Non-simple purely infinite Steinberg Algebras with applications to Kumjian-Pask algebras

TL;DR

This paper characterizes proper pure infiniteness in Steinberg algebras and Kumjian-Pask algebras, providing graph-theoretic criteria and linking algebraic properties to $C^*$-algebraic pure infiniteness.

Contribution

It offers a characterization of proper pure infiniteness for Steinberg algebras and establishes equivalences for Kumjian-Pask algebras, connecting algebraic and $C^*$-algebraic properties.

Findings

Proper pure infiniteness characterized for Steinberg algebras.

Equivalence of pure infiniteness notions for Kumjian-Pask algebras.

Pure infiniteness of Kumjian-Pask algebra implies pure infiniteness of associated $C^*$-algebra.

Abstract

In this paper, we characterize properly purely infinite Steinberg algebras $A_K(\mathcal{G})$ for strongly effective, ample Hausdorff groupoids $\mathcal{G}$. As an application, when $\Lambda$ is a strongly aperiodic $k$-graph, we show that the notions of pure infiniteness and proper pure infiniteness are equivalent for the Kumjian-Pask algebra $\text{KP}_K(\Lambda)$, which may be determined by the proper infiniteness of vertex idempotents. In particular, for unital cases, we give simple graph-theoretic criteria for the (proper) pure infiniteness of $\text{KP}_K(\Lambda)$. Furthermore, since the complex Steinberg algebra $A_\mathbb{C}(\mathcal{G})$ is a dense subalgebra of the reduced groupoid $C^*$-algebra $C^*_r(\mathcal{G})$, we focus on the problem that "when does the proper pure infiniteness of $A_\mathbb{C}(\mathcal{G})$ imply that of $C^*_r(\mathcal{G})$ in the $C^*$-sense?". In particular, we show that if the Kumjian-Pask algebra $\mathrm{KP}_{\mathbb{C}}(\Lambda)$ is purely infinite, then so is $C^*(\Lambda)$ in the sense of Kirchberg-R{\o}rdam.

Equations

$$AB:=\{\alpha \beta :\alpha \in A,\beta \in B,s(\alpha )=r(\beta )\}.$$

$$\mathrm{Iso}(\mathcal{G}):=\{\alpha \in \mathcal{G}:s(\alpha )=r(\alpha )\}.$$

$$\mathcal{G}{|}_U:=\{\alpha \in \mathcal{G}:s(\alpha ),r(\alpha )\in U\}.$$

$$(fg)(\alpha )=\sum _{r(\beta )=r(\alpha )}f(\beta )g({\beta }^{-1}\alpha )(f,g\in A_K(\mathcal{G})\mathrm{and}\alpha \in \mathcal{G}).$$

$$A_K(\mathcal{G})={\mathrm{span}}_K\left\{1_B:B\mathrm{is}\mathrm{a}\mathrm{compact}\mathrm{open}\mathrm{bisection}\right\},$$

$$I_U:=\left\{f\in A_K(\mathcal{G}):\mathrm{supp}f\subseteq {\mathcal{G}}_U\right\},$$

$$HE:=\{\lambda \in E:r(\lambda )\in H\}\mathrm{and}EH:=\{\lambda \in E:s(\lambda )\in H\},$$

$${\Lambda }^{\le n}:=\{\lambda \in \Lambda :d(\lambda )\le n,\mathrm{and}d(\lambda )+e_i\le n⟹s(\lambda ){\Lambda }^{e_i}=\varnothing \}.$$

$$\mathrm{MCE}(\mu ,\nu ):=\left\{\lambda \in {\Lambda }^{d(\mu )\vee d(\nu )}:\mu \alpha =\lambda =\nu \beta \mathrm{for}\mathrm{some}\alpha ,\beta \in \Lambda \right\}.$$

$$I_H:={\mathrm{span}}_K\left\{s_{\lambda }{s}_{{\mu }^{\ast }}:\lambda ,\mu \in \Lambda ,s(\lambda )=s(\mu )\in H\right\}$$

$$a\oplus b:=\left(\begin{array}[]{cc}a&0\\ 0&b\\ \end{array}\right)\in M_{m+n}(R).$$

$$q\oplus q$$

$$=(y\oplus y)(p\oplus p)(x\oplus x)$$

$$=(y\oplus y)(ApB)(x\oplus x)$$

$$\displaystyle=\big{(}(y\oplus y)Ax\big{)}q\big{(}yB(x\oplus x)\big{)},$$

$$a=\sum _{B\in \mathcal{F}}{r}_B{1}_B$$

$$1_V=1_V^2=y(xy)x=ypx\in I,$$

$$s(B)⊊r(B)\subseteq V.$$

$$1_{s(B)}=1_{B^{-1}B}=1_{B^{-1}}{1}_B\mathrm{and}{1}_{r(B)}=1_{B{B}^{-1}}=1_B{1}_{B^{-1}}.$$

$${\mathcal{G}}_{\Lambda }:=\left\{(\lambda x,d(\lambda )-d(\mu ),\mu x):\lambda ,\mu \in \Lambda ,s(\lambda )=s(\mu ),x\in s(\lambda ){\Lambda }^{\le \infty }\right\}$$

$$Z(\lambda {\ast }_s\mu ):=\left\{(\lambda x,d(\lambda )-d(\mu ),\mu x):x\in s(\lambda ){\Lambda }^{\le \infty }\right\}.$$

$$p\oplus p\sim x_1\oplus x_2\sim x_1+x_2\precsim x_1\sim p,$$

$$q_1\sim t_{{\mu }_1^{\ast }}{t}_{{\mu }_1}=t_w=t_{{\mu }_2^{\ast }}{t}_{{\mu }_2}\sim q_2.$$

$$1_{Z(\lambda )}=s_{\lambda }{s}_{{\lambda }^{\ast }}\sim s_{{\lambda }^{\ast }}{s}_{\lambda }=s_{s(\lambda )},$$

$$s_{\nu }{s}_{{\nu }^{\ast }}\sim s_{{\nu }^{\ast }}{s}_{\nu }=s_{s(\mu )}=s_{{\mu }^{\ast }}{s}_{\mu }\sim s_{\mu }{s}_{{\mu }^{\ast }},$$

$${\pi }_u(f){\delta }_{\alpha }:=\sum _{s(\beta )=r(\alpha )}f(\beta ){\delta }_{\beta \alpha }(f\in C_c(\mathcal{G}),\alpha \in {\mathcal{G}}_u),$$

$$\Vert f{\Vert }_r:=\underset{u\in {\mathcal{G}}^{(0)}}{\sup }\Vert {\pi }_u(f)\Vert .$$

$$g(x)=\left\{\begin{array}[]{ll}\frac{1}{\sqrt{h(x)}}&x\in\widetilde{V}\\ 0&x\in D\setminus\widetilde{V}.\end{array}\right.$$

Full text

Furthermore, since the complex Steinberg algebra $A_{\mathbb{C}}(\mathcal{G})$ is a dense subalgebra of the reduced groupoid $C^{*}$-algebra $C^{*}_{r}(\mathcal{G})$, we focus on the problem that “when does the proper pure infiniteness of $A_{\mathbb{C}}(\mathcal{G})$ imply that of $C^{*}_{r}(\mathcal{G})$ in the $C^{*}$-sense?”. In particular, we show that if the Kumjian-Pask algebra $\mathrm{KP}_{\mathbb{C}}(\Lambda)$ is purely infinite, then so is $C^{*}(\Lambda)$ in the sense of Kirchberg-Rørdam.

1. Introduction

Inspired by the Kirchberg-R${\o}$rdam’s definition of non-simple purely infinite $C^{*}$-algebras [18], Aranda Pino, Goodearl, Perera, and Siles Molina introduced the notion of purely infinite and properly purely infinite properties for (not necessarily simple) rings [6] as the nonsimple generalization of [4] (see Section 3 for definitions). In particular, it is shown in [6, Proposition 3.17] that if a $C^{*}$-algebra is purely infinite in the sense of [6], then it is purely infinite in the Kirchberg-R${\o}$rdam’s sense. These concepts are closely related, such that for s-unital rings, the properly purely infinite property implies the purely infinite one [6, Lemma 3.4(i)], and we know that the converse is also true if the ring is either exchange [6, Corollary 2.9] or simple [6, Corollary 5.8] for example.

Let $K$ be a field. Following [12], in this paper we work with strongly effective ample groupoids $\mathcal{G}$ and associated Steinberg algebras $A_{K}(\mathcal{G})$. Our aim here is to investigate (non-simple) purely infinite Steinberg algebras and Kumjian-Pask algebras. Recall that the theory of Steinberg algebras were independently introduced in [27] and [13] as a pure algebraic analogue to Renault’s groupoid $C^{*}$-algebras [24]. There has been a lot of interest in the study of Steinberg algebras, partly because they include, and give a groupoid approach for, many interesting classes of algebras such as Leavitt path [1], Kumjian-Pask [5], and inverse semigroup algebras [27] among others. Moreover, many structural properties of these algebras can be described by that of the underlying groupoids; such as, simplicity [8], primeness [28], primitivity [29], simple pure infiniteness [9], and ideal structure [12], among others.

The contents of this article are as follows. We review in Section 2 some preliminaries about groupoids, Steinberg algebras and Kumjian-Pask algebras which will be used throughout. In Section 3, we prove that a Steinberg algebra $A_{K}(\mathcal{G})$ is properly purely infinite if and only if its characteristic functions on compact open bisections are properly infinite. Then we give a sufficient condition for this property. In Section 4, we apply this result to the Kumjian-Pask algebras of strongly aperiodic $k$-graphs, and derive interesting consequences. In particular, we show that purely infinite and properly purely infinite Kumjian-Pask algebras coincide, which may be characterized by verifying the proper infiniteness of their vertex idempotents. This is the generalization of [6, Theorem 7.4] which is for the class of Leavitt path algebras, but our proof, even in the Leavitt path algebras setting, is different and simpler. Moreover, we see that a unital Kumjian-Pask algebras is (properly) purely infinite if and only if every vertex in the underlying $k$-graph receives at least one nontrivial path.

For any locally compact Hausdorff groupoid $\mathcal{G}$, we know that the complex Steinberg algebra $A_{\mathbb{C}}(\mathcal{G})$ is a dense subalgebra of the reduced groupoid $C^{*}$-algebra $C^{*}_{r}(\mathcal{G})$ (see [13, Proposition 4.2]). In [9], the authors considered [6, Problem 8.4] for simple groupoid algebras, and proved that for any second-countable ample groupoid $\mathcal{G}$, if the complex Steinberg algebra $A_{\mathbb{C}}(\mathcal{G})$ is purely infinite simple, then so is the $C^{*}$-algebra $C^{*}_{r}(\mathcal{G})$ in the sense of [16]. In Section 5, we consider second-countable and amenable groupoids $\mathcal{G}$ and generalize this result to not necessarily simple groupoid algebras with a shorter proof. To do this, we first show that a groupoid $C^{*}$-algebra $C^{*}_{r}(\mathcal{G})$ is $C^{*}$-purely infinite (in the sense of [18]) if and only if all its characteristic functions are $C^{*}$-properly infinite, which is interesting in its own right. Since the goupoid $\mathcal{G}_{\Lambda}$ associated to each $k$-graph $\Lambda$ is always amenable and second-countable, we conclude that the pure infiniteness of complex Kumjian-Pask algebra $\mathrm{KP}_{\mathbb{C}}(\Lambda)$ implies that of the $C^{*}$-algebra $C^{*}(\Lambda)$ for strongly aperiodic $\Lambda$’s.

2. Preliminaries

Here, we review some definitions and basic properties of groupoids, Steinberg algebras and Kumjian-Pask algebras.

2.1. Groupoids

By groupoid, we mean a small category $\mathcal{G}$ endowed with a composition $\mathcal{G}^{(2)}\subseteq\mathcal{G}\times\mathcal{G}\rightarrow\mathcal{G}$, by $(\alpha,\beta)\mapsto\alpha\beta$, and an inversion $\alpha\mapsto\alpha^{-1}$ on $\mathcal{G}$ satisfying the following conditions:

1. (1)

if $(\alpha,\beta),(\beta,\gamma)\in\mathcal{G}^{(2)}$, then both $(\alpha\beta,\gamma)$ and $(\alpha,\beta\gamma)$ are composable and we have $(\alpha\beta)\gamma=\alpha(\beta\gamma)$; 2. (2)

$(\alpha^{-1},\alpha)\in\mathcal{G}^{(2)}$ for all $\alpha\in\mathcal{G}$; and 3. (3)

if $(\alpha,\beta)\in\mathcal{G}^{(2)}$, then $\alpha^{-1}(\alpha\beta)=\beta$ and $(\alpha\beta)\beta^{-1}=\alpha$.

Then the source and range maps on $\mathcal{G}$ are $s(\alpha)=\alpha^{-1}\alpha$ and $r(\alpha)=\alpha\alpha^{-1}$ for all $\alpha\in\mathcal{G}$. In particular, we have $(\alpha,\beta)\in\mathcal{G}^{(2)}$ if and only if $s(\alpha)=r(\beta)$. For every two subsets $A,B$ of $\mathcal{G}$, we define the product of $A$ and $B$ as

[TABLE]

We will let $\mathcal{G}^{(0)}$ the unit space of $\mathcal{G}$, that is $\mathcal{G}^{(0)}=\{\alpha^{-1}\alpha:\alpha\in\mathcal{G}\}$. Moreover, the isotropy subgroupoid of $\mathcal{G}$ is defined by

[TABLE]

We say that a subset $U\subseteq\mathcal{G}^{(0)}$ is invariant if, for $\alpha\in\mathcal{G}$, $s(\alpha)\in U$ implies $r(\alpha)\in U$, under which we have $r(s^{-1}(U)=U=s(r^{-1}(U))$. Note that in case $U$ is an invariant subset of $\mathcal{G}^{(0)}$, then $\mathcal{G}_{U}:=s^{-1}(U)$ is a subgroupoid of $\mathcal{G}$ (with the unit space $U$), which coincides with the restriction

[TABLE]

A groupoid $\mathcal{G}$ endowed with a topology is called a topological groupoid in case the composition and inverse maps are continuous. Let $\mathcal{G}$ be a topological groupoid. A subset $B\subseteq\mathcal{G}$ is a bisection if the restrictions $r|_{B}$ and $s|_{B}$ are homeomorphisms. Then we say that $\mathcal{G}$ is ample if $\mathcal{G}$ has a basis of compact open bisections. In this case, $\mathcal{G}^{(0)}$ is an open and totally disconnected subset of $\mathcal{G}$.

Definition 2.1*.*

A topological groupoid $\mathcal{G}$ is called effective if the interior of $\mathrm{Iso}(\mathcal{G})$ is $\mathcal{G}^{(0)}$. Also, we say that $\mathcal{G}$ is strongly effective in case for every nonempty closed invariant subset $V\subseteq\mathcal{G}^{(0)}$, the restricted groupoid $\mathcal{G}_{V}$ is effective.

Clearly, when $\mathcal{G}$ is strongly effective, then it is effective as well because $\mathcal{G}^{(0)}$ is a closed invariant set. In this paper, we only consider the strongly effective ample Hausdorff groupoids.

2.2. Steinberg Algebras

Let $\mathcal{G}$ be an ample groupoid and $K$ a field. The Steinberg algebra $A_{K}(\mathcal{G})$ associated to $\mathcal{G}$ consists of all locally constant and compactly supported functions from $\mathcal{G}$ into $K$. Then it has $K$-algebra structure by considering the pointwise addition as usual and the convolution multiplication as

[TABLE]

Since $\mathcal{G}$ is assumed to be ample, we have

[TABLE]

where $1_{B}$ is the characteristic function on $B$. In particular, $A_{K}(\mathcal{G})$ has local units, in the sense that for each $f\in A_{K}(\mathcal{G})$ there is an idempotent $p\in A_{K}(\mathcal{G})$ such that $fp=pf=f$, and so is s-unital. Examples of Steinberg algebras of ample groupoids include Leavitt path algebras [13] as well as Kumjian-Pask algebras [13, 15](see Section 4 for details).

By ideal, we always mean a two-sided one. According to [12, Theorem 3.1], when $\mathcal{G}$ is a strongly effective ample Hausdorff groupoid, then every ideal of $A_{K}(\mathcal{G})$ is of the form

[TABLE]

for some open invariant subset $U$ of $\mathcal{G}^{(0)}$. Moreover, if $U$ is an open invariant subset of $\mathcal{G}^{(0)}$ and $D:=\mathcal{G}^{(0)}\setminus U$, then [12, Lemma 3.6] says that the restrictive map $q(f)=f|_{\mathcal{G}_{D}}$ is an epimorphism from $A_{K}(\mathcal{G})$ into $A_{K}(\mathcal{G}_{D})$ such that $\ker(q)=I_{U}$; thus we have $A_{K}(\mathcal{G})/I_{U}\cong A_{K}(\mathcal{G}_{D})$. We will use this result in Section 3 because the property of proper infiniteness is closely related to the structure of ideals and quotients.

2.3. Higher-rank graphs and their Kumjian-Pask algebras

Here, we recall some basic definitions of graphs of rank $k\geq 1$ and associated Kumjian-Pask algebras from [19, 5, 14], which will be needed in the results of Section 4. Let $\mathbb{N}=\{0,1,2,\ldots\}$. For fixed integer $k\geq 1$, we consider the additive semigroup $\mathbb{N}^{k}$ with the identity $0:=(0,\ldots,0)$. We write each element $n\in\mathbb{N}^{k}$ as $n=(n_{1},\ldots,n_{k})$, and the generators of $\mathbb{N}^{k}$ by $e_{1},\ldots,e_{k}$, where $e_{ij}=\delta_{i,j}$ for $1\leq i,j\leq k$. We may put a partial order $\leq$ on $\mathbb{N}^{k}$ by $m\leq n$ $\Longleftrightarrow$ $m_{i}\leq n_{i}$ for all $1\leq i\leq k$, and denote $m\vee n$ and $m\wedge n$ for the coordinate-wise maximum and minimum, respectively.

Definition 2.2* ([19]).*

A $k$-graph (or higher-rank graph) is a countable small category $\Lambda=(\Lambda^{0},\Lambda,r,s)$, where $\Lambda^{0}$ is the objects, $\Lambda$ morphisms, $r,s:\Lambda\rightarrow\Lambda^{0}$ the range and source maps, which is equipped with a degree functor $d:\Lambda\rightarrow\mathbb{N}^{k}$ satisfying the unique factorisation property: for each $\lambda\in\Lambda$ and $0\leq n\leq d(\lambda)$, there exists a unique factorisation $\lambda=\lambda(0,n)\lambda(n,d(\lambda))$ for $\lambda$ such that $d(\lambda(0,n))=n$, $d(\lambda(n,d(\lambda)))=d(\lambda)-n$.

Every $k$-graph $\Lambda$ has an edge-colored graph as its 1-skeleton, so we usually like to refer the objects of $\Lambda^{0}$ as vertices and the morphisms of $\Lambda$ as paths. If $H\subseteq\Lambda^{0}$ and $E\subseteq\Lambda^{1}$, we write

[TABLE]

and when $H=\{v\}$ is a singleton, we simply write $vE$ and $Ev$, respectively. For any $n\in\mathbb{N}^{k}$, we define $\Lambda^{n}:=\{\lambda\in\Lambda:d(\lambda)=n\}$ and

[TABLE]

Given $\mu,\nu\in\Lambda$, the set of minimal common extensions for $\mu$ and $\nu$ is denoted by $\mathrm{MCE}(\mu,\nu)$, which is

[TABLE]

A $k$-graph $\Lambda$ is called row-finite if $v\Lambda^{n}$ is finite for all $v\in\Lambda^{0}$ and $n\in\mathbb{N}^{k}$. Also, we say $\Lambda$ is locally convex in case for every $\lambda\in\Lambda^{e_{i}}$ and $1\leq j\neq i\leq k$, $r(\lambda)\Lambda^{e_{j}}\neq\emptyset$ implies $s(\lambda)\Lambda^{e_{j}}\neq\emptyset$ [22, Definition 3.10]. In this article, we consider only locally convex and row-finite $k$-graphs.

Example 2.3*.*

Let $m\in(\mathbb{N}\cup\{\infty\})^{k}$ for $k\geq 1$. Then the set $\Omega_{k,m}=\{(p,q)\in\mathbb{N}^{k}\times\mathbb{N}^{k}:p\leq q\leq m\}$ equipped with the degree map as $d(p,q)=q-p$, and the range and source maps as $r(p,q)=(p,p)$ and $s(p,q)=(q,q)$ is a row-finite $k$-graph.

A boundary path of $\Lambda$ is a degree preserving functor $x:\Omega_{k,m}\rightarrow\Lambda$ such that for every $p\in\mathbb{N}^{k}$ and $1\leq i\leq k$, $p\leq m$ and $p_{i}=m_{i}$ imply that $x(p,p)\Lambda^{e_{i}}=\emptyset$. Then we define $d(x)=m$ and usually call $x(0,0)$ range of $x$, written by $r(x)$. The set of all boundary paths of $\Lambda$ is denoted by $\Lambda^{\leq\infty}$.

There are some definitions for aperiodicity in the literature, which are equivalent for locally convex row-finite $k$-graphs. Following [22], we say that $\Lambda$ is aperiodic if for each $v\in\Lambda^{0}$, there exists $x\in v\Lambda^{\leq\infty}$ such that $\alpha\neq\beta\in\Lambda v$ implies $\alpha x\neq\beta x$.

Definition 2.4* ([5]).*

Let $\Lambda$ be a locally convex, row-finite $k$-graph and $K$ a field. The Kumjian-Pask algebra associated to $\Lambda$ is the universal $K$-algebra $\mathrm{KP}_{K}(\Lambda)$ generated by a family $\{s_{\lambda},s_{\lambda^{*}}:\lambda\in\Lambda\}$ (which is called a Kumjian-Pask $\Lambda$-family) satisfying the following conditions:

1. (KP1)

$s_{v}s_{w}=\delta_{v,w}s_{v}$ for all $v,w\in\Lambda^{0}$. 2. (KP2)

$s_{\lambda}s_{\mu}=s_{\lambda\mu}$ and $s_{\mu^{*}}s_{\lambda^{*}}=s_{(\lambda\mu)^{*}}$ for all $\lambda,\mu\in\Lambda$ with $s(\lambda)=r(\mu)$. 3. (KP3)

$s_{\lambda^{*}}s_{\mu}=\delta_{\lambda,\mu}s_{\lambda}$ for all $\lambda,\mu\in\Lambda$. 4. (KP4)

$s_{v}=\sum_{\lambda\in v\Lambda^{\leq n}}s_{\lambda}s_{\lambda^{\ast}}$ for all $v\in\Lambda^{0}$ and $n\in\mathbb{N}^{k}$.

It is show in [5, Theorem 3.4] and [14, Theorem 3.7(a)] that the $K$-algebra $\mathrm{KP}_{K}(\Lambda)$ exists and is unique up to isomorphism.

A subset $H\subseteq\Lambda^{0}$ is called hereditary if for every $\lambda\in\Lambda$, $r(\lambda)\in H$ implies $s(\lambda)\in H$, and is called saturated if for every $v\in\Lambda^{0}$ and $n\in\mathbb{N}^{k}$, $\{s(\lambda):\lambda\in v\Lambda^{\leq n}\}\subseteq H$ implies $v\in H$. Note that if $H$ is a saturated hereditary subset of $\Lambda^{0}$, then $\Lambda\setminus\Lambda H$ is a locally convex row-finite $k$-graph [22, Theorem 5.2 (b)]. We say that $\Lambda$ is strongly aperiodic in case all such quotient $k$-graphs are aperiodic. Recall from [5, Corollary 5.7] that, when $\Lambda$ is strongly aperiodic, every (two-sided) ideal of $\mathrm{KP}_{K}(\Lambda)$ is of the form

[TABLE]

for some saturated hereditary subset $H$ of $\Lambda^{0}$ (see also [14, Theorem 9.4]), and we have $\mathrm{KP}_{K}(\Lambda)/I_{H}\cong\mathrm{KP}_{K}(\Lambda\setminus\Lambda H)$ [5, Proposition 5.5].

3. Properly purely infinite Steinberg algebras

In this section, we characterize properly purely infinite Steinberg algebras $A_{K}(\mathcal{G})$ by giving necessary and sufficient conditions for it. Our result is the algebraic analogue of [7, Theorem 4.1] and the non-simple version of [9, Theorem 3.2]. Since the concept of proper purely infiniteness is closely related to the ideal and quotient structure [6, Corollary 2.9], we consider here strongly effective groupoids to apply the results of [12, Section 3]. However, verifying this concept for general groupoids seems to be more complicated, even for the $k$-graph’s ones.

Let us first recall some definitions and terminology from [6]. Suppose that $R$ is a ring. If $a\in M_{m}(R)$ and $b\in M_{n}(R)$ are two squire matrices over $R$, we denote

[TABLE]

We define $a\precsim b$ if there exist $x\in M_{m,n}(R)$ and $y\in M_{n,m}(R)$ such that $a=xby$. Note that we may consider $a$, $b$, $x$ and $y$ as elements in $M_{m+n}(R)$ by enlarging them by zeros. For $a,b\in R$, we write $a\sim b$ if there exist $r,s\in R$ such that $rs=a$ and $sr=b$, $a\leq b$ if $ab=ba=a$, and $a<b$ if $a\leq b$ and $a\neq b$. A nonzero element $a\in R$ is called infinite in case $a\oplus x\precsim a$ for some $x\in R$, and properly infinite in case $a\oplus a\precsim a$ (or equivalently, $a\oplus a\precsim a\oplus 0$). As explained after [6, Remark 2.7], if $p\in R$ is an idempotent, $p$ is infinite if and only if there is a subidempotent $p<q$ such that $p\sim q$.

Lemma 3.1**.**

Let $R$ be a ring and let $p$ and $q$ be two idempotents in $R$. If $p\sim q$ and $p$ is properly infinite, then so is $q$.

Proof.

Suppose that $p=xy$ and $q=yx$ for some $x,y\in R$. Let $p\oplus p=ApB$ where $A\in M_{2,1}(R)$ and $B\in M_{1,2}(R)$. Then we have

[TABLE]

and therefore $q\oplus q\precsim q$ as desired. ∎

Definition 3.2* ([6, Definition 3.1]).*

Let $R$ be a ring. We say that $R$ is purely infinite if

1. (1)

each quotient of $R$ is not a division ring, and 2. (2)

for every $a\in R$, $b\in RaR$ implies $b\precsim a$.

Also, we say that $R$ is properly purely infinite in case all nonzero elements of $R$ are properly infinite.

According to [6, Lemma 3.4], every properly purely infinite s-unital ring is purely infinite as well, and we know at least that the converse is also true for exchange cases [6, Corollary 5.8]. Moreover, it follows from [6, Theorem 7.4] that these notions are equivalent for the class of Leavitt path algebras, and in the next section, we will prove a similar result for the class of Kumjian-Pask algebras.

We use the following lemma to prove Theorem 3.4 below.

Lemma 3.3**.**

Let $\mathcal{G}$ be an ample Hausdorff groupoid and $K$ a field. Then the following are equivalent.

1. (1)

$\mathcal{G}$* is effective.* 2. (2)

Every nonzero ideal of $A_{K}(\mathcal{G})$ contains a nonzero idempotent $1_{V}$ for some compact open subset $V\subseteq\mathcal{G}^{(0)}$. 3. (3)

Every nonzero one-sided ideal of $A_{K}(\mathcal{G})$ contains a nonzero idempotent $p$ such that $p\sim 1_{V}$ for some compact open subset $V\subseteq\mathcal{G}^{(0)}$.

Proof.

(1) $\Longrightarrow$ (3). Let $I$ be a nonzero right ideal of $A_{K}(\mathcal{G})$. Fix a nonzero element $a\in I$. Then, by [11, Lemma 2.2], we may write

[TABLE]

where $\mathcal{F}$ is a finite collection of mutually disjoint compact open bisections. We may follow the proof of [9, Theorem 3.2] to find an idempotent $p\in I$ and a compact open $V\subseteq\mathcal{G}^{(0)}$ such that $p\sim 1_{V}$, as desired. For left ideals, we can argue similarly.

(3) $\Longrightarrow$ (2). Let $I$ be a nonzero ideal of $A_{K}(\mathcal{G})$. According to (3), $I$ contains a nonzero element $p$ such that $p\sim 1_{V}$ for some compact open $V\subseteq\mathcal{G}^{(0)}$. If $p=xy$ and $1_{V}=yx$ for some $x,y\in A_{K}(\mathcal{G})$, then we have

[TABLE]

giving (2).

(2) $\Longrightarrow$ (1). This follows from [12, Lemma 3.4]. ∎

Theorem 3.4**.**

Let $\mathcal{G}$ be a strongly effective, ample and Hausdorff groupoid and let $K$ be a field. Suppose that $\mathcal{B}$ is a basis of compact open sets for $\mathcal{G}^{(0)}$. Then the following are equivalent:

1. (1)

$A_{K}(\mathcal{G})$* is properly purely infinite.* 2. (2)

For every $V\in\mathcal{B}$, $1_{V}$ is properly infinite. 3. (3)

For every invariant open subset $U\subseteq\mathcal{G}^{(0)}$ and any nonempty relatively compact open subset $V\subseteq D:=\mathcal{G}^{(0)}\setminus U$, $1_{V}$ is an infinite idempotent in $A_{K}(\mathcal{G}_{D})$. 4. (4)

Every nonzero one-sided ideal in any quotient of $A_{K}(\mathcal{G})$ contains an infinite idempotent.

In particular, if one of the above statements holds (and so all), then $A_{K}(\mathcal{G})$ is purely infinite as well.

Proof.

(1) $\Longrightarrow$ (2) is immediate.

(2) $\Longrightarrow$ (3). Let $U$ be an invariant open subset of $\mathcal{G}^{(0)}$ and write $D:=\mathcal{G}^{(0)}\setminus U$. Given any relatively compact open $\widetilde{V}\subseteq D$, there is an open subset $V\subseteq\mathcal{G}^{(0)}$ such that $\widetilde{V}=V\cap D$. We may also suppose that $V$ is compact. Indeed, since $\mathcal{B}$ is a basis for $\mathcal{G}^{(0)}$, we can write $V=\bigcup_{C\in\mathcal{C}}C$ for some subcollection $\mathcal{C}\subseteq\mathcal{B}$. Then $\widetilde{V}=\bigcup_{C\in\mathcal{C}}C\cap D$. As $\widetilde{V}$ is compact, there are finitely many $C_{1},\ldots,C_{n}$ in $\mathcal{C}$ such that $\widetilde{V}\subseteq\bigcup_{i=1}^{n}C_{n}\cap D$. Thus, we can replace $V$ with $\bigcup_{i=1}^{n}C_{n}$ and assume that $V$ is a compact and open subset of $\mathcal{G}^{(0)}$.

Now part (2) says that $1_{V}$ is a properly infinite idempotent in $A_{K}(\mathcal{G})$. Thus, since $1_{\widetilde{V}}$ is the image of $1_{V}$ in the quotient $A_{K}(\mathcal{G}_{D})\cong A_{K}(\mathcal{G})/I_{U}$, [6, Corollary 2.9] implies that $1_{\widetilde{V}}$ is an infinite idempotent in $A_{K}(\mathcal{G}_{D})$.

(3) $\Longrightarrow$ (4). Let $I$ be an ideal of $A_{K}(\mathcal{G})$. By [12, Theorem 3.1], there is an open invariant $U\subseteq\mathcal{G}^{(0)}$ such that $I=I_{U}$, and we have $A_{K}(\mathcal{G})/I\cong A_{K}(\mathcal{G}_{D})$ for $D:=\mathcal{G}^{(0)}\setminus U$ [12, Lemma 3.6].

Now fix an arbitrary nonzero one-sided ideal $J$ of $A_{K}(\mathcal{G}_{D})$. Since $\mathcal{G}_{D}$ is effective, Lemma 3.3 implies that $J$ contains a nonzero idempotent $p$ such that $p\sim 1_{V}$ for some relatively compact open subset $V$ of $D$. Since $1_{V}$ is infinite in $A_{K}(\mathcal{G}_{D})$ [6, Corollary 2.9], then so is $p$ as desired.

(4) $\Longrightarrow$ (1) follows from [6, Proposition 3.13], completing the proof. ∎

Definition 3.5* ([2, Definition 2.1]).*

Let $\mathcal{G}$ be an ample Hausdorff groupoid. We say that $\mathcal{G}$ is locally contracting if for every compact open $V\subseteq\mathcal{G}^{(0)}$, there exists a compact open bisection $B\subseteq\mathcal{G}$ such that

[TABLE]

Note that if $s(B)\subsetneq r(B)$, then we have

[TABLE]

Using $1_{s(B)}1_{r(B)}=1_{s(B)}=1_{r(B)}1_{s(B)}$, this follows $1_{s(B)}\lneq 1_{r(B)}\sim 1_{s(B)}$, giving that $1_{r(B)}$ is an infinite idempotent in $A_{K}(\mathcal{G})$.

Corollary 3.6**.**

Let $\mathcal{G}$ be a strongly effective, ample Hausdorff groupoid and let $K$ be a field. If for every open invariant $U\subseteq\mathcal{G}^{(0)}$, the groupoid $\mathcal{G}_{\mathcal{G}^{(0)}\setminus U}$ is locally contractive, then $A_{K}(\mathcal{G})$ is properly purely infinite.

Proof.

Let $U$ be an invariant open subset of $\mathcal{G}^{(0)}$, and let $V$ be a relatively compact open subset of $D:=\mathcal{G}^{(0)}\setminus U$. Since $\mathcal{G}_{D}$ is locally contractive, there is some compact open bisection $B\subseteq\mathcal{G}^{(0)}$ such that $s(B)\subsetneq r(B)\subseteq V$. Hence $1_{r(B)}$ is infinite in $A_{K}(\mathcal{G}_{D})$ as explained above, and by $1_{r(B)}\leq 1_{V}$, then so is $1_{V}$ as well. Now (3) $\Rightarrow$ (1) in Theorem 3.4 concludes the result. ∎

4. Applications to Kumjian-Pask algebras

According to [13, Proposition 4.3] and [15, Proposition 5.4], every Kumjian-Pask algebra $\mathrm{KP}_{K}(\Lambda)$ may be considered as the Steinberg algebra $A_{K}(\mathcal{G}_{\Lambda})$ where $\mathcal{G}_{\Lambda}$ is the boundary path groupoid of $\Lambda$. So, in view of Theorem 3.4, we may investigate the proper pure infiniteness of Kumjian-Pask algebras by groupoid approach. Moreover, [6, Theorem 7.4] says that the notions of purely infinite and properly purely infinite are equivalent for Leavitt path algebras. Here, we will also generalize this result to the class of Kumjian-Pask algebras by a quite different proof.

Let us first follow [19, Section 2] and briefly review the construction of the boundary path groupoid $\mathcal{G}_{\Lambda}$ associated to $\Lambda$. (Although the groupoid $\mathcal{G}_{\Lambda}$ of [19, Section 2] is constructed for a row-finite $k$-graph with no sources, however we may easily generalize it for any row-finite $k$-graph.) So, suppose that $\Lambda$ is a fixed locally convex row-finite $k$-graph. We set

[TABLE]

and define the range and source maps by $r(x,m,y):=x$ and $s(x,m,y):=y$. If we consider the composition as $(x,m,y)(y,n,z):=(x,m+n,z)$ and the inversion as $(x,m,y)^{-1}:=(y,-m,x)$, then $\mathcal{G}_{\Lambda}$ is a groupoid. We usually identify the unit space $\mathcal{G}_{\Lambda}^{(0)}$ with the boundary path space $\Lambda^{\leq\infty}$ by $(x,0,x)\leftrightarrow x$. Moreover, we may equip $\mathcal{G}_{\Lambda}$ with a locally compact Hausdorff topology induced from that of $\Lambda^{\leq\infty}$. Indeed, for each $\lambda,\mu\in\Lambda$ with $s(\lambda)=s(\mu)$ we set

[TABLE]

When $\lambda=\mu$, we write simply $Z(\lambda)$ for $Z(\lambda*_{s}\lambda)$. Then [19, Proposition 2.8] shows that the collection $\{Z(\lambda*_{s}\mu):\lambda,\mu\in\Lambda,s(\lambda)=s(\mu)\}$ forms a basis of compact open bisections for a Hausdorff topology on $\mathcal{G}_{\Lambda}$, making it an ample groupoid. Moreover, an application of the graded uniqueness theorem implies that $\mathrm{KP}_{K}(\Lambda)\cong A_{K}(\mathcal{G}_{\Lambda})$, which maps $s_{\lambda}$ to $1_{Z(\lambda*_{s}s(\lambda))}$ and $s_{\lambda^{*}}$ to $1_{Z(s(\lambda)*_{s}\lambda)}$ (cf. [13, Proposition 4.3] and [15, Proposition 5.4]).

Before Theorem 4.2, we state a lemma.

Lemma 4.1**.**

Let $R$ be a purely infinite ring and let $p$ be an idempotent in $R$. If there are mutually orthogonal idempotents $q_{1},q_{2}\in R$ such that $p\in Rq_{i}R$, then $p$ is properly infinite.

Proof.

Since $R$ is purely infinite, $p\in Rq_{i}R$ implies $p\precsim q_{i}$ for $i\in\{1,2\}$. Suppose $p=a_{i}q_{i}b_{i}$. Putting $x_{i}:=q_{i}b_{i}a_{i}q_{i}$, then $x_{1},x_{2}$ are orthogonal idempotents in $R$ such that $x_{1}\sim p\sim x_{2}$. Note that if $x_{1}=rs$ and $x_{2}=sr$ for $r,s\in R$, then $x_{2}=x_{2}^{2}=s(rs)r=rx_{1}s$. Hence $x_{1}+x_{2}$ lies in $Rx_{1}R$, and the pure infiniteness forces $x_{1}+x_{2}\precsim x_{1}$. Moreover, for A=\left(\begin{array}[]{c}x_{1}\\ x_{2}\\ \end{array}\right) and B=\left(\begin{array}[]{cc}x_{1}&x_{2}\\ \end{array}\right), we have $AB=x_{1}\oplus x_{2}$ and $BA=x_{1}+x_{2}$. Therefore, we get

[TABLE]

establishing $p\oplus p\precsim p$. ∎

Theorem 4.2**.**

Let $\Lambda$ be a locally convex row-finite $k$-graph and $K$ a field. Suppose that $\Lambda$ is strongly aperiodic. Then the following are equivalent.

1. (1)

$\mathrm{KP}_{K}(\Lambda)$* is purely infinite.* 2. (2)

$\mathrm{KP}_{K}(\Lambda)$* is properly purely infinite.* 3. (3)

For each vertex $v\in\Lambda^{0}$, $s_{v}$ is a properly infinite idempotent.

Proof.

(1) $\Longrightarrow$ (3). Fix $v\in\Lambda^{0}$. In view of [6, Corollary 2.9], it suffices to show that the image of $s_{v}$ in any quotient of $\mathrm{KP}_{K}(\Lambda)$ is either zero or infinite. Moreover, since $\Lambda$ is strongly aperiodic, every ideal of $\mathrm{KP}_{K}(\Lambda)$ is of the form of $I_{H}$ for some saturated hereditary subset $H$ of $\Lambda^{0}$ [14, Theorem 9.4]. So, let $I_{H}$ be an ideal of $\mathrm{KP}_{K}(\Lambda)$ such that $v\notin H$. Writing $\Gamma:=\Lambda\setminus H\Lambda$, then $\Gamma$ is an aperiodic $k$-graph and we have $\mathrm{KP}_{K}(\Lambda)/I_{H}\cong\mathrm{KP}_{K}(\Gamma)$, which is purely infinite as well. In the following, for each $\gamma\in\Gamma$, we write $t_{\gamma}:=s_{\gamma}+I_{H}$ in $\mathrm{KP}_{K}(\Gamma)$. We divide our discussion into two cases:

Case I: There are two distinct vertices $v\neq u,w\in\Gamma^{0}$ such that $v\Gamma w,u\Gamma w\neq\emptyset$. Fix some $\mu_{1}\in v\Gamma w$ and $\mu_{2}\in u\Gamma w$. Then $q_{1}:=t_{\mu_{1}}t_{\mu_{1}^{*}}$ and $q_{2}:=t_{\mu_{2}}t_{\mu_{2}^{*}}$ are two orthogonal idempotents in $\mathrm{KP}_{K}(\Gamma)$ (because $s_{v}$ and $s_{u}$ are) such that

[TABLE]

Now, Lemma 4.1 implies that $t_{w}$ is properly infinite. Since $t_{v}\geq q_{1}\sim t_{\mu^{*}_{1}}t_{\mu_{1}}=t_{w}$, apply [6, Corollary 3.9(iii)] to conclude that $t_{v}$ is infinite in $\mathrm{KP}_{K}(\Gamma)$.

Case II: In the other case, there exists $w\in\Gamma^{0}$ with $v\Gamma w\neq\emptyset$ such that $w\Gamma u=\emptyset$ for all $u\in\Gamma^{0}\setminus\{w\}$. If $w\Gamma w=\{w\}$, i.e. $w$ receives no nontrivial paths, then the ideal $I_{w}=\langle t_{w}\rangle$ in $\mathrm{KP}_{K}(\Gamma)$ is a matricial algebra (cf. [20, Lemma 6.2]), which is impossible. Otherwise, since $\Gamma$ is aperiodic, [20, Lemma 6.1] implies that $w$ is the base of an (initial) cycle with an entrance. Thus $t_{w}$ is infinite (see Lemma 4.4(1) below). Now if $\mu\in v\Gamma w$, then $t_{v}\geq t_{\mu}t_{\mu^{*}}\sim t_{\mu^{*}}t_{\mu}=t_{w}$, and hence $t_{v}$ would be infinite in each case. Since $I_{H}$ was arbitrary, this follows that $s_{v}$ is a properly infinite idempotent in $\mathrm{KP}_{K}(\Lambda)$ as desired.

(3) $\Longrightarrow$ (2). To prove this part, we consider the groupoid $\mathcal{G}_{\Lambda}$ associated to $\Lambda$, and then apply Theorem 3.4. According to [19, Proposition 2.8], the collection $\mathcal{B}=\{Z(\lambda):\lambda\in\Lambda\}$ is a basis of compact and open sets for the topology on $\mathcal{G}^{(0)}$. Also, for each $\lambda\in\Lambda$, we have

[TABLE]

and hence $1_{Z(\lambda)}$ is properly infinite by Lemma 3.1. Since $\mathcal{G}_{\Lambda}$ is strongly effective [15, Proposition 6.3], Theorem 3.4 follows that $\mathrm{KP}_{K}(\Lambda)=A_{K}(\mathcal{G}_{\Lambda})$ is properly purely infinite.

(2) $\Longrightarrow$ (1) is immediate by [6, Lemma 3.4(i)]. ∎

Now, analogous to [20, Theorem 5.4], we may use the notion of generalized cycle to give a criterion under which the vertex idempotents $s_{v}$ are properly infinite. It is the $k$-graphic version of Corollary 3.6.

Definition 4.3* ([17]).*

A generalized cycle in $\Lambda$ is a pair $(\mu,\nu)$ of distinct paths $\mu,\nu\in\Lambda$ with $s(\mu)=s(\nu)$ and $r(\mu)=r(\nu)$ such that $\mathrm{MCE}(\mu\tau,\nu)\neq\emptyset$ for all $\tau\in s(\mu)\Lambda$; or equivalently $Z(\mu)\subseteq Z(\nu)$. Note that if $\mu$ is a cycle in $\Lambda\setminus\Lambda^{0}$ (in the sense $s(\mu)=r(\mu)$), then $(\mu^{i},\mu^{j})$ is a generalized cycle for every $j\leq i\in\mathbb{N}$. We say that $\tau\in s(\nu)\Lambda$ is an entrance for $(\mu,\nu)$ if $\mathrm{MCE}(\mu,\nu\tau)=\emptyset$ (i.e., $\nu\tau\Lambda^{\leq\infty}\subseteq Z(\nu)\setminus Z(\mu)$).

Following [20], if there is a path from $s(\mu)=s(\nu)$ to $v\in\Lambda^{0}$ we say that $(\mu,\nu)$ connects to $v$ or $v$ is reached from $(\mu,\nu)$.

Lemma 4.4**.**

Let $(\mu,\nu)$ be a generalized cycle in $\Lambda$ with an entrance. Then

1. (1)

$s_{s(\mu)}=s_{s(\nu)}$* is an infinite idempotent in $\mathrm{KP}_{K}(\Lambda)$.* 2. (2)

If $v\in\Lambda^{0}$ is reached from $(\mu,\nu)$, then $s_{v}$ is infinite in $\mathrm{KP}_{K}(\Lambda)$.

Proof.

[20, Lemma 4.3] says that $s_{\mu}s_{\mu^{*}}<s_{\nu}s_{\nu^{*}}$. Thus by

[TABLE]

$s_{\nu}s_{\nu^{*}}$ is an infinite idempotent, and so is $s_{\nu^{*}}s_{\nu}=s_{s(\nu)}=s_{s(\mu)}$ as well.

Statement (2) follows from (1). Indeed, if $\gamma\in v\Lambda s(\nu)$ then $s_{v}\geq s_{\gamma}s_{\gamma^{*}}\sim s_{\gamma^{*}}s_{\gamma}=s_{s(\mu)}$, and hence $s_{\gamma}s_{\gamma^{*}}$ and $s_{v}$ are infinite by part (1). ∎

Corollary 4.5**.**

Let $\Lambda$ be a strongly aperiodic, locally convex and row-finite $k$-graph and let $K$ be a field. Suppose that for each saturated hereditary $H\subseteq\Lambda^{0}$, every vertex $v\in\Lambda^{0}\setminus H=(\Lambda\setminus H\Lambda)^{0}$ is reached from a generalized cycle with an entrance in the $k$-graph $\Lambda\setminus H\Lambda$. Then $\mathrm{KP}_{K}(\Lambda)$ is properly purely infinite.

Proof.

Fix $v\in\Lambda^{0}$. For any ideal $I_{H}\trianglelefteq\mathrm{KP}_{K}(\Lambda)$ with $v\notin I_{H}$, $v$ is reached from a generalized cycle with an entrance in $\Lambda\setminus H\Lambda$, so $s_{v}+I_{H}$ is an infinite idempotent in $\mathrm{KP}_{K}(\Lambda)/I_{H}\cong\mathrm{KP}(\Lambda\setminus H\Lambda)$ by Lemma 4.4(2). Therefore, $s_{v}$ is properly infinite in $\mathrm{KP}_{K}(\Lambda)$, and Theorem 4.2 concludes the result. ∎

In the case that there are only finitely many vertices connecting to each vertex $v\in\Lambda^{0}$, we can establish simple graph-theoretic criteria for the pure infiniteness of Kumjian-Pask algebras $\mathrm{KP}_{K}(\Lambda)$. Note that this case covers all unital Kumjian-Pask algebras.

Proposition 4.6**.**

Let $\Lambda$ be a locally convex row-finite $k$-graph and $K$ a field. Suppose that $\Lambda$ is strongly aperiodic, and that $\Lambda^{0}_{\geq v}:=\{w:v\Lambda w\neq\emptyset\}$ is finite for every $v\in\Lambda^{0}$. Then the following are equivalent.

1. (1)

$v\Lambda\neq\{v\}$* for every vertex $v\in\Lambda^{0}$.* 2. (2)

Every vertex of $\Lambda$ is reached from a cycle. 3. (3)

$\mathrm{KP}_{K}(\Lambda)$* is properly purely infinite.* 4. (4)

$\mathrm{KP}_{K}(\Lambda)$* is purely infinite.*

Proof.

(1) $\Longrightarrow$ (2). Fix $v\in\Lambda^{0}$ and assume $|\Lambda^{0}_{\geq v}|=t_{0}$. Using statement (1), for each $t\geq t_{0}^{k}$ in $\mathbb{N}$, there exists $\lambda\in v\Lambda$ such that $|\lambda|:=d(\lambda)_{1}+\cdots+d(\lambda)_{k}=t$, where $d(\lambda)=(d(\lambda)_{1},\ldots,d(\lambda)_{k})$. This forces $\lambda(m)=\lambda(n)$ for some $m<n\in\mathbb{N}^{k}$, and thus the cycle $\lambda(m,n)$ connects to $v$.

(2) $\Longrightarrow$ (3). Fix some $v\in\Lambda^{0}$ and an ideal $I_{H}$ of $\mathrm{KP}_{K}(\Lambda)$ such that $s_{v}\notin I_{H}$. Then $v\notin H$. For convenience, we write $\Gamma:=\Lambda\setminus\Lambda H$, which is a strongly aperiodic $k$-graph too. We first claim $v\Gamma\neq\{v\}$. Indeed, if $v\Gamma=\{v\}$ then for each $n\in\mathbb{N}^{k}\setminus\{0\}$ we have $\{r(\lambda):\lambda\in v\Lambda^{\leq n}\}\subseteq H$. Thus $v\in H$ by the saturated property, a contradiction. Now implication (1) $\Longrightarrow$ (2) above yields that $v$ is reached from a cycle in $\Gamma$, and using [20, Lemma 6.1], is also from an (initial) cycle with an entrance. Lemma 4.4 says that $s_{v}+I_{H}$ is an infinite idempotent in $\mathrm{KP}_{K}(\Gamma)=\mathrm{KP}_{K}(\Lambda)/I_{H}$. Therefore, since $I_{H}$ was arbitrary, $s_{v}$ is properly infinite in $\mathrm{KP}_{K}(\Lambda)$, and consequently $\mathrm{KP}_{K}(\Lambda)$ is properly purely infinite by Theorem 4.2.

(3) $\Longleftrightarrow$ (4) are obtained from Theorem 4.2.

(3) and (4) $\Longrightarrow$ (1). If $v\Lambda=\{v\}$ for some $v\in\Lambda^{0}$, then the ideal $I_{v}=\langle s_{v}\rangle$ in $\mathrm{KP}_{K}(\Lambda)$ is isomorphic to the matrix algebra $M_{|v\Lambda|}(K)$ (cf. [20, Lemma 6.2]). But this is impossible because the (proper) pure infiniteness passes to ideals.

∎

5. Complex Steinberg aplgebras and groupoid $C^{\ast}$-algebras

In the last section, we investigate the relationship between the purely infinite property of Steinberg algebras and groupoid $C^{*}$-algebras. More precisely, since the complex Steinberg algebra $A_{\mathbb{C}}(\mathcal{G})$ may be regarded as a dense $*$-subalgebra of $C^{*}_{r}(\mathcal{G})$ (see [13, Proposition 4.2]), we focus on the conjecture [6, Problem 8.4] for groupoid algebras: If $A_{\mathbb{C}}(\mathcal{G})$ is properly purely infinite, is this property inherited by $C^{*}_{r}(\mathcal{G})$? It is shown in [9, Section 4] that this conjecture is true for simple groupoid algebras (when $\mathcal{G}$ is minimal and effective). However, in the nonsimple cases, the properly purely infinite property (in both algebraic and $C^{*}$-algebraic senses) is closely related to ideal structure and quotients. Here, we work with amenable groupoids in the sense of [3] to apply [3, Proposition 6.1.8] and the ideal description of [8, Corollary 5.9]. Moreover, since the groupoid $\mathcal{G}_{\Lambda}$ associated to each $k$-graph $\Lambda$ is always amenable, we deduce also an interesting result for $k$-graph algebras.

We recall the definition of reduced $C^{*}$-algebra $C^{*}_{r}(\mathcal{G})$ from [24]. Let $\mathcal{G}$ be an ample groupoid. Write $C_{c}(\mathcal{G})$ for the complex vector space consisting of compactly supported continuous functions on $\mathcal{G}$, which is an $*$-algebra with the convolution multiplication and the involution $f^{*}(\alpha):=\overline{f(\alpha^{-1})}$. Then $C^{*}_{r}(\mathcal{G})$ is the completion of $C_{c}(\mathcal{G})$ for the left regular representations on $\ell^{2}(\mathcal{G})$. Indeed, for each $u\in\mathcal{G}^{(0)}$ and $\mathcal{G}_{u}:=s^{-1}(\{u\})$, if $\pi_{u}:C_{c}(\mathcal{G})\rightarrow B(\ell^{2}(\mathcal{G}_{u}))$ is the left regular $*$-representation, defined by

[TABLE]

then the reduced $C^{*}$-algebra $C^{*}_{r}(\mathcal{G})$ is the completion of $C_{c}(\mathcal{G})$ under the reduced $C^{*}$-norm

[TABLE]

Moreover, there is a full $C^{*}$-algebra $C^{*}(\mathcal{G})$ associated to $\mathcal{G}$, which is the completion of $C_{c}(\mathcal{G})$ taken over all $\|.\|_{C_{c}(\mathcal{G})}$-decreasing representations of $\mathcal{G}$. So, $C^{*}_{r}(\mathcal{G})$ is a quotient of $C^{*}(\mathcal{G})$, and [3, Proposition 6.1.8] follows that they are equal if the underlying groupoid $\mathcal{G}$ is amenable.

Let $A$ be a $C^{*}$-algebra. For positive matrices $a\in M_{m}(A)^{+}$ and $b\in M_{n}(A)^{+}$ over $A$, we write $a\precsim_{C^{*}}b$ if there exists a sequence $\{x_{n}\}$ in $M_{n,m}(A)$ such that $x_{n}^{*}bx_{n}\rightarrow a$. As before, for $a,b\in A$ we denote $a\oplus b:=diag(a,b)$ in $M_{2}(A)$. A nonzero positive element $a\in A$ is called $C^{*}$-infinite if $a\oplus b\precsim_{C^{*}}a$ for some positive $0\neq b\in A$, and is called $C^{*}$-properly infinite if $a\oplus a\precsim_{C^{*}}a$. Following [18], we say that $A$ is $C^{*}$-purely infinite in case every nonzero positive element of $A$ is properly infinite.

Definition 5.1* (See [24, Chaper 2]).*

A groupoid $\mathcal{G}$ is called topologically principal in case the set $\{u\in\mathcal{G}^{(0)}:\{\alpha\in\mathcal{G}:r(\alpha)=s(\alpha)\}=\{u\}\}$ is dense in $\mathcal{G}^{(0)}$. We say that $\mathcal{G}$ is essentially principal if for every nonempty closed invariant subset $D$ of $\mathcal{G}^{(0)}$, the groupoid $\mathcal{G}_{D}$ is topologically principal.

Note that, when $\mathcal{G}$ is second countable, [23, Proposition 3.3] follows that $\mathcal{G}$ is strongly effective if and only if it is essentially principal.

The following is the $C^{*}$-analogue of Theorem 3.4 (see also [7, Theorem 4.1]).

Proposition 5.2**.**

Let $\mathcal{G}$ be a second countable Hausdorff ample groupoid and let $\mathcal{B}$ be a basis for $\mathcal{G}^{(0)}$ containing compact open sets. Suppose that $\mathcal{G}$ is amenable (in the sense of [3]) and essentially principal. Then $C^{*}(\mathcal{G})=C^{\ast}_{r}(\mathcal{G})$ is $C^{\ast}$-purely infinite if and only if $1_{V}$ is $C^{\ast}$-properly infinite for every $V\in\mathcal{B}$.

Remark 5.3*.*

If $\mathcal{G}$ is second countable, amenable and essentially principal (strongly effective), then [8, Corollary 5.9] implies that $C_{0}(\mathcal{G}^{(0)})$ separates closed ideals of $C^{*}(\mathcal{G})$ in the sense that for every closed ideals $I\nsubseteq J$ of $C^{*}(\mathcal{G})$, there exists $f\in C_{0}(\mathcal{G}^{(0)})$ such that $f\in I\setminus J$. Thus, in this case, [21, Proposition 2.14] implies that $C^{*}$-purely infinite, $C^{*}$-weakly purely infinite, and $C^{*}$-strongly purely infinite $C^{*}$-algebras coincide.

Proof of Proposition 5.2..

Since the “only if” implication is immediate, we prove the “if” part only. Suppose that all $1_{V}$, for $V\in\mathcal{B}$, are $C^{\ast}$-properly infinite in $C^{*}(\mathcal{G})$. Using [18, Proposition4.7], it suffices to to show that every nonzero hereditary $C^{*}$-subalgebra in any quotient of $C^{*}(\mathcal{G})$ contains an infinite projection. For this, let $I$ be a closed ideal of $C^{*}(\mathcal{G})$ and let $A$ be a nonzero hereditary $C^{*}$-subalgebra of $C^{*}(\mathcal{G})/I$. By [8, Corollary 5.9], there is an open invariant $U\subseteq\mathcal{G}^{(0)}$ such that $I=I_{U}$, and whence $C^{*}(\mathcal{G})/I\cong C^{*}(\mathcal{G}_{D})$ where $D:=\mathcal{G}^{(0)}\setminus U$. So we may assume $A\subseteq C^{*}(\mathcal{G}_{D})$.

Let $0\neq a\in A$ be a positive element. Then [10, Lemmma 3.2] says that there exists a nonzero positive element $h\in C_{0}(D)$ such that $h\precsim_{C^{*}}a$ in $C^{*}(\mathcal{G}_{D})$. Now since $\{V\cap D:V\in\mathcal{B}\}$ is a basis for the induced topology on $\mathcal{G}^{(0)}_{D}$, there is $V\in\mathcal{B}$ such that $V\cap D\neq\emptyset$ and $h(x)>0$ for all $x\in\widetilde{V}:=V\cap D$. Consider $g\in C_{0}(D)$ defined by

[TABLE]

Then we have $ghg=1_{\widetilde{V}}$, and thus $1_{\widetilde{V}}\precsim h\precsim a$ in $C^{*}(\mathcal{G}_{D})$. By the argument after [18, Proposition 2.6], we may find $y\in C^{*}(\mathcal{G}_{D})$ such that $1_{\widetilde{V}}=y^{*}ay^{*}$. If we set $z:=a^{1/2}y$, then $z^{*}z=y^{*}ay=1_{\widetilde{V}}$ and $zz^{*}=a^{1/2}yy^{*}a^{1/2}\in A$, giving $zz^{*}\sim_{C^{*}}1_{\widetilde{V}}$. Note that $1_{\widetilde{V}}$ is (properly) infinite because it is the image of $1_{V}$ in the quotient $C^{*}(\mathcal{G}_{D})\cong C^{*}(\mathcal{G})/I$. Therefore, $zz^{*}$ is an infinite projection in $A$, as desired. ∎

We now prove the main result of this section, which is the nonsimple generalization of [9, Theorem 4.7 and Corollary 4.8] with an easier way.

Proposition 5.4**.**

Let $\mathcal{G}$ be a second countable Hausdorff ample groupoid. Suppose that $\mathcal{G}$ is amenable and strongly effective. If $A_{\mathbb{C}}(\mathcal{G})$ is properly purely infinite (in the sense of Definition 3.2), then $C^{*}(\mathcal{G})=C^{*}_{r}(\mathcal{G})$ is $C^{*}$-purely infinite.

Proof.

Suppose that $A_{\mathbb{C}}(\mathcal{G})$ is properly purely infinite. According to Proposition 5.2, it suffices to show that $1_{V}$ is $C^{*}$-properly infinite for every nonempty compact open $V\subseteq\mathcal{G}^{(0)}$. So fix some such $\emptyset\neq V\subseteq\mathcal{G}^{(0)}$. Theorem 3.4 says that $1_{V}$ is a properly infinite idempotent in $A_{\mathbb{C}}(\mathcal{G})$. Thus there exists $a,b\in M_{2}(A_{\mathbb{C}}(\mathcal{G}))$ such that $a(1_{V}\oplus 0)b=1_{V}\oplus 1_{V}$. We may consider $M_{2}(A_{\mathbb{C}}(\mathcal{G}))$ as a subalgebra of $M_{2}(C^{*}(\mathcal{G}))$ in the natural way, and then apply [25, Proposition 2.4 (iii) $\Rightarrow$ (ii)] to find a sequence $\{r_{i}\}\subseteq M_{2}(C^{*}(\mathcal{G}))$ such that $r_{i}(1_{V}\oplus 0)r_{i}^{*}\rightarrow 1_{V}\oplus 1_{V}$. Therefore, we have $1_{V}\oplus 1_{V}\precsim_{C^{*}}1_{V}\oplus 0$, meaning that $1_{V}$ is $C^{*}$-properly infinite. ∎

For any locally convex row-finite $k$-graph $\Lambda$, we may associate a universal $C^{*}$-algebra $C^{*}(\Lambda)$ [19, 22], which coincides with the groupoid $C^{*}$-algebra $C^{*}_{r}(\mathcal{G}_{\Lambda})$ (see [19, Corollary 3.5(i)] for example). Hence, Proposition 5.4 deduces the following result for $k$-graph algebras.

Corollary 5.5**.**

Let $\Lambda$ be a locally convex row-finite $k$-graph, which is strongly aperiodic. If $\mathrm{KP}_{\mathbb{C}}(\Lambda)$ is purely infinite, then $C^{*}(\Lambda)$ is $C^{*}$-purely infinite.

Proof.

By Theorem 4.2, $A_{\mathbb{C}}(\mathcal{G}_{\Lambda})=\mathrm{KP}_{\mathbb{C}}(\Lambda)$ is properly purely infinite. Since the groupoid $\mathcal{G}_{\Lambda}$ is always second-countable and amenable, Proposition 5.4 follows the result. ∎