{g_i^{-1} \cdot w_i,g_i^{-1}} s_\lambda \bigg )^*\\&=\sum s_{\mu _i \lambda }\bigg (s_{\nu _i} s_{g_i^{-1} \cdot \lambda }u_{g_i^{-1} \cdot s(\lambda ),\varphi (g_i^{-1},\lambda )} \bigg )^*\\&=\sum s_{\mu _i\lambda } u_{\varphi (g_i^{-1},\lambda )^{-1}g_i^{-1} \cdot s(\lambda ),\varphi (g_i^{-1},\lambda )^{-1}} s_{\nu _i (g_i^{-1} \cdot \lambda )}^* \end{aligned}$$

where the above summations are on $\lambda \in w_i\Lambda ^{n'-d(\mu _i)}$. So, in each term of (4.1), we may assume $d(\mu _i)=n'$ and $d(\nu _i)=n'-n$. Now, for any $1\le j\le l$, (KP3) yields that

$$\begin{aligned} s_{\mu _j}^*b s_{\nu _j}=s_{\mu _j}^*\bigg ( \sum _{i=1}^lr_i s_{\mu _i}u_{w_i,g_i}s_{\nu _i}^*\bigg ) s_{\nu _j}= \sum _{i\in [j]}r_i u_{w_i,g_i}, \end{aligned}$$

where $[j]:=\{1\le i\le l:(\mu _i,\nu _i)=(\mu _j,\nu _j)\}$. Thus

$$\begin{aligned} \phi \bigg (\sum _{i\in [j]}r_i u_{w_i,g_i} \bigg )=\phi (s_{\mu _j}^*)\phi (b)\phi (s_{\nu _j})=0 \end{aligned}$$

and hypothesis forces $\sum _{i\in [j]}r_i u_{w_i,g_i}=0$. Therefore,

$$\begin{aligned} \sum _{i\in [j]}r_i s_{\mu _i}u_{w_i,g_i}s_{\nu _i}^*=s_{\mu _j}\bigg (\sum _{i\in [j]}r_i u_{w_i,g_i} \bigg ) s_{\nu _j}^*=0. \end{aligned}$$

Since the index set $\{1,\ldots ,l\}$ is a disjoint union of [j]’s, we obtain $b=0$. It follows that $\phi$ is injective. $\square$

5 ${\textrm{EP}}_R(G,\Lambda )$ as a Steinberg Algebra

In this section, we want to prove an Steinberg algebra model for ${\textrm{EP}}_R(G,\Lambda )$. Although our result will be the k-graph generalization of [9, Theorem B], note that our proof relies on the graded uniqueness theorem, Theorem 4.2, and is completely different from that of [9, Theorem B]. This gives us a much easier and shorter proof.

Let us first review some terminology about groupoids; see [19] for more details. A groupoid is a small category ${\mathcal {G}}$ with inverses. For each $\alpha \in {\mathcal {G}}$, we may define the range $r(\alpha ):=\alpha \alpha ^{-1}$ and the source $s(\alpha ):=\alpha ^{-1}\alpha$ satisfying $r(\alpha )\alpha =\alpha =\alpha s(\alpha )$. It follows that for every $\alpha ,\beta \in {\mathcal {G}}$, the composition $\alpha \beta$ is well defined if and only if $s(\alpha )=r(\beta )$. The unit space of ${\mathcal {G}}$ is ${\mathcal {G}}^{(0)}:=\{\alpha ^{-1}\alpha : \alpha \in {\mathcal {G}}\}$. Throughout the paper we work with topological groupoids, which are ones equipped with a topology such that the maps r and s are continuous. Then a bisection is a subset $B\subseteq {\mathcal {G}}$ such that both restrictions $r|_B$ and $s|_B$ are homeomorphisms. In case ${\mathcal {G}}$ has a basis of compact open bisections, ${\mathcal {G}}$ is called an ample groupoid.

Let $(G,\Lambda )$ be a self-similar k-graph. We also recall the groupoid ${\mathcal {G}}_{G,\Lambda }$ introduced in [15]. Let $C({\mathbb {N}}^k,G)$ be the group of all maps form ${\mathbb {N}}^k$ to G with the pointwise multiplication. For $f,g\in C({\mathbb {N}}^k,G)$, define the equivalence relation $f\sim g$ in case there exists $n_0\in {\mathbb {N}}^k$ such that $f(n)=g(n)$ for all $n\ge n_0$. Write $Q({\mathbb {N}}^k,G):=C({\mathbb {N}}^k,G)/\sim$. Also, for each $z\in {\mathbb {Z}}^k$, let ${\mathcal {T}}_z: C({\mathbb {N}}^k,G)\rightarrow C({\mathbb {N}}^k,G)$ be the automorphism defined by

$$\begin{aligned} {\mathcal {T}}_z(f)(n)=\left\{ \begin{array}{ll} f(n-z) &{} n-z\ge 0 \\ e_G &{} \textrm{otherwise} \end{array} \right. \qquad (f\in C({\mathbb {N}}^k,G),~ n\in {\mathbb {N}}^k). \end{aligned}$$

Then ${\mathcal {T}}_z$ induces an automorphism, denoted again by ${\mathcal {T}}_z$, on $Q({\mathbb {N}}^k,G)$, which is ${\mathcal {T}}_z([f])=[{\mathcal {T}}_z(f)]$. So, ${\mathcal {T}}:{\mathbb {Z}}^k\rightarrow \textrm{Aut}Q({\mathbb {N}}^k,G)$ is a homomorphism and we consider the semidirect product group $Q({\mathbb {N}}^k,G)\rtimes _{{\mathcal {T}}} {\mathbb {Z}}^k$.

Note that for every $g\in G$ and $x\in \Lambda ^\infty$, one may define $\varphi (g,x)\in C({\mathbb {N}}^k,G)$ by

$$\begin{aligned} \varphi (g,x)(n):=\varphi (g,x(0,n)) \qquad (n\in {\mathbb {N}}^k). \end{aligned}$$

Moreover, [15, Lemma 3.7] says that there exists a unique action $G\curvearrowright \Lambda ^\infty$ by defining

$$\begin{aligned} (g\cdot x)(m,n):=\varphi (g,x(0,m)) \cdot x(m,n) \qquad ((m,n)\in \Omega _k) \end{aligned}$$

for every $g\in G$ and $x\in \Lambda ^\infty$.

Definition 5.1

A self-similar k-graph $(G,\Lambda )$ is said to be pseudo-free if for any $g\in G$ and $\mu \in \Lambda$, $g\cdot \mu =\mu$ and $\varphi (g,\mu )=e_G$ imply $g=e_G$.

According to [15, Lemma 5.6], in case $(G,\Lambda )$ is pseudo-free, then we have

$$\begin{aligned} g\cdot \mu =h \cdot \mu ~~ \textrm{and} ~~ \varphi (g,\mu )=\varphi (h,\mu ) ~~ \Longrightarrow ~~ g=h \end{aligned}$$

for every $g,h\in G$ and $\mu \in \Lambda$.

Definition 5.2

Associated to $(G,\Lambda )$ we define the subgroupoid

$$\begin{aligned} {\mathcal {G}}_{G,\Lambda }&:=\left\{ \bigg (\mu (g\cdot x);{\mathcal {T}}_{d(\mu )}([\varphi (g,x)]),d(\mu )-d(\nu );\nu x\bigg ) :g\in G,\right. \\&\qquad \left. ~\mu ,\nu \in \Lambda ,~ s(\mu )=g\cdot s(\nu ) \right\} \end{aligned}$$

of $\Lambda ^\infty \times \bigg (Q({\mathbb {N}}^k,G)\rtimes _{{\mathcal {T}}} {\mathbb {Z}}^k\bigg ) \times \Lambda ^\infty$ with the range and source maps

$$\begin{aligned} r(x;[f],n-m;y)=x \quad \textrm{ and }\quad s(x;[f],n-m;y)=y. \end{aligned}$$

Note that if we set

$$\begin{aligned} Z(\mu ,g,\nu ):=\left\{ \bigg (\mu (g\cdot x);{\mathcal {T}}_{d(\mu )}([\varphi (g,x)]),d(\mu )-d(\nu );\nu x\bigg ):x\in s(\nu )\Lambda ^\infty \right\} , \end{aligned}$$

then the basis

$$\begin{aligned} {\mathcal {B}}_{G,\Lambda }:=\left\{ Z(\mu ,g,\nu ):\mu ,\nu \in \Lambda ,g\in G,s(\mu )=g\cdot s(\nu )\right\} \end{aligned}$$

induces a topology on ${\mathcal {G}}_{G,\Lambda }$. In case $(G,\Lambda )$ is pseudo-free, [16, Proposition 3.11] shows that ${\mathcal {G}}_{G,\Lambda }$ is a Hausdorff groupoid with compact open base ${\mathcal {B}}_{G,\Lambda }$.

Definition 5.3

Let $(G,\Lambda )$ be a pseudo-free self-similar k-graph and R a unital commutative $*$-ring. Then the Steinberg algebra associated to $(G,\Lambda )$ is the R-algebra

$$\begin{aligned} A_R({\mathcal {G}}_{G,\Lambda }):=\textrm{span}_R\{1_B: B ~ \mathrm {is ~ a ~ compact ~ open ~ bisection}\} \end{aligned}$$

endowed with the pointwise addition, the multiplication $fg(\gamma ):=\sum _{\alpha \beta =\gamma }f(\alpha ) g(\beta )$, and the involution $f^*(\gamma ):=f(\gamma ^{-1})^*$ for all $\gamma \in {\mathcal {G}}_{G,\Lambda }$.

To prove Theorem 5.5, we need the following lemma.

Lemma 5.4

Let $(G,\Lambda )$ be a pseudo-free self-similar k-graph. Let $v,w\in \Lambda ^0$ and $g,h\in G$ with $g\cdot v=h\cdot v=w$. Then $Z(v,g,w)\cap Z(v,h,w)= \emptyset$ whenever $g\ne h$.

Proof

Suppose that $(g\cdot x;[\varphi (g,x)],0;x)=(h \cdot y,[\varphi (h,y)],0;y)\in Z(v,g,w)\cap Z(v,h,w)$ where $x,y\in Z(w)$. Then $y=x$, $g\cdot x=h \cdot x$ and $[\varphi (g,x)]=[\varphi (h,x)]$. Since $(G,\Lambda )$ is pseudo-free, [15, Corollary 5.6] implies that $g=h$. $\square$

Theorem 5.5

Let $(G,\Lambda )$ be a pseudo-free self-similar k-graph. Then there is a (unique) $*$-algebra isomorphism $\phi :{\textrm{EP}}_R(G,\Lambda )\rightarrow A_R({\mathcal {G}}_{G,\Lambda })$ such that

$$\begin{aligned} \phi (s_\mu )=1_{Z(\mu , e_G, s(\mu ))} \quad \textrm{ and }\quad \phi (u_{v,g})=1_{Z(v,g,g^{-1} \cdot v )} \end{aligned}$$

for every $\mu \in \Lambda$, $v\in \Lambda ^0$, and $g\in G$. In particular, the elements $rs_\mu$ and $ru_{v,g}$ with $r\in R{\setminus }\{0\}$ are all nonzero.

Proof

For each $v\in \Lambda ^0$, $\mu \in \Lambda$ and $g\in G$, define

$$\begin{aligned} S_\mu :=1_{Z(\mu , e_G, s(\mu ))} \quad \textrm{ and }\quad U_{v,g}:=1_{Z(v,g,g^{-1} \cdot v )}. \end{aligned}$$

Since $S_\mu ^*=1_{Z(\mu ,e_G,s(\mu ))^{-1}}=1_{Z(s(\mu ),e_G,\mu )}$ and $U_{v,g}^*=1_{Z(v,g,g^{-1} \cdot v )^{-1}}=1_{Z(g^{-1} \cdot v,g^{-1},v)}$, a long but straightforward computation shows that $\{S_\mu ,U_{v,g}\}$ is a $(G,\Lambda )$-family in $A_R({\mathcal {G}}_{G,\Lambda })$. Then, by the universal property, such $*$-homomorphism $\phi$ exists.

[16, Proposition 3.11] says that ${\mathcal {G}}_{G,\Lambda }$ is ample with compact open base ${\mathcal {B}}_{G,\Lambda }$. Since each element $Z(\mu ,g,\nu )$ of ${\mathcal {B}}_{G,\Lambda }$ can be written as

$$\begin{aligned} Z(\mu ,g,\nu )=Z(\mu ,e_G,s(\mu ))Z(s(\mu ),g,s(\nu ))Z(\nu ,e_G,s(\nu ))^{-1}, \end{aligned}$$

$\phi$ is surjective.

We will show the injectivity of $\phi$ by applying the graded uniqueness theorem. Note that the continuous 1-cocycle $c:{\mathcal {G}}_{G,\Lambda }\rightarrow {\mathbb {Z}}^k$, defined by $c(\mu (g\cdot x);[f],d(\mu )-d(\nu );\nu x):=d(\mu )-d(\nu )$, induces a ${\mathbb {Z}}^k$-grading on $A_R({\mathcal {G}}_{G,\Lambda })$. Also, $\phi$ preserves the ${\mathbb {Z}}^k$-grading because it does on the generators. Now, to apply Theorem 4.2, we assume $\phi (a)=0$ for an element of the form $a=\sum _{i=1}^l r_i u_{v,g_i}$ with $g_i^{-1} \cdot v =g_j^{-1} \cdot v$ for $1\le i,j\le l$. We may also assume that the $g_i$’s are distinct (otherwise, combine the terms with same $g_i$’s). We then have

$$\begin{aligned} \phi (a)=\sum _{i=1}^l r_i1_{Z(v,g_i,g_i^{-1} \cdot v )}=0. \end{aligned}$$

Lemma 5.4 says that the bisections $Z(v,g_i,g_i^{-1} \cdot v )$ are pairwise disjoint. Hence, for each i, if we pick some $\alpha \in Z(v,g_i,g_i^{-1} \cdot v )$, then $r_i=\phi (a)(\alpha )=0$. Therefore $a=0$, and Theorem 4.2 concludes that $\phi$ is injective. We are done. $\square$

Combining [20, Theorem 6.7], [15, Theorem 5.9], and Theorem 5.5 gives the next corollary. (Although in [15] it is supposed $|\Lambda ^0|<\infty$, but [15, Theorem 5.9] holds also for $\Lambda$ with infinitely many vertices.)

Corollary 5.6

Let $(G,\Lambda )$ be a pseudo-free self-similar k-graph over an amenable group G. Then the complex algebra $\textrm{EP}_{\mathbb {C}}(G,\Lambda )$ is a dense subalgebra of ${\mathcal {O}}_{G,\Lambda }$ introduced in [16].

In the following, we see that the Kumjian-Pask algebra $\textrm{KP}_R(\Lambda )$ from [2] can be embedded in ${\textrm{EP}}_R(G,\Lambda )$.

Corollary 5.7

Let $(G,\Lambda )$ be a pseudo-free self-similar k-graph. Let the Kumjian-Pask algebra $\textrm{KP}_R(\Lambda )$ be generated by a Kumjian-Pask $\Lambda$-family $\{t_\mu :\mu \in \Lambda \}$. Then the map $t_\mu \mapsto s_\mu$ embeds $\textrm{KP}_R(\Lambda )$ into ${\textrm{EP}}_R(G,\Lambda )$ as a $*$-subalgebra.

Proof

We know that $\textrm{KP}_R(\Lambda )$ is ${\mathbb {Z}}^k$-graded by the homogenous components

$$\begin{aligned} \textrm{KP}_R(\Lambda )_n:=\textrm{span}_R \left\{ t_\mu t_\nu ^*:\mu ,\nu \in \Lambda , d(\mu )-d(\nu )=n\right\} . \end{aligned}$$

for all $n\in {\mathbb {Z}}^k$. Then, the universal property of Kumjian-Pask algebras gives a graded $*$-algebra homomorphism $\phi :\textrm{KP}_R(\Lambda )\rightarrow {\textrm{EP}}_R(G,\Lambda )$ such that $\phi (t_\mu ):=s_\mu$ and $\phi (t_\mu ^*):=s_\mu ^*$ for every $\mu \in \Lambda$. Moreover, Theorem 5.5 shows that $\phi (rt_\mu )=rs_\mu \ne 0$ for all $r\in R{\setminus } \{0\}$ and $\mu \in \Lambda$. Therefore, the graded uniqueness theorem for Kumjian-Pask algebras [2, Theorem 4.1] implies that $\phi$ is injective. $\square$

Definition 5.8

Let ${\mathcal {G}}$ be a topological groupoid. We say that ${\mathcal {G}}$ is topologically principal if the set of units with trivial isotropy group, that is $\{u\in {\mathcal {G}}^{(0)}: s^{-1}(u)\cap r^{-1}(u)=\{u\}\}$, is dense in ${\mathcal {G}}^{(0)}$.

The analogue of the topologically principal property for self-similar k-graphs is G-aperiodicity (see [15, Proposition 6.5]).

Definition 5.9

Let $(G,\Lambda )$ be a self-similar k-graph. Then $\Lambda$ is said to be G-aperiodic if for every $v\in \Lambda ^0$, there exists $x\in v\Lambda ^\infty$ with the property that

$$\begin{aligned} x(p,\infty )=g\cdot x(q,\infty ) ~~\Longrightarrow ~~ g=e_G ~~ \textrm{and} ~~ p=q \qquad (\forall g\in G,~ \forall p,q\in {\mathbb {N}}^k). \end{aligned}$$

Theorem 5.10

(The Cuntz–Krieger uniqueness) Let $(G,\Lambda )$ be a pseudo-free self-similar k-graph. Let $(G,\Lambda )$ be also G-aperiodic. Suppose that $\phi :{\textrm{EP}}_R(G,\Lambda )\rightarrow A$ is a $*$-algebra homomorphism from ${\textrm{EP}}_R(G,\Lambda )$ into a $*$-algebra A such that $\phi (rs_v)\ne 0$ for all $0\ne r\in R$ and $v\in \Lambda ^0$. Then $\phi$ is injective.

Proof

First note that ${\mathcal {G}}_{G,\Lambda }$ is a Hausdorff ample groupoid by [16, Proposition 3.11], and that ${\mathcal {B}}_{G,\Lambda }$ is a basis for ${\mathcal {G}}_{G,\Lambda }$ consisting compact open bisections. Also, [16, Lemma 3.12] says that ${\mathcal {G}}_{G,\Lambda }$ is topologically principal (so is effective in particular). So, we may apply [5, Theorem 3.2].

Denote by $\psi :{\textrm{EP}}_R(G,\Lambda )\rightarrow A_R({\mathcal {G}}_{G,\Lambda })$ the isomorphism of Theorem 5.5. If on the contrary $\phi$ is not injective, then neither is ${\widetilde{\phi }}:=\phi \circ \psi ^{-1}:A_R({\mathcal {G}}_{G,\Lambda })\rightarrow A$. Thus, by [5, Theorem 3.2], there exists a compact open subset $K\subseteq {\mathcal {G}}_{G,\Lambda }^{(0)}$ and $r\ne 0$ such that ${\widetilde{\phi }}(r1_K)=0$. Since K is open, there is a unit $U=Z(\mu ,e_G,\mu )\in {\mathcal {B}}_{G,\Lambda }$ such that $U\subseteq K$. So we get

$$\begin{aligned} \phi (rs_\mu s_\mu ^*)={\widetilde{\phi }}(r1_U)={\widetilde{\phi }}(r1_{U\cap K})={\widetilde{\phi }}(r1_K) {\widetilde{\phi }}(1_U)=0, \end{aligned}$$

and hence

$$\begin{aligned} \phi (rs_{s(\mu )})=\phi (s_\mu ^*)\phi (rs_\mu s_\mu ^*)\phi (s_\mu )=0. \end{aligned}$$

This contradicts the hypothesis, and therefore $\phi$ is injective. $\square$

6 Ideal Structure

By an ideal we mean a two-sided and self-adjoint one. In this section, we characterize basic, ${\mathbb {Z}}^k$-graded and diagonal-invariant ideals of ${\textrm{EP}}_R(G,\Lambda )$, which are exactly all basic $Q({\mathbb {N}}^k,G)\rtimes _{{\mathcal {T}}}{\mathbb {Z}}^k$-graded ones.

Let $(G,\Lambda )$ be a pseudo-free self-similar k-graph. Since ${\mathcal {G}}_{G,\Lambda }$ is a Hausdorff ample groupoid [15, Theorem 5.8], ${\mathcal {G}}_{G,\Lambda }^{(0)}$ is both open and closed, and for every $f\in A_R({\mathcal {G}}_{G,\Lambda })$ the restricted function $f|_{{\mathcal {G}}_{G,\Lambda }^{(0)}}=f\chi _{{\mathcal {G}}_{G,\Lambda }^{(0)}}$ lies again in $A_R({\mathcal {G}}_{G,\Lambda })$. Then $A_R({\mathcal {G}}_{G,\Lambda }^{(0)})$ is a $*$-subalgebra of $A_R({\mathcal {G}}_{G,\Lambda })$ and there is a conditional expectation ${\mathcal {E}}:A_R({\mathcal {G}}_{G,\Lambda })\rightarrow A_R({\mathcal {G}}_{G,\Lambda }^{(0)})$ defined by ${\mathcal {E}}(f)=f|_{{\mathcal {G}}_{G,\Lambda }^{(0)}}$ for $f\in A_R({\mathcal {G}}_{G,\Lambda })$. Let $D:=\textrm{span}_R\{s_\mu s_\mu ^*:\mu \in \Lambda \}$ be the diagonal of ${\textrm{EP}}_R(G,\Lambda )$. In light of Theorem 5.5, it is easy to check that the expectation is ${\mathcal {E}}:{\textrm{EP}}_R(G,\Lambda )\rightarrow D$ defined by

$$\begin{aligned} {\mathcal {E}}\bigg (s_\mu u_{s(\mu ),g}s_\nu ^*\bigg )=\delta _{\mu ,\nu }\delta _{g,e_G} s_\mu s_\mu ^* \qquad (\mu ,\nu \in \Lambda ,~s(\mu )=g\cdot s(\nu )). \end{aligned}$$

Definition 6.1

An ideal I of ${\textrm{EP}}_R(G,\Lambda )$ is called diagonal-invariant whenever ${\mathcal {E}}(I)\subseteq I$. Also, I is said to be basic if $rs_v\in I$ implies $s_v\in I$ for all $v\in \Lambda ^0$ and $r\in R{\setminus } \{0\}$.

Definition 6.2

Let $(G,\Lambda )$ be a self-similar k-graph. A subset $H\subseteq \Lambda ^0$ is called

• (1) G-hereditary if $r(\mu )\in H ~~ \Longrightarrow ~~ g\cdot s(\mu )\in H$ for all $g\in G$ and $\mu \in \Lambda$;

• (2) G-saturated if $v\in \Lambda ^0$ and $s(v\Lambda ^n) \subseteq H~~ \mathrm {for ~ some ~} n\in {\mathbb {N}}^k ~~ \Longrightarrow ~~ v\in H$.

In the following, given any $H\subseteq \Lambda ^0$, we denote by $I_H$ the ideal of ${\textrm{EP}}_R(G,\Lambda )$ generated by $\{s_v:v\in H\}$. Also, for each ideal I of ${\textrm{EP}}_R(G,\Lambda )$, we define $H_I:=\{v\in \Lambda ^0: s_v\in I\}$.

To prove Theorem 6.8 we need some structural lemmas about the ideals $I_H$ and associated quotients ${\textrm{EP}}_R(G,\Lambda )/I_H$.

Lemma 6.3

If I is an ideal of ${\textrm{EP}}_R(G,\Lambda )$, then $H_I:=\{v\in \Lambda ^0:s_v\in I\}$ is a G-saturated G-hereditary subset of $\Lambda ^0$.

Proof

The proof is straightforward. $\square$

Lemma 6.4

Let H be a G-saturated G-hereditary subset of $\Lambda ^0$ and $I_H$ the ideal of ${\textrm{EP}}_R(G,\Lambda )$ generated by $\{s_v:v\in H\}$. Then we have

$$\begin{aligned} I_H=\textrm{span}_R\left\{ s_\mu u_{s(\mu ),g}s_\nu ^*:g\in G, ~ s(\mu )=g\cdot s(\nu )\in H\right\} , \end{aligned}$$

(6.1)

and $I_H$ is a ${\mathbb {Z}}^k$-graded diagonal-invariant ideal.

Proof

Denote by J the right-hand side of (6.1). The identity

$$\begin{aligned} s_\mu u_{s(\mu ),g}s_\nu ^*=s_\mu (s_{s(\mu )})u_{s(\mu ),g}s_\nu ^* \end{aligned}$$

yields $J\subseteq I_H$. Also, using the description of ${\textrm{EP}}_R(G,\Lambda )$ in Proposition 2.7, it is straightforward to check that J is an ideal of ${\textrm{EP}}_R(G,\Lambda )$. So, by $s_v=s_vu_{v,e_G}s_v^*$, J contains all generators of $I_H$, and we have proved (6.1).

Now, (6.1) says that $I_H$ is spanned by its homogenous elements, hence it is a graded ideal. Moreover, let $a=\sum _{i=1}^l s_{\mu _i}u_{s(\mu _i),g_i}s_{\nu _i}^*\in I_H$ such that $s(\mu _i)=g\cdot s(\nu _i)\in H$. Then, in particular, each term of a with $g_i=e_G$ belongs to $I_H$. Therefore, ${\mathcal {E}}(a)\in I_H$, and $I_H$ is diagonal-invariant. $\square$

Let H be a G-saturated G-hereditary subset of $\Lambda ^0$ and consider the k-subgraph $\Lambda \setminus \Lambda H$. Then the restricted action $G\curvearrowright \Lambda \setminus \Lambda H$ is well defined, and hence $(G,\Lambda {\setminus } \Lambda H, \varphi |_{G\times \Lambda {\setminus } \Lambda H})$ is also a self-similar k-graph. So we have:

Lemma 6.5

Let $(G,\Lambda )$ be a pseudo-free self-similar k-graph. If H is a G-saturated G-hereditary subset of $\Lambda ^0$, then $(G,\Lambda \setminus \Lambda H)$ is a pseudo-free self-similar k-graph.

Proof

The proof is straightforward. $\square$

Lemma 6.6

Let H be a G-saturated G-hereditary subset of $\Lambda ^0$. For every $v\in \Lambda ^0$ and $r\in R{\setminus }\{0\}$, $rs_v\in I_H$ implies $v\in H$.

Proof

Let $\{t_\mu , w_{v,g}\}$ be the generators of $\textrm{EP}_R(G,\Lambda \setminus \Lambda H)$. If we define

$$\begin{aligned} S_\mu :=\left\{ \begin{array}{ll} t_\mu &{} s(\mu )\notin H \\ 0 &{} \textrm{otherwise} \end{array} \right. \quad \text { and } \quad U_{v,g}:=\left\{ \begin{array}{ll} w_{v,g} &{} v\notin H \\ 0 &{} \textrm{otherwise,} \end{array} \right. \end{aligned}$$

then $\{S_\mu ,U_{v,g}\}$ is a $(G,\Lambda )$-family in $\textrm{EP}_R(G,\Lambda \setminus \Lambda H)$, and by the universality, there is a $*$-homomorphism $\psi :{\textrm{EP}}_R(G,\Lambda )\rightarrow \textrm{EP}_R(G,\Lambda {\setminus }\Lambda H)$ such that $\psi (s_\mu )=S_\mu$ and $\psi (u_{v,g})=U_{v,g}$ for all $\mu \in \Lambda$, $v\in \Lambda ^0$ and $g\in G$. Since $\psi (s_v)=0$ for every $v\in H$, we have $I_H\subseteq \ker \psi$. On the other hand, Theorem 5.5 implies that all $\psi (rs_v)=rt_v$ are nonzero for $v\in \Lambda ^0{\setminus } H$ and $r\in R{\setminus }\{0\}$.

Now assume $rs_v\in I_H$ for some $v\in \Lambda ^0$ and $r\in R{\setminus }\{0\}$. If $v\in \Lambda ^0{\setminus } H$, then $\psi (rs_v)=rt_v\ne 0$, and we get $rs_v\notin \ker \psi \supseteq I_H$, a contradiction. $\square$

In fact, Lemma 6.6 says that $I_H$ is a basic ideal with $H_{I_H}=H$ for every G-saturated G-hereditary subset H of $\Lambda ^0$.

Proposition 6.7

Let H be a G-saturated G-hereditary subset of $\Lambda ^0$. Let $\{t_\mu ,w_{v,g}\}$ be the $(G,\Lambda \setminus \Lambda H)$-family generating $\textrm{EP}_R(G,\Lambda \setminus \Lambda H)$. Then the map $\psi : \textrm{EP}_R(G,\Lambda {\setminus } \Lambda H) \rightarrow {\textrm{EP}}_R(G,\Lambda )/I_H$ defined by

$$\begin{aligned} \psi (t_\mu w_{s(\mu ),g} t_\nu ^*):=s_\mu u_{s(\mu ),g} s_\nu ^* +I_H \qquad (\mu ,\nu \in \Lambda \setminus \Lambda H,~ g\in G) \end{aligned}$$

is an (R-algebra) $*$-isomorphism.

Proof

If we set $T_\mu := s_\mu +I_H$ and $W_{v,g}:=u_{v,g}+I_H$ for every $v\in \Lambda ^0,~\mu \in \Lambda$, and $g\in G$, then $\{T_\mu ,W_{v,g}\}$ is a $(G,\Lambda \setminus \Lambda H)$-family in ${\textrm{EP}}_R(G,\Lambda )/I_H$ (the relations of Definition 2.5 for $\{T_\mu , W_{v,g}\}$ immediately follow from those for $\{s_\mu ,u_{v,g}\}$). So, the universality of $\textrm{EP}_R(G,\Lambda \setminus \Lambda H)$ gives such $*$-homomorphism $\psi$. Note that $s_\mu \in I_H$ for each $\mu \in \Lambda H$ by (6.1), which gives the surjectivity of $\psi$.

To prove the injectivity, we apply the graded uniqueness theorem, Theorem 4.2. First, since $I_H$ is a ${\mathbb {Z}}^k$-graded ideal, ${\textrm{EP}}_R(G,\Lambda )/I_H$ has a natural ${\mathbb {Z}}^k$-grading and $\psi$ is a graded homomorphism. Thus, we fix an element in $\textrm{EP}_R(G,\Lambda \setminus \Lambda H)$ of the form $a=\sum _{i=1}^l r_iw_{v,g_i}$ such that $v\in \Lambda ^0{\setminus } H$ and $g_i^{-1} \cdot v =g_j^{-1} \cdot v$ for all $1\le i,j\le l$. Without loss of generality, we may also suppose that the $g_i$’s are distinct. If $\psi (a)=0$, then $\psi (a)=\sum _{i=1}^l r_iu_{v,g_i}+I_H=I_H$ and $\sum _{i=1}^l r_iu_{v,g_i}\in I_H$. Thus, for each $1\le j\le l$, we have

$$\begin{aligned} \bigg (\sum _{i=1}^l r_iu_{v,g_i}\bigg )u_{g_j^{-1} \cdot v, g_j^{-1}}=\sum _{i=1}^l r_iu_{v,g_ig_j^{-1}}\in I_H \qquad (\mathrm {by ~ eq.~ (5) ~ in~ Definition~ 2.5}) \end{aligned}$$

and since $I_H$ is diagonal-invariant,

$$\begin{aligned} r_js_v=r_ju_{v,e_G}={\mathcal {E}}\left( \sum _{i=1}^l r_iu_{v,g_ig_j^{-1}}\right) \in I_H. \end{aligned}$$

As $v\notin H$, Lemma 6.6 forces $r_j=0$ for each $1\le j\le l$, hence $a=0$. Now Theorem 4.2 implies that $\psi$ is an isomorphism. $\square$

Theorem 6.8

Let $(G,\Lambda )$ be pseudo-free self-similar k-graph. Then $H\mapsto I_H$ is a one-to-one correspondence between G-saturated G-hereditary subsets of $\Lambda ^0$ and basic, ${\mathbb {Z}}^k$-graded and diagonal-invariant ideals of ${\textrm{EP}}_R(G,\Lambda )$, with inverse $I\mapsto H_I$.

Proof

The injectivity of $H\mapsto I_H$ follows from Lemma 6.6. Indeed, if $I_H=I_K$ for G-saturated G-hereditary subsets $H,K\subseteq \Lambda ^0$, then Lemma 6.6 yields that $H=H_{I_H}=H_{I_K}=K$.

To see the surjectivity, we take a basic, ${\mathbb {Z}}^k$-graded and diagonal-invariant ideal I of ${\textrm{EP}}_R(G,\Lambda )$, and then prove $I=I_{H_I}$. Write $J:=I_{H_I}$ for convenience. By Proposition 6.7 we may consider $\textrm{EP}_R(G,\Lambda \setminus \Lambda H_I)\cong {\textrm{EP}}_R(G,\Lambda )/J$ as a $*$-R-algebra. Let $\{s_\mu ,u_{v,g}\}$ and $\{t_\mu ,w_{v,g}\}$ be the generators of ${\textrm{EP}}_R(G,\Lambda )$ and ${\textrm{EP}}_R(G,\Lambda )/J$, respectively. Since $J\subseteq I$, we may define the quotient map $q:{\textrm{EP}}_R(G,\Lambda )/J\rightarrow {\textrm{EP}}_R(G,\Lambda )/I$ such that

$$\begin{aligned} q(t_\mu )=s_\mu +I \quad \textrm{ and }\quad q(w_{v,g})=u_{v,g}+I \end{aligned}$$

for all $\mu \in \Lambda$, $v\in \Lambda ^0$ and $g\in G$. Notice that q preserves the grading because I is a ${\mathbb {Z}}^k$-graded ideal. So, we can apply Theorem 4.2 to show that q is an isomorphism. To do this, fix an element of the form $a=\sum _{i=1}^l r_iw_{v,g_i}$ with $v\in \Lambda ^0{\setminus } H_I$ such that $g_i^{-1} \cdot v =g_j^{-1} \cdot v$ for $1\le i,j\le l$ and $q(a)=0$. Then $\sum _{i=1}^l r_iu_{v,g_i}\in I$. As before, we may also assume that the $g_i$’s are distinct. Thus, for each $1\le j\le l$, we have

$$\begin{aligned} b_j:=\bigg (\sum _{i=1}^l r_iu_{v,g_i}\bigg )\bigg (u_{g_j^{-1} \cdot v,g_j^{-1}}\bigg )=\sum _{i=1}^l r_iu_{v,g_ig_j^{-1}}\in I. \end{aligned}$$

Since I is diagonal-invariant and basic, the case $r_j\ne 0$ yields $r_js_v=r_ju_{v,e_G}={\mathcal {E}}(b_j)\in I$, and thus $s_v\in I$ and $v\in H_I$, which contradicts the choice of v. It follows that $r_j$’s are all zero, and hence $a=0$. Now Theorem 4.2 implies that q is injective, or equivalently $I=J=I_{H_I}$ as desired. $\square$

In the end, we remark the following about $Q({\mathbb {N}}^k,G)\rtimes _{{\mathcal {T}}}{\mathbb {Z}}^k$-graded ideals of ${\textrm{EP}}_R(G,\Lambda )$.

Remark 6.9

Let $(G,\Lambda )$ be a pseudo-free self-similar k-graph and ${\mathcal {G}}_{G,\Lambda }$ be the associated groupoid. Denote by $\Gamma :=Q({\mathbb {N}}^k,G)\rtimes _{{\mathcal {T}}}{\mathbb {Z}}^k$ the group introduced in Sect. 5. If we define $c:{\mathcal {G}}_{G,\Lambda }\rightarrow \Gamma$ by $c(x;\gamma ;y):=\gamma$, then c is a cocycle on ${\mathcal {G}}_{G,\Lambda }$ because $c(\alpha \beta )=c(\alpha )*_\Gamma c(\beta )$ for all $\alpha ,\beta \in {\mathcal {G}}_{G,\Lambda }$ with $s(\alpha )=r(\beta )$. Hence, it induces a $\Gamma$-grading on $A_R({\mathcal {G}}_{G,\Lambda })={\textrm{EP}}_R(G,\Lambda )$ with the homogenous components

$$\begin{aligned} A_\gamma :=\textrm{span}_R\{1_V:V\subseteq c^{-1}(\gamma ) \mathrm {~is~a~compact~open~bisection~}\} \end{aligned}$$

(see [6, Proposition 5.1] for example). By a similar argument as in [6, §6.5] and combining Theorem 5.5 and [6, Theorem 5.3], we may obtain that the ideals of the form $I_H$, described in Theorem 6.8 above, are precisely the basic, $\Gamma$-graded ideals of ${\textrm{EP}}_R(G,\Lambda )$.