1.1. Inverse semigroups

We recall the basic notions about inverse semigroups. See [Reference Lawson4] or [Reference Paterson8] for more details. An inverse semigroup $S$ is a semigroup where for every $s\in S$ there exists a unique $s^{*}\in S$ such that $s=ss^{*}s$ and $s^{*}=s^{*}ss^{*}$. We denote the set of all idempotents in $S$ by $E(S)\colon \!=\{e\in S\mid e^{2}=2\}$. It is known that $E(S)$ is a commutative subsemigroup of $S$. An inverse semigroup which consists of idempotents is called a (meet) semilattice. A zero element is a unique element $0\in S$ such that $0s=s0=0$ holds for all $s\in S$. A unit is a unique element $1\in S$ such that $1s=s1=s$ holds for all $s\in S$. In this paper, we assume that every inverse semigroup always has a zero element, although it does not necessarily have a unit. An inverse semigroup with a unit is called an inverse monoid. By a subsemigroup of $S$, we mean a subset of $S$ which is closed under the product and inverse of $S$. For $s,\,t\in S$, we write $s\leq t$ if $ts^{*}s=s$ holds. Then this defines a partial order on $S$. Note that $e\leq f$ holds if and only if $ef=e$ holds for $e,\,f\in E(S)$. A pair $s,\,t\in S$ is said to be compatible if $s^{*}t,\, st^{*}\in E(S)$ holds. Notice that $s,\,t$ are compatible if there exists $u\in S$ such that $s,\,t\leq u$. A subsemigroup of an inverse semigroup $S$ is said to be wide if it contains $E(S)$. A subset $I\subset E(S)$ is called an ideal if $e\in I$ and $f\leq e$ implies $f\in I$. A subset $C\subset I$ of an ideal $I\subset E(S)$ is called a cover if for every $e\in I{\setminus} \{0\}$ there exists $c\in C$ such that $ec\not =0$.

For a topological space $X$, we denote by $I_X$ the set of all homeomorphisms between open sets in $X$. Then $I_X$ is an inverse semigroup with respect to the product defined by the composition of maps. For $f,\,g\in I_X$, note that $f\leq g$ holds if and only if $\operatorname {\mathrm {dom}} f\subset \operatorname {\mathrm {dom}} g$ and $f(x)=g(x)$ hold for all $x\in \operatorname {\mathrm {dom}} f$.

1.2. Étale groupoids

We recall the basic notions on étale groupoids. See [Reference Paterson8, Reference Sims11] for more details.

A groupoid is a set $G$ together with a distinguished subset $G^{(0)}\subset G$, source and range maps $d,\,r\colon G\to G^{(0)}$ and a multiplication

\begin{align*} G^{(2)}\colon\!= \{(\alpha,\beta)\in G\times G\mid d(\alpha)=r(\beta)\}\ni (\alpha,\beta)\mapsto \alpha\beta \in G \end{align*}

such that

1. (1) for all $x\in G^{(0)}$, $d(x)=x$ and $r(x)=x$,

2. (2) for all $\alpha \in G$, $\alpha d(\alpha )=r(\alpha )\alpha =\alpha$,

3. (3) for all $(\alpha,\,\beta )\in G^{(2)}$, $d(\alpha \beta )=d(\beta )$ and $r(\alpha \beta )=r(\alpha )$,

4. (4) if $(\alpha,\,\beta ),\,(\beta,\,\gamma )\in G^{(2)}$, we have $(\alpha \beta )\gamma =\alpha (\beta \gamma )$,

5. (5) every $\gamma \in G$, there exists $\gamma '\in G$ which satisfies $(\gamma ',\,\gamma ),\, (\gamma,\,\gamma ')\in G^{(2)}$ and $d(\gamma )=\gamma '\gamma$ and $r(\gamma )=\gamma \gamma '$.

Since the element $\gamma '$ in (5) is uniquely determined by $\gamma$, $\gamma '$ is called the inverse of $\gamma$ and denoted by $\gamma ^{-1}$. We call $G^{(0)}$ the unit space of $G$. A subgroupoid of $G$ is a subset of $G$ which is closed under the inversion and multiplication. A subgroupoid of $G$ is said to be wide if it contains $G^{(0)}$.

A topological groupoid is a groupoid equipped with a topology where the multiplication and the inverse are continuous. A topological groupoid is said to be étale if the source map is a local homeomorphism. Note that the range map of an étale groupoid is also a local homeomorphism. An étale groupoid is said to be ample if it has an open basis which consists of compact sets. In this paper, we mainly treat ample groupoids.

A topological groupoid $G$ is said to be topologically principal if the set

\begin{align*} \{x\in G^{(0)}\mid G(x)=\{x\}\} \end{align*}

is dense in $G^{(0)}$, where $G(x)$ is the isotropy group at $x\in G^{(0)}$ :

\begin{align*} G(x)\colon\!= \{\alpha\in G\mid d(\alpha)=r(\alpha)=x\}. \end{align*}

1.3. Étale groupoids associated with inverse semigroup actions

An étale groupoid arises from an action of an inverse semigroup to a topological space. We recall how to construct an étale groupoid from an inverse semigroup action. We begin with the definition of an inverse semigroup action.

Let $X$ be a topological space. Recall that $I_X$ is an inverse semigroup of homeomorphisms between open sets in $X$. An action $\alpha \colon S\curvearrowright X$ is a semigroup homomorphism $S\ni s\mapsto \alpha _s\in I_X$. In this paper, we always assume that every action $\alpha$ satisfies $\bigcup _{e\in E(S)}\operatorname {\mathrm {dom}}(\alpha _e)=X$ and $\operatorname {\mathrm {dom}}(\alpha _0)=\emptyset$. For $e\in E(S)$, we denote the domain of $\alpha _e$ by $D_e^{\alpha }$. Then $\alpha _s$ is a homeomorphism from $D_{s^{*}s}^{\alpha }$ to $D_{ss^{*}}^{\alpha }$. We often omit $\alpha$ of $D_{e}^{\alpha }$ if there is no chance to confuse.

For an action $\alpha \colon S\curvearrowright X$, we associate an étale groupoid $S\ltimes _{\alpha }X$ as the following. First we put the set $S*X\colon \!= \{(s,\,x) \in S\times X \mid x\in D^{\alpha }_{s^{*}s}\}$. Then we define an equivalence relation $\sim$ on $S*X$ by declaring that $(s,\,x)\sim (t,\,y)$ holds if

\begin{align*} x=y\text{ and there exists }e\in E(S)\text{ such that }x\in D^{\alpha}_e\text{ and }se=te. \end{align*}

Set $S\ltimes _{\alpha }X\colon \!= S*X/{\sim }$ and denote the equivalence class of $(s,\,x)\in S*X$ by $[s,\,x]$. The unit space of $S\ltimes _{\alpha }X$ is $X$, where $X$ is identified with the subset of $S\ltimes _{\alpha }X$ via the injection

\begin{align*} X\ni x\mapsto [e,x] \in S\ltimes_{\alpha}X, x\in D^{\alpha}_e. \end{align*}

The source map and range maps are defined by

\begin{align*} d([s,x])=x, r([s,x])=\alpha_s(x) \end{align*}

for $[s,\,x]\in S\ltimes _{\alpha }X$. The product of $[s,\,\alpha _t(x)],\,[t,\,x]\in S\ltimes _{\alpha }X$ is $[st,\,x]$. The inverse should be $[s,\,x]^{-1}=[s^{*},\,\alpha _s(x)]$. Then $S\ltimes _{\alpha }X$ is a groupoid in these operations. For $s\in S$ and an open set $U\subset D_{s^{*}s}^{\alpha }$, define

\begin{align*} [s, U]\colon\!= \{[s,x]\in S\ltimes_{\alpha}X\mid x\in U\}. \end{align*}

These sets form an open basis of $S\ltimes _{\alpha }X$. In these structures, $S\ltimes _{\alpha }X$ is an étale groupoid.

2.1. Correspondence between subsemigroups and subgroupoids

We begin with the definition of a strongly tight action.

We remark that if there exists a strongly tight action $\alpha \colon S\curvearrowright X$, then $X$ is totally disconnected and $S\ltimes _{\alpha }X$ is an ample groupoid.

Strongly tight actions are related with the actions on tight spectrums of inverse semigroups, which are investigated in [Reference Exel and Pardo3]. We will see a relation between strongly tight actions and tight groupoids in § 5.

We construct subsemigroups from wide groupoids.

Definition 2.1.2 Let $S$ be an inverse semigroup, $X$ be a locally compact Hausdorff space and $\alpha \colon S\curvearrowright X$ be an action. Put $G\colon \!= S\ltimes _{\alpha }X$. For a wide subgroupoid $H\subset G$, we define

\begin{align*} T_H\colon\!= \{s\in S\mid [s,D_{s^{*}s}]\subset H\}. \end{align*}

Proof. For $e\in E(S)$, $[e,\,D_e]\subset G^{(0)}\subset H$ holds. Hence $T_H$ contains $E(S)$.

Next, we show that $T_H$ is a subsemigroup of $S$. We show $st\in T_H$ for $s,\,t\in T_H$. For $x\in D_{(st)^{*}st}$, it follows that $[s,\,\alpha _t(x)],\,[t,\,x]\in H$ from $s,\,t\in T_H$. Thus, we obtain

\begin{align*} [st,x]=[s,\alpha_t(x)][t,x]\in H. \end{align*}

Therefore, we have $[st,\,D_{(st)^{*}st}]\subset H$ and $st\in T_H$.

It is clear that $T_H$ is closed under the inverse. Hence $T_H$ is a wide subsemigroup of $S$.

We define a class of subsemigroups which corresponds to open wide subgroupoids (c.f. Theorem 2.1.10).

Proof. Take $s\in S$ and assume that there exists a finite set $F\subset E(S)$ such that $sf\in T_H$ for all $f\in F$ and $D_{s^{*}s}\subset \bigcup _{f\in F}D_f$. It suffices to show $s\in T_H$. For $x\in D_{s^{*}s}$, there exists $f\in F$ with $x\in D_f$. Since we have $sf\in T_H$, it follows

\begin{align*} [s,x]=[sf,x]\in H. \end{align*}

Thus, we obtain $[s,\,D_{s^{*}s}]\subset H$ and therefore $s\in T_H$.

The proof of the next proposition is left to the reader.

Proposition 2.1.7 Let $S$ be an inverse semigroup, $X$ be a locally compact Hausdorff space and $\alpha \colon S\curvearrowright X$ be an action. For a wide subsemigroup $T\subset S,$ the map

\begin{align*} T\ltimes_{\alpha}X\ni [t,x]\mapsto [t,x]\in S\ltimes_{\alpha}X \end{align*}

is an open map and an isomorphism onto its image.

Via the map in the previous proposition, $T\ltimes _{\alpha }X$ is identified with the wide open subgroupoid of $S\ltimes _{\alpha }X$.

Proof. Assume that $[s,\,x]\in T_H\ltimes _{\alpha }X$. Then there exists $t\in T_H$ such that $[s,\,x]=[t,\,x]$. Now we have

\begin{align*} [s,x]=[t,x]\in [t,D_{t^{*}t}]\subset H. \end{align*}

Next, we show the other inclusion $T_H\ltimes _{\alpha }X\supset H$ under the assumption that $\alpha$ is strongly tight and $H$ is open. Take $[s,\,x]\in H$. Since $H$ is open and $\alpha$ is strongly tight, there exists $e\in E(S)$ such that $x\in D_e\subset D_{s^{*}s}$ and $[s,\,D_e]\subset H$. One can see $[se,\,D_{(se)^{*}se}]\subset H$, so we have $se\in T_H$. Therefore, it follows $[s,\,x]=[se,\,x]\in T_H\ltimes _{\alpha }X$.

The next theorem follows from Lemmas 2.1.8 and 2.1.9.

Theorem 2.1.10 Let $S$ be an inverse semigroup, $X$ be a locally compact Hausdorff space and $\alpha \colon S\curvearrowright X$ be an action. Assume that $\alpha \colon S\curvearrowright X$ is strongly tight and put $G\colon \!= S\ltimes _{\alpha }X$. Let $\mathcal {T}$ denote the set of all wide $\alpha$-join closed subsemigroups of $S$. In addition, let $\mathcal {H}$ denote the set of all wide open subgroupoids of $G$. Then maps

\begin{align*} \mathcal{T}\ni T\to T\ltimes_{\alpha}X\in \mathcal{H} \end{align*}

and

\begin{align*} \mathcal{H}\ni H\to T_H\in \mathcal{T} \end{align*}

are inverse maps of each other.

2.2. Closedness of subgroupoid

We give conditions where $T\ltimes _{\alpha }X$ is closed in $S\ltimes _{\alpha }X$.

Definition 2.2.1 Let $S$ be an inverse semigroup and $T\subset S$ be a wide subsemigroup. For $s\in S$, we define $\mathcal {J}_s^{T}\subset E(S)$ as

\begin{align*} \mathcal{J}_s^{T}\colon\!= \{e\in E(S)\mid se\in T \text{ and } e\leq s^{*}s\}. \end{align*}

We remark that

\begin{align*} \mathcal{J}_s^{E(S)}=\{e\in E(S)\mid e\leq s\} \end{align*}

holds. This ideal appears in [Reference Exel and Pardo3, Definition 3.11].

Assume that an action $\alpha \colon S\curvearrowright X$ is given. For an ideal $\mathcal {J}\subset E(S)$, we define $D(\mathcal {J})\colon \!=\bigcup _{e\in \mathcal {J}}D_e$. The next lemma is a slight generalization of [Reference Exel and Pardo3, Proposition 3.14].

Lemma 2.2.3 Let $S$ be an inverse semigroup, $X$ be a locally compact Hausdorff space and $\alpha \colon S\curvearrowright X$ be an action. Assume that we are given a wide subsemigroup $T\subset S$. Then the formula

\begin{align*} [s,D(\mathcal{J}^{T}_s)]=[s,D_{s^{*}s}]\cap (T\ltimes_{\alpha}X) \end{align*}

holds for all $s\in S$.

Proof. Take $[s,\,x]\in [s,\, D(\mathcal {J}^{T}_s)]$. Then there exists $e\in \mathcal {J}^{T}_s$ with $x\in D_e$. By the definition of $\mathcal {J}^{T}_s$, we have $se\in T$ and $e\leq s^{*}s$. Hence we obtain

\begin{align*} [s,x]=[se,x]\in [s,D_{s^{*}s}]\cap (T\ltimes_{\alpha}X). \end{align*}

Now we have shown $[s,\,D(\mathcal {J}^{T}_s)]\subset [s,\,D_{s^{*}s}]\cap (T\ltimes _{\alpha }X)$. To show the reverse inclusion, take $[s,\,x] \in [s,\,D_{s^{*}s}]\cap (T\ltimes _{\alpha }X)$. Since $[s,\,x]$ belongs to $T\ltimes _{\alpha }X$, there exists $t\in T$ and $f\in E(S)$ such that $sf=tf$ and $x\in D_f$ hold. Since we have $ss^{*}sf=sf=tf\in T$, $s^{*}sf$ belongs to $\mathcal {J}^{T}_s$. Since we also have $x\in D_{s^{*}sf}\subset D(\mathcal {J}^{T}_s)$, we obtain $[s,\,x] \in [s,\,D(\mathcal {J}_s^{T})]$.

Proof. First, we show that (1) implies (2). By Lemma 2.2.3 and (1), $[s,\,D(\mathcal {J}_s^{T})]$ is a closed subset of $[s,\,D_{s^{*}s}]$. Since the restriction of the domain map $d\colon [s,\,D_{s^{*}s}]\to D_{s^{*}s}$ is a homeomorphism, $d([s,\,D(\mathcal {J}_s^{T})])=D(\mathcal {J}_s^{T})$ is closed in $D_{s^{*}s}$. Next, we show that (2) implies (1). It follows that $[s,\,D(\mathcal {J}^{T}_s)]$ is a closed subset of $[s,\,D_{s^{*}s}]$ from the same argument in the above and (2). We have that $[s,\,D_{s^{*}s}]{\setminus} [s,\,D(\mathcal {J}^{T}_s)]$ is open in $S\ltimes _{\alpha }X$ since $[s,\,D_{s^{*}s}]$ is open in $S\ltimes _{\alpha }X$ and $[s,\,D(\mathcal {J}^{T}_s)]$ is closed in $[s,\,D_{s^{*}s}]$. One can see that the formula

\begin{align*} S\ltimes_{\alpha}X{\setminus} T\ltimes_{\alpha}X=\bigcup_{s\in S}([s,D_{s^{*}s}]{\setminus} [s,D(\mathcal{J}^{T}_s)]) \end{align*}

holds. Hence $S\ltimes _{\alpha }X{\setminus} T\ltimes _{\alpha }X$ is open in $S\ltimes _{\alpha }X$, which implies $T\ltimes _{\alpha }X$ is closed in $S\ltimes _{\alpha }X$.

The next Lemma is essentially same as the [Reference Exel and Pardo3, Proposition 3.7]. We give a proof for the reader's convenience.

Now we obtain the characterization about the closedness of open wide subgroupoids.

Wide clopen subgroupoids arise from partial group homomorphisms. We observe this fact in the remainder of this subsection.

Let $S$ be an inverse semigroup and $\Gamma$ be a group. Put $S^{\times }\colon \!= S{\setminus} \{0\}$. A map $\sigma \colon S^{\times }\to \Gamma$ is called a partial homomorphism if $\sigma (st)=\sigma (s)\sigma (t)$ holds for any pair $s,\,t\in S^{\times }$ with $st\not =0$. A partial homomorphism gives us a suitable subsemigroup as follows.

Proof. First, we show (1). One can see that $\ker \sigma$ is a wide subsemigroup of $S$ in a straightforward way. We show $\ker \sigma$ is $\alpha$-join closed. Take $s\in S$ and assume that there exists a finite set $F\subset E(S)$ such that $sF\subset \ker \sigma$ and $D_{s^{*}s}\subset \bigcup _{f\in F}D_f$. It suffices to show $s\in \ker \sigma$. We may assume that $s\not =0$. Then there exists $f\in F$ such that $D_{s^{*}s}\cap D_f\not =\emptyset$, which implies $sf\not =0$. Since we have $sf\in \ker \sigma$, it follows

\begin{align*} \sigma(s)=\sigma(s)e=\sigma(s)\sigma(f)=\sigma(sf)=e. \end{align*}

Hence $s\in \ker \sigma$.

Next, we show (2). Although it is possible to apply Proposition 2.2.4, we show (2) using a cocycleFootnote ¹ on a groupoid. We define the map $c_{\sigma }\colon S\ltimes _{\alpha }X\to \Gamma$ by

\begin{align*} c_{\sigma}([s,x])=\sigma(s),\quad [s,x]\in S\ltimes_{\alpha}X. \end{align*}

Then $c_{\sigma }$ is a continuous cocycle. One can see that

\begin{align*} \ker\sigma\ltimes_{\alpha}X= c_{\sigma}^{{-}1}(e) \end{align*}

holds. Hence $\ker \sigma \ltimes _{\alpha }X$ is closed in $S\ltimes _{\alpha }X$.

3.1. Inverse semigroups of compact bisections

Let $G$ be an ample étale groupoid. Recall that an open set $U\subset G$ is called a bisection if the restrictions $d|_U$ and $r|_U$ are homeomorphisms onto the images. Let $I(G)$ denote the set of all compact bisections of $G$. For $U,\,V\in I(G)$, their product $UV$ is defined by

\begin{align*} UV\colon\!=\{\alpha\beta\in G\mid \alpha\in U, \beta\in V, d(\alpha)=r(\beta)\}. \end{align*}

Then $UV$ belongs to $I(G)$. It is known that $I(G)$ becomes an inverse semigroup. Note that the inverse of $U\in I(G)$ is given by

\begin{align*} U^{{-}1}=\{\alpha^{{-}1}\in G\mid \alpha\in U\}. \end{align*}

The order of $I(G)$ as an inverse semigroup coincides with the order defined by inclusion. A pair $U,\,V\in I(G)$ is said to be compatible if $U^{-1}V$ and $UV^{-1}$ belong to $E(I(G))$. If $U,\,V\in I(G)$ are compatible, $U\cup V$ is an element of $I(G)$. Note that $U\cup V$ is the least upper bound of $\{U,\,V\}$. Thus, $I(G)$ admits joins of compatible pairs in $I(G)$. A subsemigroup $T\subset I(G)$ is said to be join closed if all joins of compatible pair of $T$ also belongs to $T$.

For $U\in I(G)$, we have a homeomorphism $\rho _U\colon d(U)\to r(U)$ defined by

\begin{align*} \rho_U(d(\alpha))=r(\alpha), \alpha\in U. \end{align*}

Then the map $U\mapsto \rho _U$ defines an action $\rho \colon I(G)\curvearrowright G^{(0)}$. One can see that $\rho$ is strongly tight. The following theorem is essentially same as [Reference Matsnev and Resende7, Theorem 2.8].

Proof. For $\alpha \in G$, there exists $U_{\alpha }\in I(G)$ such that $\alpha \in U_{\alpha }$ since $G$ is ample. Then $[U_{\alpha },\, d(\alpha )]\in I(G)\ltimes _{\rho }G^{(0)}$ is independent of the choice of $U_{\alpha }$. Thus, we obtain the map

\begin{align*} \Phi\colon G\ni \alpha\mapsto [U_{\alpha},d(\alpha)]\in I(G)\ltimes_{\rho}G^{(0)}. \end{align*}

One can see that $\Phi$ is an isomorphism as a morphism between étale groupoids. Indeed, the map $\Psi \colon I(G)\ltimes _{\rho }G^{(0)}\to G$ defined by $\Psi ([U,\,x])= d_U^{-1}(x)$ is the inverse map of $\Phi$.

Proof. Assume that $T\subset I(G)$ is join closed. Take $U\in I(G)$ and there exists a finite set $\mathcal {F}\subset E(I(G))$ such that $U \mathcal {F}\in T$ and $D_{U^{*}U}^{\rho }\subset \bigcup _{O\in \mathcal {F}}D_O^{\rho }$. Observe that elements in $U\mathcal {F}$ are pairwisely compatible and $\bigvee _{O\in \mathcal {F}} UO=U$ holds. Since $T$ is join closed, $U$ belongs to $T$.

To show the converse, assume that $T\subset I(G)$ is $\rho$-join closed. Let $U,\,V\in T$ be compatible. Put $\mathcal {F}\colon \!= \{U^{-1}U,\, V^{-1}V\}\subset E(I(G))$. Then one can see that $(U\cup V) \mathcal {F}\!=\!\{U,\,V\}\!\subset T$ hold. Since we have

\begin{align*} \operatorname{\mathrm{dom}}(\rho_{U\cup V})=d(U)\cup d(V)=\operatorname{\mathrm{dom}} \rho_U\cup \operatorname{\mathrm{dom}}\rho_V, \end{align*}

we obtain $U\cup V\in T$ by the $\rho$-closedness of $T$.

Theorem 2.1.10, Theorem 3.1.1 and Lemma 3.1.2 yield the next corollary.

3.2. Polycyclic monoids

We apply Theorem 2.1.10 to the polycyclic monoids $P_n$. See [Reference Lawson5] or [Reference Paterson8, Example 3 in Chapter 4.2] for details on the polycyclic monoids. Remark that the polycyclic monoids are called the Cuntz semigroups in [Reference Paterson8].

Definition 3.2.1 Let $n\geq 2$ be a natural number. The polycyclic monoid $P_n$ is an inverse monoid defined by

\begin{align*} P_n\colon\!= i< s_1,s_2,\dots,s_n \mid s_i^{*}s_j=\delta_{i,j} 1>. \end{align*}

Set $\Sigma _n\colon \!= \{1,\,2,\,\dots,\,n\}$ and

\begin{align*} \Sigma_n^{\mathbb{N}}\colon\!=\{(x_i)_{i=1}^{\infty}\mid x_i\in \Sigma_n \text{ for all $i\in\mathbb{N}$}\}. \end{align*}

It follows that $\Sigma _n^{\mathbb {N}}$ is a compact Hausdorff space from Tychonoff's theorem. We write a finite sequence on $\Sigma _n$ like $\mu =(\mu _1,\,\mu _2,\,\dots,\,\mu _l)$, where each $\mu _j$ is an element of $\Sigma _{n}$. Here, $l\in \mathbb {N}$ is called the length of $\mu$, which we denote by $\lvert \mu \rvert$. The only element of length zero is denoted by $\varepsilon$, which is called the empty word. We denote the set of all finite sequence on $\Sigma _n$ by $\Sigma _n^{*}$. For $\mu \in \Sigma _n^{*}$, we define a cylinder set $C(\mu )\subset \Sigma _n^{\mathbb {N}}$ as the set of all infinite sequences which begin with $\mu$ :

\begin{align*} C(\mu)\colon\!= \{(x_i)_{i=1}^{\infty}\in \Sigma_n^{\mathbb{N}}\mid x_i=\mu_i \text{ for all $i=1,\,2,\,\dots,\,\lvert\mu\rvert$}\}. \end{align*}

We represent an element of $C(\mu )$ as $\mu x$ with $x\in \Sigma _n^{\mathbb {N}}$. Each $C(\mu )$ is a compact open set of $\Sigma _n^{\mathbb {N}}$ and the family of all $C(\mu )$ is a basis of $\Sigma _n^{\mathbb {N}}$. For $\mu \in \Sigma _n^{*}$, we define $s_{\mu }\in P_n$ as

\begin{align*} s_{\mu}\colon\!= s_{\mu_1}s_{\mu_2}\cdots s_{\mu_{\lvert\mu\rvert}}. \end{align*}

For the empty word $\varepsilon \in \Sigma _n^{\mathbb {N}}$, we define $s_{\varepsilon }=1$. It is known that an element of $P_n{\setminus} \{0\}$ is represented as the form $s_{\mu }s_{\nu }^{*}$ for unique $\mu,\,\nu \in \Sigma _n^{*}$.

Now we define an action $\beta \colon P_n\curvearrowright \Sigma _n^{\mathbb {N}}$. For $s_{\mu }s_{\nu }^{*}\in P_n$, define $\beta _{s_{\mu }s_{\nu }^{*}}\colon C(\mu )\to C(\nu )$ by

\begin{align*} \beta_{s_{\mu}s_{\nu}^{*}}(\nu x)=\mu x , x\in\Sigma_n^{\mathbb{N}}. \end{align*}

Then the map $s_{\mu }s_{\nu }^{*}\mapsto \beta _{s_{\mu }s_{\nu }^{*}}$ defines an action $\beta \colon P_n\curvearrowright \Sigma _n^{\mathbb {N}}$. Since the domain of $s_{\mu }s_{\mu }^{*}$ coincides with $C(\mu )$, the action $\beta$ is strongly tight.

For $k,\,l\in \mathbb {N}$, we define

\begin{align*} P_n^{k,l}\colon\!=\{s_{\mu}s_{\nu}^{*}\in P_n\mid\lvert\mu\rvert=k,\lvert\nu\rvert=l\}. \end{align*}

Observe that

\begin{align*} M_n\colon\!= \bigcup_{k\in\mathbb{N}}P_n^{k,k}\cup\{0\}=\{s_{\mu}s_{\nu}^{*}\in P_n \mid \lvert{\mu}\rvert=\lvert\nu\rvert\}\cup\{0\} \end{align*}

is a wide subsemigroup of $P_n$.

We investigate $\beta$-join closed subsemigroups $T\subset P_n$ such that $M_n\subset T$. For $m\in \mathbb {N}$, define

\begin{align*} P_n^{m}\colon\!= \bigcup_{k-l\in m\mathbb{Z}} P_n^{k,l}\cup\{0\}. \end{align*}

Then one can see that $P_n^{m}$ is a $\beta$-join closed subsemigroup which contains $M_n$. Notice that $P_n^{0}=M_n$. Conversely, we obtain the following proposition.

In order to prove this proposition, we prepare a few lemmas. The next lemma follows from straightforward calculations.

Lemma 3.2.3 For $i,\,j,\,k,\,l\in \mathbb {N},$ we have

\begin{align*} P_n^{i,j}P_n^{k,l}= \begin{cases} P_n^{i+k-j,l}\cup\{0\} & (k\geq j),\\ P_n^{i,j-k+l}\cup\{0\} & (k\leq j). \end{cases} \end{align*}

Proof.

1. (1) Assume $\lvert \mu \rvert =\lvert \mu '\rvert$ and $\lvert \nu \rvert =\lvert \nu '\rvert$ hold for $\mu ',\,\nu '\in \Sigma _n^{\mathbb {N}}$. Then we have $s_{\mu '}s_{\mu }^{*},\, s_{\nu }s_{\nu '}^{*}\in M_n\subset T$. Since we assume $s_{\mu }s_{\nu }^{*}\in T$, it follows

   \begin{align*} s_{\mu'}s_{\nu'}=s_{\mu'}s_{\mu}^{*}s_{\mu}s_{\nu}^{*}s_{\nu}s_{\nu'}^{*}\in T. \end{align*}
   Hence we have $P_n^{\lvert \mu \rvert,\lvert \nu \rvert }\subset T$.

2. (2) is clear, so we show (3) next. Take $s_{\mu }s_{\nu }^{*}\in P_n^{k,l}$ and $x,\,y\in \Sigma _n$ arbitrarily. Then we have

   \begin{align*} s_{\mu x}s_{\nu x}^{*}=s_{\mu }s_{\nu}^{*} s_{\nu x}s_{\nu x}^{*}\in T, \end{align*}
   where we use the fact $s_{\nu x}s_{\nu x}^{*}\in M_n\subset T$. Using (1) in the above, we obtain $P_n^{k+1.l+1}\subset T$.

Finally we show (4) under the assumption that $T$ is $\beta$-join closed. Take $s_{\mu }s_{\nu }^{*}\in P_n^{k-1,l-1}$. For each $x\in \Sigma _n$, we have

\begin{align*} s_{\mu}s_{\nu}^{*}s_{\nu x}s_{\nu x}^{*}=s_{\mu x}s_{\nu x}^{*}\in T, \end{align*}

since we assume $P_n^{k,l}\subset T$. Observe that

\begin{align*} D^{\beta}_{(s_{\mu}s_{\nu}^{*})^{*}s_{\mu}s_{\nu}^{*} }=D^{\beta}_{s_{\nu}s_{\nu}^{*}}=\bigcup_{x\in \Sigma_n}D^{\beta}_{s_{\nu x}s_{\nu x}^{*}}({=}C(\nu)). \end{align*}

Since $T$ is $\beta$-join closed, we have $s_{\mu }s_{\nu }^{*}\in T$. Hence we have shown $P_n^{k-1,l-1}\subset T$.

Proof of Proposition 3.2.2. We may assume that $M_n\subsetneq T$. We define

\begin{align*} m\colon\!= \min \{\left\lvert\lvert\mu\rvert- \lvert\nu\rvert \right\rvert\in\mathbb{N}\mid s_{\mu}s_{\nu}^{*}\in T{\setminus} M_n \}({>}0). \end{align*}

We show $T=P_n^{m}$. By the definition of $m$, there exists $s_{\mu }s_{\nu }^{*}\in T$ such that $\lvert \lvert \mu \rvert -\lvert \nu \rvert \rvert =m$. Since $T$ is closed under the inverse, we may assume $\lvert \mu \rvert -\lvert \nu \rvert =m$. Using (1) of Lemma 3.2.4, we have $P_m^{\lvert \mu \rvert,\lvert \nu \rvert }\subset T$. Applying (4) of Lemma 3.2.4 repeatedly, we obtain $P_n^{m,0}\subset T$ and it follows $P_n^{0,m}\subset T$ from (2) of Lemma 3.2.4. Now one can see that $P_n^{k,l}\subset T$ holds for $k,\,l$ with $k-l\in m\mathbb {Z}$. Hence we obtain $P_n^{m}\subset T$.

Next, we show $T\subset P_n^{m}$. Assume that there exists $s_{\mu }s_{\nu }^{*}\in T$ such that $s_{\mu }s_{\nu }^{*}\not \in P_n^{m}$. We may assume that $\lvert \mu \rvert >\lvert \nu \rvert$. Take $a,\,b\in \mathbb {N}$ such that $\lvert \mu \rvert -\lvert \nu \rvert =am+b$ and $1\leq b\leq m-1$. We have $P_n^{\lvert \mu \rvert,\lvert \nu \rvert }\subset T$ by (1) of Lemma 3.2.4. Using (4) of Lemma 3.2.4 repeatedly, we have $P_n^{am+b,0}\subset T$. Since we have $P_n^{m,0}\subset T$, it follows

\begin{align*} P_n^{(a-1)m+b,0}\subset P_n^{am+b,0}P_n^{m,0}\subset T, \end{align*}

where we used Lemma 3.2.3. Repeating this argument inductively, we obtain $P_n^{b,0}\subset T$. This contradicts to the minimality of $m$. Now we have shown $T=P_n^{m}$.

By Theorem 2.1.10 and Proposition 3.2.2, an open proper intermediate subgroupoid between $P_n\ltimes _{\alpha }\Sigma _n^{\mathbb {N}}$ and $M_n\ltimes _{\beta }\Sigma _n^{\mathbb {N}}$ is given by the form $P_n^{m}\ltimes _{\beta }\Sigma _n^{\mathbb {N}}$ for some $m\in \mathbb {N}$. Now we see $P_n^{m}\ltimes _{\beta }\Sigma _n^{\mathbb {N}}$ is closed. Observe that $P_n\ltimes _{\beta }\Sigma _n^{\mathbb {N}}$ has a continuous cocycle $c\colon P_n\ltimes _{\beta } \Sigma _n^{\mathbb {N}}\to \mathbb {Z}$ defined by $c([s_{\mu }s_{\nu }^{*},\,x])=\lvert \mu \rvert -\lvert \nu \rvert$. Since we have $P_n^{m}\ltimes _{\beta }\Sigma _n^{\mathbb {N}}=c^{-1}(m\mathbb {Z})$, it follows that $P_n^{m}\ltimes _{\beta }\Sigma _n^{\mathbb {N}}$ is a closed subset of $P_n\ltimes _{\beta }\Sigma _n^{\mathbb {N}}$. Hence we obtain the next proposition.

It follows from Corollary 5.2.5 that $P_n\ltimes _{\beta }\Sigma _{n}^{\mathbb {N}}$ is isomorphic to the tight groupoid of $P_n$. See § 5 for more details.

4.1. Analysis of cartan intermediate subalgebras by using inverse semigroups

In this section, we explain a correspondence between Cartan intermediate subalgebras and certain subsemigroups of an inverse semigroup.

We investigate a certain class of intermediate C*-subalgebras between Cartan pairs defined as follows.

Renault's cerebrated theorem states that a Cartan pair arises from a twisted groupoid. We refer to [Reference Brown, Exel, Fuller, Pitts and Reznikoff1, Reference Renault10, Reference Sims11] for twists of étale groupoids. A twisted groupoid over $G$ is a topological groupoid $\Sigma$ with the central extension

\begin{align*} G^{(0)}\times \mathbb{T} \hookrightarrow \Sigma \overset{q}{\twoheadrightarrow} G, \end{align*}

where $\mathbb {T}$ is the circle group. In this paper, this twist is abbreviated to $q\colon \Sigma \to G$. We denote the reduced C*-algebra of the twist $q\colon \Sigma \to G$ by $C^{*}_{\lambda }(\Sigma )$. Recall that $C^{*}_{\lambda }(\Sigma )$ contains $C_0(G^{(0)})$ as a subalgebra. We denote the reduced C*-algebra of $G$ by $C^{*}_{\lambda }(G)$, which is isomorphic to the reduced C*-algebra of the trivial twist $G\times \mathbb {T}\to G$.

From now on, we identify $C^{*}_{\lambda }(\Sigma )$ and $C_0(G^{(0)})$ with $A$ and $D$ respectively for a Cartan pair $(A,\,D)$.

Let $q\colon \Sigma \to G$ be a twist and $H\subset G$ be a wide open subgroupoid. [Reference Brown, Exel, Fuller, Pitts and Reznikoff1, Lemma 3.2] states that $\Sigma _H\colon \!= q^{-1}(H)$ naturally becomes a twist over $H$ and there exists a natural inclusion $C^{*}_{\lambda }(\Sigma _H)\subset C^{*}_{\lambda }(\Sigma )$. The authors in [Reference Brown, Exel, Fuller, Pitts and Reznikoff1] showed this map $H\mapsto C^{*}_{\lambda }(\Sigma _H)$ gives a certain correspondence as follows.

Combining Theorem 4.1.5 with Theorem 2.1.10, we obtain the next Corollary.

Example 4.1.7 We investigate certain subalgebras of the Cuntz algebras by using the polycyclic monoids here. For $n\in \mathbb {N}$ with $n\geq 2$, the Cuntz algebra $O_n$ is the universal unital C*-algebra generated by isometries $S_1,\,\dots,\,S_n$ which satisfy Cuntz relation as follows:

\begin{align*} S_i^{*}S_j=\delta_{i,j}1, \sum_{i=1}^{n}S_iS_i^{*}=1. \end{align*}

For a finite sequence $\mu =(\mu _1,\,\dots,\,\mu _l)$ on $\{1,\,\dots,\,n\}$, we define

\begin{align*} S_{\mu}\colon\!= S_{\mu_1}S_{\mu_2}\cdots S_{\mu_l}. \end{align*}

Then $O_n$ is the closure of the linear span of $\{S_{\mu }S_{\nu }^{*}\}_{\mu,\nu }$, where $\mu$ and $\nu$ are taken over the all finite sequences on $\{1,\,\dots,\,n\}$. Let $D_n$ be the subalgebra of $O_n$ generated by $\{S_{\mu }S_{\mu }^{*}\}_{\mu }$, where $\mu$ is taken over the all finite sequences on $\{1,\,\dots,\,n\}$. We denote the gauge action by $\tau \colon \mathbb {T}\curvearrowright O_n$. Note that the gauge action satisfies $\tau _z(S_i)=zS_i$ for all $z\in \mathbb {T}$ and $i=1,\,2,\,\dots,\,n$. We denote the fixed point algebra of $\tau$ by

\begin{align*} O_{n}^{\tau}\colon\!= \bigcap_{z\in\mathbb{T}}\{x\in O_n\mid \tau_z(x)=x\}. \end{align*}

Then $O_{n}^{\tau }$ is the closure of the linear span of

\begin{align*} \{S_{\mu}S_{\nu}^{*}\in O_n\mid \lvert\mu\rvert=\lvert\nu\rvert\}, \end{align*}

where $\lvert \mu \rvert$ denotes the length of $\mu$.

The polycyclic monoids have strongly tight actions $\beta \colon P_n\curvearrowright \Sigma _n^{\mathbb {N}}$, described in § 3.2. Put $G_n\colon \!= P_n\ltimes _{\beta }\Sigma _n^{\mathbb {N}}$. Then $G_n$ is a topologically principal locally compact Hausdorff second countable ample groupoid. For $s_i\in P_n$, let $\chi _{[s_i,D_{s_i^{*}s_i}]}$ denote the characteristic function on $[s_i,\,D_{s_i^{*}s_i}]\subset G_n$. Then $\{\chi _{[s_i,D_{s_i^{*}s_i}]}\}_{i=1}^{n}$ are elements of $C^{*}_{\lambda }(G_n)$ and generate $C^{*}_{\lambda }(G_n)$. Since $\{\chi _{[s_i,D_{s_i^{*}s_i}]}\}_{i=1}^{n}$ satisfies the Cuntz relation, $O_n$ and $C^{*}_{\lambda }(G_n)$ are isomorphic via the unique isomorphism $\Phi \colon O_n\to C^{*}_{\lambda }(G_n)$ such that $\Phi (S_i)=\chi _{[s_i,D_{s_i^{*}s_i}]}$ holds for all $i=1,\,\dots,\,n$. One can see that

\begin{align*} \Phi(D_n)=C(\Sigma_n^{\mathbb{N}}) \text{ and } \Phi(O_n^{\tau})= C^{*}_{\lambda}(M_n\ltimes_{\beta}\Sigma_n^{\mathbb{N}}) \end{align*}

hold. Define $O_n^{m}\subset O_n$ to be the subalgebra generated by

\begin{align*} \{S_{\mu}S_{\nu}^{*}\in O_n\mid \lvert\mu\rvert-\lvert\nu\rvert\in m\mathbb{Z}\}. \end{align*}

One can see that

\begin{align*} \Phi(O_n^{m})=C^{*}_{\lambda}(P_n^{m}\ltimes_{\beta}\Sigma_n^{\mathbb{N}}) \end{align*}

holds. Therefore, it follows from Proposition 3.2.2 that a Cartan intermediate subalgebra $O_n^{\tau }\subset B\subset O_n$ coincides with $O_n^{m}$ for some $m\in \mathbb {N}$. Moreover, every Cartan intermediate subalgebra between $O_n^{\tau }$ and $O_n$ admits a conditional expectation from $O_n$ by Proposition 3.2.5 and Theorem 4.1.5.

We note that $O_n^{m}$ is isomorphic to $O_{n^{m}}$. Indeed, $\{S_{\mu }\}_{\lvert \mu \rvert =m}$ generates $O_n^{m}$ and satisfies the Cuntz relation.

5.1. Tight groupoids

First, we recall the definition of tight groupoids. Refer to [Reference Exel2] or [Reference Exel and Pardo3] for more details. Let $S$ be an inverse semigroup. A character on $E(S)$ is a non-zero semigroup homomorphism from $E(S)$ to $\{0,\,1\}$, where $\{0,\,1\}$ is equipped with the usual multiplication. We denote the set of all characters on $E(S)$ by $\widehat {E}(S)$. Letting $\widehat {E}(S)$ be equipped with the pointwise convergence topology, $\widehat {E}(S)$ is a locally compact Hausdorff space. For a $\xi \in \widehat {E}(S)$, $\xi ^{-1}(\{1\})\subset E(S)$ is a proper filter in the following sense :

1. (1) $\xi ^{-1}(\{1\})$ does not contain $0$,

2. (2) if $e$ and $f$ belongs to $\xi ^{-1}(\{1\})$, then $ef$ also belongs to $\xi ^{-1}(\{1\})$,

3. (3) if $e\in \xi ^{-1}(\{1\})$ and $f\geq e$ hold, then $f$ belongs to $\xi ^{-1}(\{1\})$.

A character $\xi \in \widehat {E}(S)$ is called an ultracharacter if $\xi ^{-1}(\{1\})$ is a maximal proper filter. A character $\xi \in \widehat {E}(S)$ is an ultracharacter if and only if there is no character $\eta \in \widehat {E}(S)$ such that $\xi <\eta$ holds. The set of all ultracharacters on $E(S)$ is denoted by $\widehat {E}_{\infty }(S)$. The closure of $\widehat {E}_{\infty }(S)$ in $\widehat {E}(S)$ is denoted by $\widehat {E}_{\mathrm {tight}}(S)$. An element in $\widehat {E}_{\mathrm {tight}}(S)$ is called a tight character.

We define the spectral action $\beta \colon S\curvearrowright \widehat {E}(S)$. For $e\in E(S)$, put

\begin{align*} D_e^{\beta}\colon\!=\{\xi\in\widehat{E}(S)\mid \xi(e)=1\}. \end{align*}

Note that $D_e^{\beta }$ is a compact open set of $\widehat {E}(S)$. For $s\in S$ and $\xi \in D_{s^{*}s}^{\beta }$, define $\beta _s(\xi )\in D_{ss^{*}}^{\beta }$ by

\begin{align*} \beta_s(\xi)(e)\colon\!= \xi(s^{*}es), e\in E(S). \end{align*}

Then $\beta _s\colon D_{s^{*}s}^{\beta }\to D_{ss^{*}}^{\beta }$ is a homeomorphism. The map $s\mapsto \beta _s$ defines an action $\beta \colon S\curvearrowright \widehat {E}(S)$. It is known that $\widehat {E}_{\infty }(S)$ and $\widehat {E}_{\mathrm {tight}}(S)$ are $\beta$-invariant (see [Reference Exel2, Proposition 12.11]). The restrictions of $\beta$ to $\widehat {E}_{\infty }(S)$ and $\widehat {E}_{\mathrm {tight}}(S)$ are denoted by

\begin{align*} \theta_{\infty}\colon S\curvearrowright \widehat{E}_{\infty}(S) \text{ and } \theta\colon S\curvearrowright \widehat{E}_{\mathrm{tight}}(S) \end{align*}

respectively. The tight groupoid of $S$ is defined as $G_{\mathrm {tight}}(S)\colon \!= S\ltimes _{\theta } \widehat {E}_{\mathrm {tight}}(S)$.

5.2. Characterization of tight groupoids

In this subsection, we characterize strongly tight actions with non-empty domains. The proof of the next proposition is left to the reader.

Proposition 5.2.1 Let $S$ be an inverse semigroup, $X$ be a locally compact Hausdorff space and $\alpha \colon S\curvearrowright X$ be an action. For $x\in X,$ we define $\xi _x\colon E(S)\to \{0,\,1\}$ by

\begin{align*} \xi_x(e)\colon\!= \begin{cases} 1 & (x\in D^{\alpha}_e), \\ 0 & (x\not\in D^{\alpha}_e). \end{cases} \end{align*}

Then $\xi _x\in \widehat {E}(S)$.

Strongly tight actions with non-empty domains are characterized as the following theorem.

Theorem 5.2.2 Let $S$ be an inverse semigroup, $X$ be a locally compact Hausdorff space and $\alpha \colon S\curvearrowright X$ be a strongly tight action such that $D_e\not =\emptyset$ holds for each $e\in E(S){\setminus} \{0\}$. Then the map $X\ni x\mapsto \xi _x\in \widehat {E}(S)$ in Proposition 5.2.1 gives a homeomorphism

\begin{align*} X\ni x\mapsto \xi_x\in\widehat{E}_{\infty}(S) \end{align*}

which induces an isomorphism

\begin{align*} S\ltimes_{\alpha}X\ni [s,x]\mapsto [s,\xi_x]\in S\ltimes_{\theta_{\infty}}\widehat{E}_{\infty}(S). \end{align*}

Remark 5.2.3 It seems difficult to drop the assumption that $D_e\not =\emptyset$ for $e\in E(S){\setminus} \{0\}$. Define matrices

\begin{align*} p= \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, q= \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} . \end{align*}

Then $E\colon \!=\{0,\,p,\,q,\,1\}$ is a semilattice with respect to the usual multiplication. Let $X=\{x\}$ be a singleton. Define an action $\alpha \colon E\curvearrowright X$ by declaring $D_1=X$ and $D_p=D_q=D_0=\emptyset$. Then $\alpha$ is strongly tight, although $\widehat {E}_{\infty }$ is not homeomorphic to $X$. Note that $\xi _x$, which is defined in the proof of Theorem 5.2.2, is not an ultracharacter. Therefore, it seems difficult to find a natural map between $X$ and $\widehat {E}_{\infty }$.

The author in [Reference Lawson6] showed the following theorem.

Theorem 5.2.2 and Theorem 5.2.4 yield the following characterization of tight groupoids.

Remark 5.2.6 The implication $(3)\Rightarrow (2)$ in Corollary 5.2.5 dose not necessarily hold in general. Let $E$ be a semilattice generated by $0$ and $\{p_i\}_{i\in \mathbb {N}}$ with the relation

\begin{align*} p_ip_j= \begin{cases} p_i & (i=j),\\ 0 & (i\not=j). \end{cases} \end{align*}

Then $\widehat {E}_{\infty }=\widehat {E}_{\mathrm {tight}}$ holds, although $\widehat {E}_{\infty }$ is not compact. Indeed $\widehat {E}_{\infty }$ is homeomorphic to $\mathbb {N}$.