A remark on the freeness condition of Suzuki's correspondence theorem for intermediate ${\rm C}^\ast$-algebras

TL;DR

This paper generalizes Suzuki's correspondence theorem for intermediate C*-algebras to non-free actions of groups with the approximation property, establishing a one-to-one correspondence under certain conditions.

Contribution

It extends Suzuki's theorem to non-free actions by proving a correspondence between intermediate C*-algebras for group actions with the AP.

Findings

Established a bijective correspondence for intermediate C*-algebras in non-free actions.

Generalized Suzuki's theorem beyond free actions.

Applicable to groups satisfying the approximation property.

Abstract

Let $\Gamma$ be a discrete group satisfying the approximation property (AP). Let $X$, $Y$ be $\Gamma$-spaces and $\pi \colon Y \to X$ be a proper factor map which is injective on the non-free part. We prove the one-to-one correspondence between intermediate ${\rm C}^\ast$-algebras of $C_0(X) \rtimes_r \Gamma \subset C_0(Y) \rtimes \Gamma$ and intermediate $\Gamma$-${\rm C}^\ast$-algebras of $C_0(X) \subset C_0(Y)$. This is a generalization of Suzuki's theorem that proves the statement for free actions.

Equations

$$C_0(Z)\mapsto C_0(Z){⋊}_r\Gamma$$

$$\displaystyle E(fu_{g})=\left\{\begin{array}[]{l}f\ \ \mbox{if}\,g=e\\ 0\ \ \mbox{if}\,g\neq e,\end{array}\right.$$

$$X{⋊}_r\Gamma :=\overline{\mathrm{span}}\{x{u}_g\mid x\in X,g\in \Gamma \}.$$

$$\mathcal{D}\mapsto \mathcal{D}+C=:{\mathcal{D}}^{\wedge }$$

$$\mathcal{D}\mapsto \mathcal{D}\cap B=:{\mathcal{D}}^{\vee }.$$

$$C_0(Z)\mapsto C_0(Z){⋊}_r\Gamma$$

$$\Phi (a)$$

$$=\sum _{k=0}^n{f}_k{g}_k{u}_{t_k}=f_0+\sum _{k=1}^n{f}_k{g}_k{u}_{t_k}$$

$${\approx }_{(\epsilon /n)\cdot n}{f}_0+\sum _{k=1}^n\widetilde{f_k}{g}_k{u}_{t_k}$$

$$E(a)$$

$$\mathcal{D}$$

$$=(\tilde{E}({\mathcal{D}}^{\wedge }){⋊}_r\Gamma )\cap (C_0(Y){⋊}_r\Gamma )$$

$$=(\tilde{E}({\mathcal{D}}^{\wedge })\cap C_0(Y)){⋊}_r\Gamma .$$

$$X\ni z$$

$$Y\ni (z,w)$$

$$X=(U\cap V)\bigsqcup ((\overline{V}\cap (U\cap V{)}^c)\cup V^c).$$

$$(\beta \phi )(\beta Y\setminus Y)=\partial X\text{ and }\beta \phi (Y)=X.$$

$$C(\tilde{Y})/C_0(Y)\cong \beta {\phi }_{\ast }(C(\partial X)).$$

$$C(\tilde{Y})/C_0(Y)$$

$$\cong (\beta \phi {)}_{\ast }(C(\tilde{X}))/(\beta \phi {)}_{\ast }(C(\tilde{X}))\cap C_0(Y)$$

$$=(\beta \phi {)}_{\ast }(C(\tilde{X}))/{\phi }_{\ast }(C_0(X))$$

$$\cong \beta {\phi }_{\ast }(C(\partial X)).$$

$$C(\partial Y)\cong \beta {\phi }_{\ast }(C(\partial X))\text{ and }\beta \phi (\partial Y)\cong \partial X.$$

$${\mathcal{R}}_{T,\Gamma }:=\{(x,x)\mid x\in \partial {\mathbb{F}}_d\}\cup \{(g.t^{\infty },g.t^{-\infty })\mid g\in \Gamma ,t\in T\cup T^{-1}\}.$$

$$(\rho :\partial {\mathbb{F}}_d\to Z)\mapsto S_{\rho },$$

$$S\mapsto ({\rho }_S:\partial {\mathbb{F}}_d\to \partial {\mathbb{F}}_d/{\mathcal{R}}_{S,\Gamma }),$$

$$S\mapsto C(\partial {\mathbb{F}}_d/{\mathcal{R}}_{S,\Gamma }){⋊}_r\Gamma$$

Full text

1. Introduction

Let $\Gamma$ be a discrete group and $X$, $Y$ be locally compact spaces on which $\Gamma$ acts. Let $\pi\colon Y\to X$ be a proper factor map. We study the relation between intermediate $\Gamma$-${\rm C}^{\ast}$-algebras of $C_{0}(X)\subset C_{0}(Y)$ and intermediate ${\rm C}^{\ast}$-algebras of $C_{0}(X)\rtimes_{r}\Gamma\subset C_{0}(Y)\rtimes_{r}\Gamma$.

Inclusions of operator algebras play an important role in many subjects including operator theory and knot theory. Structures of subalgebras of ${\rm C}^{\ast}$-algebras have been studied by many hands ([ILP98], [Izu02], [GK96], [Zac01], [Zsi00], etc.).

A Galois correspondence theorem in operator algebras refers to a type of structure results for subalgebras of crossed products and fixed point subalgebras of operator algebras. This is proved in many cases. More precisely, a Galois correspondence is that for an operator algebra $M$ on which a compact group $G$ (or a discrete group $\Gamma$) acts, there exists a one-to-one correspondence between intermediate operator algebras of $M^{G}\subset M$ and closed subgroups of $G$ (or a one-to-one correspondence between intermediate operator algebras of $M\subset M\rtimes\Gamma$ and subgroups of $\Gamma$). Izumi, Longo and Popa [ILP98] prove the Galois correspondence for a factor $M$ on which a compact group $G$ acts minimally (or a discrete group $\Gamma$ acts outerly). In [Izu02], Izumi proves the Galois correspondence for a simple $\sigma$-unital ${\rm C}^{\ast}$-algebra on which a finite group acts outerly.

Ge and Kadison [GK96] prove the tensor splitting theorem that for every factor $M$ and every von Neumann algebra $N$, every von Neumann subalgebra of $M\overline{\otimes}N$ which contains $M$ is of the form $M\overline{\otimes}N_{0}$ for some von Neumann subalgebra $N_{0}$ of $N$. In the case of simple ${\rm C}^{\ast}$-algebras, the tensor splitting theorem is established under some conditions (see [Zac01], [Zsi00]).

Suzuki proves the following theorem among others in [Suz18].

Theorem 1** (Suzuki, [Suz18], Main Theorem (${\rm C}^{\ast}$-case)).**

Let $\Gamma$ be a discrete group satisfying the AP. Let $X$, $Y$ be $\Gamma$-spaces on which $\Gamma$ acts freely and $\pi$ be a proper factor map from $Y$ to $X$. Then the map

[TABLE]

gives a lattice isomorphism between the lattice of intermediate extensions of $\pi$ and that of intermediate ${\rm C}^{\ast}$-algebras of $C_{0}(X)\rtimes_{r}\Gamma\subset C_{0}(Y)\rtimes_{r}\Gamma$.

The freeness condition cannot be removed in general (see [Suz18, Proposition 2.6]). The following theorem generalizes the above result of Suzuki by relaxing the freeness condition.

Theorem 2** (Theorem 9).**

Let $\Gamma$ be a discrete group satisfying the AP. Let $X$, $Y$ be $\Gamma$-spaces and $\pi$ be a proper factor map from $Y$ to $X$ with the following condition$\colon$for every element $x$ in $X$ fixed by some non-neutral element of $\Gamma$, one has $|\pi^{-1}(x)|=1$. Then the map

[TABLE]

gives a lattice isomorphism between the lattice of intermediate extensions of $\pi$ and that of intermediate ${\rm C}^{\ast}$-algebras of $C_{0}(X)\rtimes_{r}\Gamma\subset C_{0}(Y)\rtimes_{r}\Gamma$.

Notation

Throughout this paper, $\Gamma$ denotes a discrete group. The symbol ‘$\rtimes_{r}$’ stands for the reduced ${\rm C}^{\ast}$-crossed product. For $g\in\Gamma$, denote by $\lambda_{g}$ the unitary element of the reduced group ${\rm C}^{\ast}$-algebra ${\rm C}_{r}^{*}\Gamma$ correponding to $g$. For a unital $\Gamma$-${\rm C}^{\ast}$-algebra $A$ and $g\in\Gamma$, denote $u_{g}$ the canonical implementing unitary element of $g$ in the reduced crossed product of $A\rtimes_{r}\Gamma$. For two elements $a$, $b$ of a ${\rm C}^{\ast}$-algebra, we denote $a\approx_{\varepsilon}b$ if $\|a-b\|<\varepsilon$. We denote by $\mathbb{N}$ the set of positive integers. Also, we denote by $\mathbb{N}_{0}$ the union of $\mathbb{N}$ and $\{0\}$.

Acknowledgements

The author would like to thank his supervisor, Professor Narutaka Ozawa for his support and many valuable comments. He also thanks Professor Yuhei Suzuki for valuable comments and suggesting Example 23 and the arguments following it.

2. Preliminaries

We say $X$ is a $\Gamma$-space if $X$ is a locally compact space equipped with a $\Gamma$-action by homeomorphisms.

Definition 3**.**

For a $\Gamma$-space $X$ and an element $g$ of $\Gamma$, we denote by ${\rm Fix}_{X}(g)$ the set of fixed points of $g$, i.e. $\{x\in X\mid g.x=x\}$. Also, we denote by $\mathcal{S}_{X}$ the set of all elements in $X$ which have non-trivial stabilizers.

Definition 4**.**

Let $X$, $Y$ be $\Gamma$-spaces. A map $\pi$ from $Y$ to $X$ is said to be a factor map if it is a $\Gamma$-equivariant quotient map from $Y$ to $X$. We also refer to $\pi$ as an extension. A factor map is proper if the preimage of every compact set is compact.

The approximation property (AP) has been introduced for locally compact groups by Haagerup–Kraus [HK94]. In the discrete case, the AP is weaker than weak amenability and stronger than exactness. See [HK94] and Section 12 of [BO08] for details.

Let $X$ be a $\Gamma$-space. There exists a canonical conditional expectation $E$ from $C_{0}(X)\rtimes_{r}\Gamma$ onto $C_{0}(X)$ defined by

[TABLE]

for $f\in C_{0}(X)$ and $g\in\Gamma$. Note that $E$ is faithful (see [BO08], Chapter 4.1).

The following proposition plays an important role in the proof of the main theorem.

Proposition 5** ([Suz17], Proposition 3.4).**

Let $\Gamma$ be a group satisfying the ${\rm AP}$. Let $A$ be a $\Gamma$-${\rm C}^{\ast}$-algebra and let $X$ be a closed subspace of $A$. Let $x$ be an element of $A\rtimes_{r}\Gamma$ satisfying $E(xu_{g}^{*})\in X$ for all $g\in\Gamma$. Then $x$ is contained in the closed subspace

[TABLE]

3. Main theorem

In this section, we prove the Main Theorem (Theorem 9).

Proposition 6**.**

Let $X$, $Y$ be $\Gamma$-spaces and $\pi$ be a $\Gamma$-equivariant quotient map from $Y$ to $X$. Assume for every element $x\in\mathcal{S}_{X}$, one has $|\pi^{-1}(x)|=1$. Then for every subset $C$ of $\mathcal{S}_{Y}$, one has $\pi^{-1}\circ\pi(C)=C$.

Proof.

Let $\varphi\colon\mathcal{S}_{Y}\to\mathcal{S}_{X}$ be the restriction of $\pi$ to $\mathcal{S}_{Y}$. It suffices to show that $\varphi$ is bijective. For every $x\in\mathcal{S}_{X}$, there exists $e\neq g\in\Gamma$, $x=g.x$. Since $\pi$ is surjective, there exists $y\in Y$ satisfying $\pi(y)=x$. If $g.y\neq y$, then one has $|\pi^{-1}(x)|\geq 2$, which contradicts the assumption. Hence, $g.y=y$, that is $y\in\mathcal{S}_{Y}$. Thus, $\varphi$ is surjective. Injectivity follows from the assumption. ∎

Lemma 7**.**

Let $X$, $Y$ be compact Hausdorff spaces and $\pi$ be a continuous map from $Y$ to $X$. Let $K$ be a compact subset of $Y$ satisfying $K=\pi^{-1}\circ\pi(K)$ and $U$ be an open neighborhood of $K$. Then there exists an open neighborhood $V$ of $\pi(K)$ satisfying $\pi^{-1}(V)\subset U$.

Proof.

Suppose there exists no such open neighborhood. Since $X$ is Hausdorff, there exists a decreasing net $\{V_{i}\}_{i\in I}$ of open sets such that $\bigcap_{i}V_{i}=\pi(K)$. By the assumption, for every $i\in I$, one has $\pi^{-1}(V_{i})\not\subset U$. Hence, there exist $x_{i}\in V_{i}$ and $y_{i}\in Y$ such that $\pi(y_{i})=x_{i}$ and $y_{i}\not\in U$. Since $Y$ is compact, there exists a subnet $\{y_{j}\}_{j\in J}$ and $y\in Y$ such that $y_{j}\to y$. Since $\pi$ is continuous, one has $\pi(y_{j})\to\pi(y)$. Then one has $\pi(y)\in\pi(K)$. Hence, one has $y\in K$. Since $U$ is an open neighborhood of $K$, there exists $j_{0}\in J$ such that $y_{j_{0}}\in U$. This contradicts $y_{j}\not\in U$ for every $j\in J$. ∎

We say that $A\subset B$ is a non-degenerate inclusion of ${\rm C}^{\ast}$-algebras if every (or, equivalently, some) approximate unit of $A$ is an approximate unit of $B$. If $A\subset B$ is non-degenerate, the inclusion extends to an inclusion $M(A)\subset M(B)$ (see [Lan95, Proposition 2.1]).

We use the following lemma to show the non-compact case of Theorem 9.

Lemma 8**.**

Let $A\subset B$ be a non-degenerate inclusion of ${\rm C}^{\ast}$-algebras. Let $C$ be a ${\rm C}^{\ast}$-subalgebra of the multiplier algebra $M(A)$. The map

[TABLE]

defines a one-to-one correspondence between intermediate ${\rm C}^{\ast}$-algebras $A\subset\mathcal{D}\subset B$ and intermediate ${\rm C}^{\ast}$-algebras $A+C\subset\mathcal{D}\subset B+C$, with its inverse map given by

[TABLE]

Proof.

Let $\mathcal{D}$ be an intermediate ${\rm C}^{\ast}$-algebra of $A\subset B$. We first note that $C\subset M(\mathcal{D})$ and $\mathcal{D}^{\wedge}=\mathcal{D}+C=Q_{\mathcal{D}}^{-1}\circ Q_{\mathcal{D}}(C)\subset M(\mathcal{D})$ is a ${\rm C}^{\ast}$-subalgebra, where $Q_{\mathcal{D}}\colon M(\mathcal{D})\to M(\mathcal{D})/\mathcal{D}$ is the quotient map. Let $\{u_{i}\}$ be an approximate unit of $A$. By the non-degeneracy of the inclusion, $\{u_{i}\}$ is an approximate unit of $B$.

Since $\mathcal{D}\subset B$, one has $\mathcal{D}\subset(\mathcal{D}^{\wedge})^{\vee}$. Let $a$ be an element of $(\mathcal{D}^{\wedge})^{\vee}$. Since $\mathcal{D}^{\wedge}=\mathcal{D}+C$, there exist $d\in\mathcal{D}$ and $c\in C$ such that $a=d+c$. Since $u_{i}c\in A$, one has $u_{i}(d+c)\in\mathcal{D}+A=\mathcal{D}$. Since $\{u_{i}\}$ is an approximate unit for $B$, one has $a\in\mathcal{D}$. Hence, one has $(\mathcal{D}^{\wedge})^{\vee}=\mathcal{D}$.

Let $\mathcal{D}$ be an intermediate ${\rm C}^{\ast}$-algebra of $A+C\subset B+C$. Since $C\subset\mathcal{D}$, one has $(\mathcal{D}^{\vee})^{\wedge}\subset\mathcal{D}$. Let $a$ be an element of $\mathcal{D}$. Since $a\in B+C$, there exist $b\in B$ and $c\in C$ such that $a=b+c$. Since $u_{i}b\in B$ and $u_{i}c\in A$, one has $u_{i}(b+c)\in B\cap\mathcal{D}$. Since $c-u_{i}c\in C+A$, one has $u_{i}b+c=u_{i}(b+c)+(c-u_{i}c)\in(\mathcal{D}\cap B)+C=(\mathcal{D}^{\vee})^{\wedge}$. Hence, one has $(\mathcal{D}^{\vee})^{\wedge}$. ∎

Theorem 9**.**

Let $\Gamma$ be a discrete group satisfying the AP. Let $X$, $Y$ be $\Gamma$-spaces and $\pi$ be a proper factor map from $Y$ to $X$ such that $|\pi^{-1}(x)|=1$ for every element $x$ of $\mathcal{S}_{X}$. We regard $C_{0}(X)$ as a $\Gamma$-${\rm C}^{\ast}$-subalgebra of $C_{0}(Y)$ via $\pi$. Then the map

[TABLE]

gives a lattice isomorphism between the lattice of intermediate extensions of $\pi$ and that of intermediate ${\rm C}^{\ast}$-algebras of $C_{0}(X)\rtimes_{r}\Gamma\subset C_{0}(Y)\rtimes_{r}\Gamma$.

Proof.

We first show this theorem when $X$ and $Y$ are compact. Let $E$ be the canonical conditional expectation from $C(Y)\rtimes_{r}\Gamma$ to $C(Y)$. Let $a$ be an element of $C(Y)\rtimes_{r}\Gamma$. Let $\varepsilon>0$ be given. There exist $n\in\mathbb{N}\cup\{0\}$, $t_{k}\in\Gamma$, and $f_{k}\in C(Y)$ $(k\in\{0,\ldots,n\})$ such that $\|a-\sum_{k=0}^{n}f_{k}u_{t_{k}}\|<\varepsilon$, $t_{0}=e$, and $t_{k}\neq e$ $(k\in\{1,\ldots,n\})$.

Since ${\rm Fix}_{X}(t)$ is closed for every $t\in\Gamma$, for each $k\in\{1,2,\ldots,n\}$, there exists $\tilde{f_{k}}\in C(X)$ such that $f_{k}=\pi_{\ast}(\tilde{f_{k}})$ on ${\rm Fix}_{Y}(t_{k})$ by the Tietze extension theorem and Lemma 6.

For each $k\in\{1,\dots,n\}$, since ${\rm Fix}_{Y}(t_{k})$ is compact, there exists an open set $W_{k}$ of $Y$ such that ${\rm Fix}_{Y}(t_{k})\subset W_{k}$ and $|f_{k}-\pi_{*}(\tilde{f_{k}})|<\varepsilon/n$ on $W_{k}$. By Proposition 6 and Lemma 7, there exists an open neighborhood $U_{k}^{0}$ of $\pi({\rm Fix}_{Y}(t_{k}))$ in $X$ such that $\pi^{-1}(U_{k}^{0})\subset W_{k}$.

For each $k\in\{1,2,\ldots,n\}$, $t_{k}$ acts on $X\setminus U_{k}^{0}$ freely. Hence, there exist a finite subset $J_{k}^{\prime}\subset\mathbb{N}$ and a finite open covering $\{U_{k}^{j}\}_{j\in J_{k}^{\prime}}$ of $X\setminus U_{k}^{0}$ such that for every $j\in J_{k}^{\prime}$, $t_{k}U_{k}^{j}\cap U_{k}^{j}=\emptyset$. Let $J_{k}=J_{k}^{\prime}\cup\{0\}$ and $I=J_{1}\times J_{2}\times\cdots\times J_{n}$. For every ${\bf i}=(j_{1},j_{2},\ldots,j_{n})\in I$, we define $V_{\bf i}$ to be $U_{1}^{j_{1}}\cap U_{2}^{j_{2}}\cap\cdots\cap U_{n}^{j_{n}}$. We remark that for every $k\in\{1,2,\ldots,n\}$ and every ${\bf i}=(j_{1},j_{2},\ldots,j_{n})\in I$ with $j_{k}\neq 0$, one has $t_{k}U_{\bf i}\cap U_{\bf i}=\emptyset$. We also have $\bigcup_{\begin{subarray}{c}{\bf i}=(j_{1},j_{2},\ldots,j_{n})\\ j_{k}=0\end{subarray}}U_{\bf i}=U_{k}^{0}$. For the open covering $\{V_{\bf i}\}_{{\bf i}\in I}$ of $X$, there exists $h_{\bf i}\in C(X)^{+}$ such that $\sum_{{\bf i}\in I}h_{\bf i}=1$ and ${\rm supp}(h_{\bf i})\subset V_{\bf i}$ $({\bf i}\in I)$. We remark that for every $k\in\{1,2,\ldots,n\}$ and every ${\bf i}=(j_{1},j_{2},\ldots,j_{n})\in I$ with $j_{k}\neq 0$, one has $h_{\bf i}(t_{k}.h_{\bf i})=0$. We define a c.c.p map $\Phi\colon C(Y)\rtimes_{r}\Gamma\to C(X)\rtimes_{r}\Gamma$ by the map $a\mapsto\sum_{{\bf i}\in I}h_{\bf i}^{\frac{1}{2}}ah_{\bf i}^{\frac{1}{2}}$. For each $k\in\{1,2,\ldots,n\}$, we define $g_{k}:=\Phi(u_{t_{k}})u_{t_{k}}^{*}=\sum_{{\bf i}\in I}h_{\bf i}^{\frac{1}{2}}(t_{k}.h_{\bf i}^{\frac{1}{2}})\in C(X)$. For $k\neq 0$, since ${\rm supp}(g_{k})\subset U_{k}^{0}$ and $\|g_{k}\|\leq 1$, one has $f_{k}g_{k}\approx_{\varepsilon/n}\tilde{f_{k}}g_{k}$. Therefore,

[TABLE]

Since $\Phi(a)\in C^{*}(a,C(X))$ and $\sum_{k=1}^{n}\tilde{f_{k}}g_{k}u_{t_{k}}\in C(X)\rtimes_{r}\Gamma$, one has

[TABLE]

Since $\varepsilon>0$ was arbitrary, one has $E(a)\in C^{*}(a,C(X)\rtimes_{r}\Gamma)$.

Let $\mathcal{D}$ be an intermediate ${\rm C}^{\ast}$-algebra of $C(X)\rtimes_{r}\Gamma\subset C(Y)\rtimes_{r}\Gamma$. Then, by the result shown in the previous paragraph, one has $E(\mathcal{D})\subset\mathcal{D}$. By Proposition 5, for every intermediate ${\rm C}^{\ast}$-algebra $\mathcal{D}$ of $C(X)\rtimes_{r}\Gamma\subset C(Y)\rtimes_{r}\Gamma$, one has $\mathcal{D}=E(\mathcal{D})\rtimes_{r}\Gamma$.

Next, we show this theorem in the case where $X$ and $Y$ are noncompact. Let $\tilde{X}=X\sqcup\{x_{\infty}\}$, $\tilde{Y}=Y\sqcup\{y_{\infty}\}$ be the one-point compactifications of $X$, $Y$ respectively. Let $\tilde{\pi}\colon\tilde{Y}\to\tilde{X}$ denote the continuous extension of $\pi$. Since $\tilde{\pi}^{-1}(\{x_{\infty}\})=\{y_{\infty}\}$, $\tilde{\pi}$ satisfies the assumption of this theorem.

We will use a one-to-one correspondence between intermediate ${\rm C}^{\ast}$-algebras of $C_{0}(X)\rtimes_{r}\Gamma\subset C_{0}(Y)\rtimes_{r}\Gamma$ and that of $C(\tilde{X})\rtimes_{r}\Gamma\subset C(\tilde{Y})\rtimes_{r}\Gamma$. Let $\tilde{E}$ be the canonical conditional expectation from $C(\tilde{Y})\rtimes_{r}\Gamma$ to $C(\tilde{Y})$. Let $\mathcal{D}$ be an intermediate ${\rm C}^{\ast}$-algebra of $C_{0}(X)\rtimes_{r}\Gamma\subset C_{0}(Y)\rtimes_{r}\Gamma$. Since $\mathcal{D}^{\wedge}:=\mathcal{D}+{\rm C}_{r}^{*}\Gamma$ is an intermediate ${\rm C}^{\ast}$-algebra of $C(\tilde{X})\rtimes_{r}\Gamma\subset C(\tilde{Y})\rtimes_{r}\Gamma$, one has $\mathcal{D}^{\wedge}=\tilde{E}(\mathcal{D}^{\wedge})\rtimes_{r}\Gamma$. Let $(\mathcal{D}^{\wedge})^{\vee}=\mathcal{D}^{\wedge}\cap(C_{0}(Y)\rtimes_{r}\Gamma)$. By Lemma 8 and Proposition 5, one has

[TABLE]

∎

4. Examples

Example 10** (branched covering).**

Let $X$ be a complex plane $\mathbb{C}$. Let $k$ be an integer greater than or equal to $2$ and $Y=\{(z,w)\in\mathbb{C}\times\mathbb{C}\mid w^{k}=z\}$ be a Riemann surface of the $k$-th square root. Let $\pi$ be a projection from $Y$ to $X$, i.e., $Y\ni(z,w)\mapsto z\in X$. Let $\alpha\in\mathbb{R}\setminus\mathbb{Q}$. We define $\mathbb{Z}$-actions on $X$, $Y$ by the following: for each $n\in\mathbb{Z}$,

[TABLE]

Then $\pi$ is a $\mathbb{Z}$-equivariant proper quotient map and for every $n\in\mathbb{Z}$, one has ${\rm Fix}_{Y}(n)=\{(0,0)\}$ and ${\rm Fix}_{X}(n)=\{0\}$.

We show that the assumption of $\pi$ in Theorem 9 is closed under taking the direct product. It follows from the Whitehead lemma [Whi48] that $\pi_{1}\times\pi_{2}$ in Lemma 12 is quotient. For the reader’s convenience, we include the proof.

Lemma 11**.**

Let $Y$ be a topological space and $X$ be a locally compact space. Let $\pi$ be a surjective proper continuous map and $U$ be a subset of $X$. If $\pi^{-1}(U)$ is open, $U$ is open.

Proof.

Take an element $x$ in $U$. Let $V$ be a relative compact open neighborhood of $x$. It suffices to show $U\cap V$ is open. Since $\pi^{-1}(U\cap V)=\pi^{-1}(U)\cap\pi^{-1}(V)$, $\pi^{-1}(U\cap V)$ is open. By continuity and properness, $\overline{V}\cap(U\cap V)^{c}=\pi(\pi^{-1}(\overline{V})\cap\pi^{-1}(U\cap V)^{c})$ is compact. We remark that

[TABLE]

Since $\overline{V}\cap(U\cap V)^{c}$ and $V^{c}$ is closed, $((\overline{V}\cap(U\cap V)^{c})\cup V^{c}$ is closed. Then $U\cap V$ is open. ∎

Lemma 12**.**

For each $i\in\{1,2\}$, let $X_{i}$, $Y_{i}$ be $\Gamma$-spaces and $\pi\colon Y_{i}\to X_{i}$ be a proper factor maps. We see $X_{1}\times X_{2}$ and $Y_{1}\times Y_{2}$ are $\Gamma$-spaces with diagonal actions. Then $\pi:=\pi_{1}\times\pi_{2}\colon Y_{1}\times Y_{2}\to X_{1}\times X_{2}$ is a proper factor map.

Proof.

It suffices to check properness and quotientness. We will show properness. Let $C\subset Y_{1}\times Y_{2}$ be a compact subset. For each $i\in\{1,2\}$, let $p_{i}\colon Y_{1}\times Y_{2}\to Y_{i}$ be $i$-th projections. By the $\pi_{i}$’s continuity, for each $i\in\{1,2\}$, $p_{i}(C)$ is compact. Since for each $i$, $\pi_{i}^{-1}(p_{i}(C))$ is compact, $\pi^{-1}(C)\subset\pi_{1}^{-1}(p_{1}(C))\times\pi_{2}^{-1}(p_{2}(C))$ is compact.

We will show quotientness. Let $U\subset X$. Since $\pi$ is continuous, if $U$ is open, then $\pi^{-1}(U)$ is open. By Lemma 11, if $\pi^{-1}(U)$ is open, then $U$ is open. ∎

Example 13**.**

For each $i\in\{1,2\}$, let $X_{i}$, $Y_{i}$ be $\Gamma$-spaces and $\pi\colon Y_{i}\to X_{i}$ be a $\Gamma$-equivariant proper factor map. By the above lemma, if for every $i\in\{1,2\}$, $\pi_{i}$ is a proper factor map such that $|\pi_{i}^{-1}(x)|=1$ for every element $x$ in $\mathcal{S}_{X_{i}}$, then so is $\pi_{1}\times\pi_{2}\colon Y_{1}\times Y_{2}\to X_{1}\times X_{2}$.

We show that some compactifications satisfy the assumption in the main theorem. For a locally compact space $X$, we denote by $\beta X$ the Stone-$\check{{\rm C}}$ech compactification of $X$. The following proposition is easily seen from the universal property of the Stone-$\check{{\rm C}}$ech compactification and Lemma 11.

Proposition 14**.**

Let $\varphi\colon Y\to X$ be a proper quotient map between locally compact spaces $X$ and $Y$. Let $\tilde{X}=X\cup\partial X$ be a compactification of $X$. Then there exists a quotient map $\beta\varphi\colon\beta Y\to\tilde{X}$ extending $\varphi$.

Remark 15**.**

By the above proposition, one has

[TABLE]

It follows that $(\beta\varphi)_{*}(C(\tilde{X}))\cap C_{0}(Y)=(\beta\varphi)_{*}(C_{0}(X))$.

Proposition 16**.**

With the same assumptions as the previous proposition, we define $\tilde{Y}$ to be the character space of the ${\rm C}^{\ast}$-subalgebra of $C(\beta Y)$ generated by $(\beta\varphi)_{*}(C(\tilde{X}))$ and $C_{0}(Y)$. Then one has

[TABLE]

Proof.

[TABLE]

∎

Remark 17**.**

Let $\partial Y$ to be $\tilde{Y}\setminus Y$. Then one has

[TABLE]

For a compact space $X$ and an open subset $U$ of $X$, we consider the ${\rm C}^{\ast}$-subalgebra of $\ell_{\infty}(X)$ generated by $C(X)$ and the characteristic function $\chi_{\overline{U}}$.

Proposition 18**.**

Let $X$ be a compact space and $U$ be an open subset of $X$. Let $p=\chi_{\overline{U}}\in l_{\infty}(X)$, Then one has l_{\infty}(X)\supset\mbox{{\rm C}^{\ast}}(C(X),p)\cong C(\overline{U})\oplus C(X\setminus U)

Proof.

Let $V$ be a subset of $X$. Let $q=\chi_{V}$ be the characteristic function in $\ell_{\infty}(X)$. We will show $C(X)q\cong C(\overline{V})$. We define $\varphi\colon C(X)q\to C(\overline{V})$ by $fq\mapsto f|_{\overline{V}}$ for $f\in C(X)$. The well-definedness and the injectivity of $\varphi$ follows from continuity. The surjectivity of $\varphi$ follows from the Tietze’s extension theorem. Hence, $\varphi$ is an isomorphism.

Hence, one has \mbox{{\rm C}^{\ast}}(C(X),p)\cong C(X)p\oplus C(X)(1-p)\cong C(\overline{U})\oplus C(X\setminus U). ∎

Remark 19**.**

Under the above assumption, let $\tilde{X}$ to be the character space of \mbox{{\rm C}^{\ast}}(C(X),p) and $\iota\colon C(X)\to C(\tilde{X})$ be the inclusion of $C(X)$ to $C(\tilde{X})$. By the above proposition, one has $\tilde{X}\cong\overline{U}\sqcup(X\setminus U)$. Hence, $\pi:=\iota_{*}\colon\tilde{X}\to X$ is a proper quotient map, and for every $x\in X\setminus\partial U$, one has $|\pi^{-1}(x)|=1$. Furthermore, for every $x\in X$, $x$ belong to $\partial U$ if and only if one has $|\pi^{-1}(x)|=2$.

Example 20**.**

Let $X$ be a compact $\Gamma$-space, $U$ be an open subset of $X$ and $p=\chi_{\overline{U}}\in l_{\infty}(X)$. Let compact $\Gamma$-space $\tilde{X}$ to be the character space of \mbox{{\rm C}^{\ast}}(C(X),\Gamma\cdot p) , $\iota\colon C(X)\to C(\tilde{X})$ be the inclusion of $C(X)$ to $C(\tilde{X})$ and $\pi:=\iota_{*}\colon\tilde{X}\to X$. Then $\pi$ is a proper factor map. If $\mathcal{S}_{X}\cap\Gamma\cdot\partial U=\emptyset$, then for every $x\in\mathcal{S}_{X}$, one has $|\pi^{-1}(x)|=1$.

Proof.

We prove $|\pi^{-1}(x)|=1$ for every $x\in\mathcal{S}_{X}$. It suffices to show that for every $x\in X$, one has $x\in\Gamma\cdot\partial U$ if and only if $|\pi^{-1}(x)|\geq 2$.

Let $x\in\Gamma\cdot\partial U$. There exists $g\in\Gamma$ s.t. $x\in g.\partial U$. Let $X_{1}$ to be the character space of \mbox{{\rm C}^{\ast}}(C(X),g.p). Let $\iota_{1}$ be the inclusion of $C(X)$ to \mbox{{\rm C}^{\ast}}(C(X),g.p) and $\iota_{2}$ be the inclusion of \mbox{{\rm C}^{\ast}}(C(X),g.p) to $C(\tilde{X})$. Let $\pi_{1}=(\iota_{1})_{*}\colon X_{1}\to X$ and $\pi_{2}=(\iota_{2})_{*}\colon\tilde{X}\to X_{1}$. Since $\pi^{-1}(x)=\pi_{2}^{-1}(\pi_{1}^{-1}(x))$, one has $|\pi^{-1}(x)|=|\pi_{2}^{-1}(\pi_{1}^{-1}(x))|\geq|\pi_{1}^{-1}(x)|=2$ by the above remark.

Let $x$ be an element of $X$ such that $|\pi^{-1}(x)|\geq 2$. There exist distinct elements $y_{1}$ and $y_{2}$ in $\tilde{X}$ such that $\pi(y_{1})=\pi(y_{2})=x$. Let $\Gamma=\{g_{n}\}_{n\in\mathbb{N}}$. Let $A_{n}$ be a ${\rm C}^{\ast}$-subalgebra of $C(\tilde{X})$ generated by $C(X)$ and $\{g_{k}.p\}_{k=1}^{n}$. Then $\cup_{n}A_{n}$ is dense in $C(\tilde{X})$. So, there exist $n\in\mathbb{N}$ and $a\in A_{n}$ such that $y_{1}(a)\neq y_{2}(a)$. We regard $y_{1},\,y_{2}$ as characters on $A_{n}$. By the above remark, there exists $g\in\Gamma$ such that $x\in g.\partial U$. ∎

The assumption in the above remark is satisfied in many cases including the following case.

Example 21**.**

Let $X$ be a compact metric $\Gamma$-space. Assume $\mathcal{S}_{X}$ is separable and the Lebesgue covering dimension of $\mathcal{S}_{X}$ is zero. Let $A$, $B$ be disjoint closed subsets in $X$. Then, by the second separation theorem (see [Eng78, 1.5.13]), there exists an open set $U$ in $X$ s.t. $A\subset U$, $B\subset U^{c}$ and $\partial U\cap\mathcal{S}_{X}=\emptyset$. Since $\mathcal{S}_{X}$ is $\Gamma$-invariant, one has $\mathcal{S}_{X}\cap\Gamma\cdot\partial U=\emptyset$.

We construct another example. We are grateful to Yuhei Suzuki for letting us know the following example. We say an element $g$ in a free group is indivisible if it is not a proper power of some element in the free group. For every element $x$ in a free group, we denote by $x^{\infty}$ the limit of $x^{n}$ in the Gromov compactification of the free group.

Remark 22**.**

We consider the free group $\mathbb{F}_{d}$ of rank $d$, where $1<d<\infty$, and its action on its Gromov boundary $\partial\mathbb{F}_{d}$. Let $x$ be an element of $\mathbb{F}_{d}$. The stabilizer group of $x^{\infty}$ is a cyclic group and we denote by $y$ its generator. Then $y$ is indivisible. Also, one has either $x^{\infty}=y^{\infty}$ or $x^{\infty}=y^{-\infty}$. By replacing $y$ with $y^{-1}$, we can choose an element $y$ satisfying $x^{\infty}=y^{\infty}$. Also, for an indivisible element $x$, every element in $\mathbb{F}_{d}$ which fixes $x^{\infty}$ is represented by the power of $x$.

For a subgroup $\Lambda$ of $\Gamma$ and an element $t$ of $\Gamma$, we denote by $C_{\Lambda}(t)$ the $\Lambda$-conjugacy class of $t$.

Example 23**.**

Let $\mathbb{F}_{d}:=\langle a_{1},\ldots,a_{d}\rangle$ be the free group of rank $d$, where $1<d<\infty$. Let $\partial\mathbb{F}_{d}$ denote the Gromov boundary. Let $\Gamma$ be a non-trivial normal subgroup of $\mathbb{F}_{d}$. Then the $\Gamma$-action on $\partial\mathbb{F}_{d}$ is topologically free and minimal. Let $T$ be a subset of $\mathbb{F}_{d}$ which consists of indivisible elements such that $|\{C_{\mathbb{F}_{d}}(t)\mid t\in T\}|<\infty$. Assume that for every non-zero $n\in\mathbb{Z}$ and for every $t\in T$, $t^{n}\notin\Gamma$. Let

[TABLE]

Then $\mathcal{R}_{T,\Gamma}$ is a $\Gamma$-invariant equivalence relation on $\partial\mathbb{F}_{d}$. Also, $\partial\mathbb{F}_{d}/\mathcal{R}_{T,\Gamma}$ is Hausdorff.

Furthermore, the quotient map $\pi\colon\partial\mathbb{F}_{d}\to\partial\mathbb{F}_{d}/\mathcal{R}_{T,\Gamma}$ is a proper quotient map satisfying for every $x\in\partial\mathbb{F}_{d}/\mathcal{R}_{T,\Gamma}$ with a non-trivial stabilizer, $|\pi^{-1}(x)|=1$.

Proof.

Since the $\mathbb{F}_{d}$-action on $\partial\mathbb{F}_{d}$ is topologically free, the $\Gamma$-action on $\partial\mathbb{F}_{d}$ is topologically free. We will show the minimality of the $\Gamma$-action. Let $S:=\{a_{1},\ldots,a_{d}\}$, which is a finite generating set of $\partial\mathbb{F}_{d}$. We regard $\partial\mathbb{F}_{d}$ as the set of infinite reduced words of $\mathbb{F}_{d}$ (see [BO08, 5.1]). Let $x=x_{1}x_{2}\cdots$ and $y=y_{1}y_{2}\cdots$ be elements in $\partial\mathbb{F}_{d}$, where $x_{i},y_{i}\in S\cup S^{-1}$. Let $\gamma\in\Gamma$ be a non-trivial element. For each $n\in\mathbb{N}$, there exists $z_{n}\in S\cup S^{-1}$ such that $|y_{n}z_{n}|>|y_{n}|$ and $|z_{n}\gamma|>|\gamma|$, where $|\cdot|$ is the length function on $\Gamma$ determined by $S$. Let $w_{n}=(y_{1}\cdots y_{n})z_{n}\gamma z_{n}^{-1}(y_{1}\cdots y_{n})^{-1}\in\mathbb{F}_{d}$. Then one has $w_{n}x\to y$. Also, by the normality of $\Gamma$, $w_{n}$ belongs to $\Gamma$. Hence, the $\Gamma$-action on $\partial\mathbb{F}_{d}$ is minimal.

Since $|\{C_{\mathbb{F}_{d}}(t)\mid t\in T\}|<\infty$, there exists a finite subset $T^{\prime}$ of $T$ such that $\{C_{\mathbb{F}_{d}}(t)\mid t\in T\}=\{C_{\mathbb{F}_{d}}(t)\mid t\in T^{\prime}\}$. So, one has $\mathcal{R}_{T,\Gamma}\subset\mathcal{R}_{T^{\prime},\mathbb{F}_{d}}$. Hence, in the similar way as [Suz17, Lemma 4.4], we can show that $\mathcal{R}_{T,\Gamma}$ is a $\Gamma$-invariant equivalence relation on $\partial\mathbb{F}_{d}$ and $\partial\mathbb{F}_{d}/\mathcal{R}_{T,\Gamma}$ is Hausdorff.

We will show that $\pi$ satisfies that for every $x\in\partial\mathbb{F}_{d}/\mathcal{R}_{T,\Gamma}$ with a non-trivial stabilizer, one has $|\pi^{-1}(x)|=1$. Let $x\in\partial\mathbb{F}_{d}/\mathcal{R}_{T,\Gamma}$ satisfying $\gamma.x=x$ for some non-neutral element $\gamma\in\Gamma$. We remark each equivalence class of $\mathcal{R}_{T,\Gamma}$ contains at most two elements. Suppose there exist $g\in\Gamma$ and $t\in T\cup T^{-1}$ such that $\pi^{-1}(\{x\})=\{g.t^{\infty},g.t^{-\infty}\}$. Then one has $\gamma.g.t^{\infty}\in\{g.t^{\infty},g.t^{-\infty}\}$. Since there exists no element $h\in\mathbb{F}_{d}$ with $h.t^{\infty}=t^{-\infty}$, we may assume $\gamma.g.t^{\infty}=g.t^{\infty}$. Since $g^{-1}\gamma g$ fixes $t^{\infty}$, there exist a non-zero integer $n$ such that $g^{-1}\gamma g=t^{n}$ by Remark 22. This contradicts the assumption. Hence, one has $|\pi^{-1}(x)|=1$. ∎

By the following proposition and Theorem 9, one has a one-to-one correspondence between intermediate ${\rm C}^{\ast}$-algebras of $C(\partial\mathbb{F}_{d}/\mathcal{R}_{T,\Gamma})\rtimes_{r}\Gamma\subset C(\partial\mathbb{F}_{d})\rtimes_{r}\Gamma$ and subsets of $T$.

Proposition 24**.**

In addition to the above condition, assume for distinct elements $s$, $t\in T$, one has $C_{\Gamma}(s)\cap\{t,t^{-1}\}=\emptyset$. Then the map

[TABLE]

where $S_{\rho}:=\{t\in T\mid|\rho^{-1}\circ\rho(t)|=2\}$, gives a one-to-one correspondence between intermediate extensions of $\pi$ and subsets of $T$ with its inverse map given by

[TABLE]

where $\rho_{S}\colon\partial\mathbb{F}_{d}\to\partial\mathbb{F}_{d}/\mathcal{R}_{S,\Gamma}$ is the quotient map.

Proof.

Let ($\rho$, $\rho^{\prime}$) be an intermediate extension of $\pi$, where $\rho\colon\partial\mathbb{F}_{d}\to Z$ and $\rho^{\prime}\colon Z\to\partial\mathbb{F}_{d}/\mathcal{R}_{T,\Gamma}$ are factor maps such that $\rho^{\prime}\circ\rho=\pi$. Since $\rho_{S_{\rho}}$ is surjective, we define the $\Gamma$-equivariant map $f\colon\partial\mathbb{F}_{d}/\mathcal{R}_{S_{\rho},\Gamma}\to Z$ by $x\mapsto\rho(y)$ for some $y\in\partial\mathbb{F}_{d}$ with $\rho_{S_{\rho}}(y)=x$. We first check this map is well-defined. Let $x\in\partial\mathbb{F}_{d}/\mathcal{R}_{S_{\rho},\Gamma}$. Let $y_{1}$ and $y_{2}$ be elements in $\partial\mathbb{F}_{d}$ such that $\rho_{S_{\rho}}(y_{1})=\rho_{S_{\rho}}(y_{2})=x$. We may assume $y_{1}\neq y_{2}$. By the definition of $\mathcal{R}_{S_{\rho},\Gamma}$, we may assume $y_{1}=g.s^{\infty}$ and $y_{2}=g.s^{-\infty}$ for some $g\in\Gamma$ and some $s\in S_{\rho}$. By the definition of $\mathcal{R}_{S_{\rho},\Gamma}$, one has $(g.s^{\infty},g.s^{-\infty})\in\mathcal{R}_{S_{\rho},\Gamma}$. Hence, one has $\rho(y_{1})=\rho(y_{2})$. Hence, $f$ is well-defined. Similarly, we can check $f$ is $\Gamma$-equivariant.

We will show $f$ is a continuous bijective map, that is an isomorphism. By the definition of $f$, one has $\rho=f\circ\rho_{S_{\rho}}$. Hence, $f$ is surjective. Also, since $\rho$ and $\rho_{S_{\rho}}$ are quotient maps, $f$ is continuous. We will show the injectivity of $f$. Let $x_{1}$, $x_{2}$ be elements in $\partial\mathbb{F}_{d}/\mathcal{R}_{S_{\rho},\Gamma}$ with $f(x_{1})=f(x_{2})$. There exist $y_{1}$ and $y_{2}$ in $\partial\mathbb{F}_{d}$ such that $\rho_{S_{\rho}}(y_{i})=x_{i}$ for each $i\in\{1,2\}$. Then one has $\rho(y_{1})=\rho(y_{2})$. We may assume $y_{1}\neq y_{2}$. By the definition of $S_{\rho}$, one has $(y_{1},y_{2})\in\mathcal{R}_{S_{\rho},\Gamma}$. So, one has $x_{1}=x_{2}$. Hence, $f$ is injective. Hence, $\partial\mathbb{F}_{d}/\mathcal{R}_{S_{\rho},\Gamma}\cong Z$ as an intermediate extension of $\pi$.

Let $S$ be a subset of $T$. We will show that $S=S_{\rho_{S}}$. Let $s\in S$. Since $(s^{\infty},s^{-\infty})\in\mathcal{R}_{S,\Gamma}$, one has $s\in S_{\rho_{S}}$. Let $s\in S_{\rho_{S}}$. Then one has $(s^{\infty},s^{-\infty})\in\mathcal{R}_{S,\Gamma}$. So, there exist $g\in\Gamma$ and $s^{\prime}\in S$ such that $s^{\prime\infty}=g.s^{\infty}$ or $s^{\prime\infty}=g.s^{-\infty}$. Suppose $s\neq s^{\prime}$. Since $gsg^{-1}$ fixes $g.s^{\infty}$ and $g.s^{-\infty}$, and $s^{\prime}$ is indivisible, then there exist $n\in\mathbb{Z}\setminus\{0\}$ such that $s^{\prime n}=gsg^{-1}$ by Remark 22. Since $s$ is indivisible, one has $n=\pm 1$. This contradicts the assumption, that is $C_{\Gamma}(s)\cap\{s^{\prime},s^{\prime-1}\}=\emptyset$, where $[s]$ is the $\Gamma$-conjugacy class of $s$. Hence, one has $s=s^{\prime}\in S$. ∎

Corollary 25**.**

In the above conditions, the map

[TABLE]

gives a lattice isomorphism between the lattice of subsets of $T$ and that of intermediate ${\rm C}^{\ast}$-algebras of $C(\partial\mathbb{F}_{d}/\mathcal{R}_{T,\Gamma})\rtimes_{r}\Gamma\subset C(\partial\mathbb{F}_{d})\rtimes_{r}\Gamma$.

In many cases, we can construct an infinite subset $T$ of $\mathbb{F}_{d}$ satisfying the above conditions as follows.

Proposition 26**.**

Let $\mathbb{F}_{d}$ be the free group of rank $d$ $(d>1)$. Let $\Gamma$ be a normal subgroup of $\mathbb{F}_{d}$ such that the quotient group $\mathbb{F}_{d}/\Gamma$ is not virtually cyclic. Then there exists an infinite subset $T$ satisfying the following properties:

1. (1)

Every element of $T$ is indivisible. 2. (2)

For every $0\neq n\in\mathbb{Z}$ and $t\in T$, one has $t^{n}\notin\Gamma$. 3. (3)

For every distinct elements $s$, $t\in T$, one has $C_{\Gamma}(s)\cap\{t,t^{-1}\}=\emptyset$. 4. (4)

$|\{C_{\mathbb{F}_{d}}(t)\mid t\in T\}|<\infty$. 5. (5)

$|\{C_{\Gamma}(t)\mid t\in T\}|=\infty$.

Proof.

Since $\mathbb{F}_{d}/\Gamma$ is not virtually cyclic, there exists an indivisible element $t$ of $\mathbb{F}_{d}$ such that for every $0\neq n\in\mathbb{Z}$, $t^{n}\notin\Gamma$. We denote by $\langle t\rangle$ the subgroup of $\mathbb{F}_{d}$ generated by $t$. Let $S$ be a subset of $\mathbb{F}_{d}$ such that $\bigsqcup_{s\in S}C_{\Gamma}(sts^{-1})=C_{\mathbb{F}_{d}}(t)$. We show $|S|=[\mathbb{F}_{d}/\Gamma\colon\langle t\rangle\Gamma]$, where $[\mathbb{F}_{d}/\Gamma\colon\langle t\rangle\Gamma]$ is the index of $\langle t\rangle\Gamma$ in $\mathbb{F}_{d}/\Gamma$. Let $s$, $s^{\prime}\in\mathbb{F}_{d}$. Let $C_{\Gamma}(sts^{-1})=C_{\Gamma}(s^{\prime}ts^{\prime-1})$. Then there exists $g\in\Gamma$ such that $gsts^{-1}g^{-1}=s^{\prime}ts^{\prime-1}$. Hence, one has $s^{\prime-1}gst=ts^{\prime-1}gs$. Since the centralizer of $t$ is $\langle t\rangle$, one has $s^{\prime-1}gs\in\langle t\rangle$. Hence, one has $s\Gamma=s^{\prime}\langle t\rangle\Gamma$. Hence, one has $|S|=[\mathbb{F}_{d}/\Gamma\colon\langle t\rangle\Gamma]$. Since $\mathbb{F}_{d}/\Gamma$ is not virtually cyclic, one has $[\mathbb{F}_{d}/\Gamma\colon\langle t\rangle\Gamma]=\infty$. Hence, one has $|S|=\infty$.

Let $T=\{sts^{-1}\mid s\in S\}$. By construction, $T$ satisfies the properties (4) and (5). Since $\Gamma$ is normal and $t$ is indivisible, $T$ satisfies the properties (1) and (2). We show $T$ satisfies the property (3). Let distinct elements $s$, $t\in S$. By construction, one has $t\notin C_{\Gamma}(s)$. Suppose there exists $g\in\Gamma$ such that $t^{-1}=gst(gs)^{-1}$. Then one has $t^{-\infty}=gs.t^{\infty}$, which is a contradiction, since there exists no element $h$ of $\mathbb{F}_{d}$ such that $h.t^{\infty}=t^{-\infty}$, ∎