# Simplicity of crossed products of the actions of totally disconnected
locally compact groups on their boundaries

Ryoya Arimoto  RIMS, Kyoto University, 606-8502 Japan
<EMAIL_ADDRESS>
###### Abstract.

We prove that if a totally disconnected locally compact group admits a
topologically free boundary, then the reduced crossed product of continuous
functions on its Furstenberg boundary by the group is simple. We also prove a
partial converse of this result.

###### Key words and phrases:

reduced crossed product, simplicity, Furstenberg boundary, topologically free.

###### 1991 Mathematics Subject Classification:

Primary 37A55; Secondary 46L05, 22D25

##  1\. Introduction

One of the fundamental constructions of operator algebras is the construction
from groups or group actions and this gives rise to a natural question of
which group or group action produces simple operator algebras. In this paper,
we will restrict our attention to the question for the reduced group
$\mathrm{C}^{*}$-algebras and the reduced crossed products. The reduced group
$\mathrm{C}^{*}$-algebra for a locally compact group $G$ is a norm closure of
$\lambda(C_{c}(G))$ in $B(L^{2}(G))$ where $\lambda\colon C_{c}(G)\to
B(L^{2}(G))$ is the left regular representation; $\lambda(f)\xi:=f*\xi$ for
$f\in C_{c}(G)$ and $\xi\in L^{2}(G)$. For the definition of the reduced
crossed products, see Section 2.3.

For discrete groups, this question has long been studied. The first example of
$\mathrm{C}^{*}$-simple groups, i.e., a group whose reduced group
$\mathrm{C}^{*}$-algebra is simple, was found by Powers in 1975 ([13]); his
result states that the free group of rank two is $\mathrm{C}^{*}$-simple.
Since then, many groups, including the Baumslag–Solitar groups ([7]), were
shown to be $\mathrm{C}^{*}$-simple using Powers’ argument. Finally, in 2017,
Kalantar and Kennedy proved that a discrete group is $\mathrm{C}^{*}$-simple
if and only if the group acts on its Furstenberg boundary topologically freely
([11]). Breuillard, Kalantar, Kennedy, and Ozawa found that the extremal
disconnectedness of the Furstenberg boundary of a discrete group ensures that
topological freeness is equivalent to freeness for the action of a discrete
group on its Furstenberg boundary.

As for the reduced crossed products, in 1990, Kawamura and Tomiyama showed
that for an amenable discrete group $\Gamma$ with an action on a compact
Hausdorff space $X$, the reduced crossed product $C(X)\rtimes_{r}\Gamma$ is
simple if and only if the action is minimal and topologically free ([10]). In
1994, Archbold and Spielberg showed that for a discrete group $\Gamma$ with an
action on a $\mathrm{C}^{*}$-algebra $A$, if the action is minimal and
topologically free (both notions are adjusted to a non-commutative setting),
then $A\rtimes_{r}\Gamma$ is simple ([2]). They also extended the result of
Kawamura and Tomiyama to amenable actions with simpler proof. Kalantar and
Kennedy showed that for a discrete group $\Gamma$, the simplicity of
$C(\partial_{F}\Gamma)\rtimes_{r}\Gamma$ implies that the topological freeness
of the action on $\partial_{F}\Gamma$ ([11]) where $\partial_{F}\Gamma$ is the
Furstenberg boundary of $\Gamma$.

In conclusion, the following theorem is known.

######  Theorem (See [11, Theorem 6.2.] and [3, Theorem 3.1.]).

Let $\Gamma$ be a discrete group. The following conditions are equivalent.

  1. (i)

The reduced group $\mathrm{C}^{*}$-algebra $\mathrm{C}^{*}_{\lambda}(\Gamma)$
of $\Gamma$ is simple.

  2. (ii)

The reduced crossed product $C(\partial_{F}\Gamma)\rtimes_{r}\Gamma$ is
simple.

  3. (iii)

The action $\Gamma\curvearrowright\partial_{F}\Gamma$ is free.

  4. (iv)

The action of $\Gamma$ on some $\Gamma$-boundary is topologically free.

Recently, some progress has been made in that question for locally compact
groups. Raum proved that any $\mathrm{C}^{*}$-simple group is totally
disconnected ([15]). The existence of a non-discrete $\mathrm{C}^{*}$-simple
group was confirmed by Suzuki ([18]) and he also showed that the class of
$\mathrm{C}^{*}$-simple groups is abundant ([19]). Raum also proved that the
Schlichting completion of the Baumslag–Solitar group is
$\mathrm{C}^{*}$-simple, using Suzuki’s result ([16]).

Progress from another point of view has also been made. In 2022, Caprace, Le
Boudec, and Matte Bon proved that totally disconnected groups with a piecewise
minimal-strongly-proximal action admit a topologically free boundary ([5]). In
particular, it is shown that the Neretin groups admit a topologically free
boundary (see [5, Section 4] for the Neretin groups). Recently, Le Boudec and
Tsankov showed that a locally compact group admits a topologically free
boundary if and only if the group acts on its Furstenberg boundary freely
([12]). Combining these two results (since the latter result was announced
beforehand, this conclusion is written in [5]), we obtain that the Neretin
groups act on its Furstenberg boundary freely. This is the first example of a
non-discrete compactly generated simple locally compact group acting on its
Furstenberg boundary. It is still not known whether the Neretin groups are
$\mathrm{C}^{*}$-simple or not.

Our main theorems and their proofs are motivated by the work of Kawamura and
Tomiyama ([10]) and the work of Archbold and Spielberg ([2]).

######  Theorem A (Theorem 3.1).

Let $G$ be a totally disconnected locally compact group and $\partial_{F}G$ be
its Furstenberg boundary. If $G$ admits a topologically free boundary, then
$C(\partial_{F}G)\rtimes_{r}G$ is simple.

A converse of Theorem A holds under some technical conditions. See the text
before Theorem 3.4 for these conditions.

######  Theorem B (Theorem 3.4).

Let $G$ be a totally disconnected locally compact group and $X$ be a compact
$G$-space. Assume that the action $G\curvearrowright X$ is amenable and the
stabilizer map $X\ni x\mapsto G_{x}\in\operatorname{Sub}{(G)}$ is continuous.
If $C(X)\rtimes_{r}G$ is simple, then the action $G\curvearrowright X$ is
free.

Theorem A applies to two examples; the Neretin groups $\mathcal{N}_{d,k}$ and
the HNN extensions, especially the Schlichting completions of the
Baumslag–Solitar groups. In particular, our result states that
$C(\partial_{F}\mathcal{N}_{d,k})\rtimes_{r}\mathcal{N}_{d,k}$ is simple where
$\mathcal{N}_{d,k}$ is the Neretin group.

### Acknowledgements

The author is grateful to Professor Narutaka Ozawa for his valuable comments.
This work was supported by Grant-in-Aid for JSPS Fellows Grant Number
JP23KJ1186.

##  2\. Preliminaries

###  2.1. Totally disconnected locally compact groups

In this paper, locally compact groups are assumed to be second countable. A
totally disconnected locally compact group is a topological group whose
underlying topological space is totally disconnected and locally compact. By
van Dantzig’s theorem ([20]), a locally compact group is totally disconnected
if and only if it admits a neighborhood basis of identity consisting of
compact open subgroups.

###  2.2. Group actions and the Furstenberg boundary

Let $G$ be a locally compact group. For a compact Hausdorff space $X$,
$\operatorname{Prob}{X}$ denotes the space of Radon probability measures on
$X$ with weak-$*$ topology. We identify elements of $X$ with the corresponding
Dirac measures in $\operatorname{Prob}{X}$. An action $G\curvearrowright X$ of
$G$ on a compact Hausdorff space $X$ is said to be minimal if there is no non-
trivial $G$-invariant closed subset, is said to be strongly proximal if
$\overline{G.\mu}$ contains a element of $X$ for every
$\mu\in\operatorname{Prob}{X}$, and is said to be topologically free if a
subset $\\{x\in X\mid G_{x}=\\{e\\}\\}$ is dense in $X$.

A compact $G$-space is called a $G$-boundary if the action is minimal and
strongly proximal. The Furstenberg boundary $\partial_{F}G$ of $G$ is the
universal $G$-boundary, that is, for every $G$-boundary $X$, there is a
(unique) $G$-quotient map $\partial_{F}G\twoheadrightarrow X$. For any locally
compact groups, such a boundary exists (see [8, Proposition 4.6.]). For a
totally disconnected locally compact group $G$ and its compact open subgroup
$K$, $C(K\backslash\partial_{F}G)$ is an injective $\mathrm{C}^{*}$-algebra
(see [14, Remark 4.8.]).

###  2.3. Crossed products

Let $G$ be a locally compact group with a fixed left Haar measure $\mu$,
$\Delta$ be the modular function of $G$, and $X$ be a compact $G$-space. Then
$C(X)$ is a $G$-$\mathrm{C}^{*}$-algebra with the action $\alpha$ defined by
$[\alpha_{g}f](x)=f(g^{-1}.x)$. Let $C_{c}(G,C(X))$ be the set of continuous
functions $f$ from $G$ to $C(X)$ whose support $\overline{\\{g\in G\mid
f(g)\neq 0\\}}$ is compact. This becomes a $*$-algebra by the following
operations:

$\displaystyle f_{1}*f_{2}(g)$
$\displaystyle:=\int_{G}f_{1}(h)\alpha_{h}(f_{2}(h^{-1}g))\,dy$ $\displaystyle
f^{*}(g)$ $\displaystyle:=\Delta(g^{-1})\alpha_{g}(f(g^{-1})^{*})$

for $f,f_{1},f_{2}\in C_{c}(G,C(X))$. For a faithful representation $\pi\colon
C(X)\hookrightarrow B(H)$, define a $*$-representation $\Pi\colon
C_{c}(G,C(X))\to B(L^{2}(G)\otimes H)$ by

$\Pi(f)\xi(x)=\int_{G}\pi(\alpha_{x^{-1}}(f(y)))\xi(y^{-1}x)\,dy,$

here we identify $L^{2}(G)\otimes H$ with $L^{2}(G,H)$ and $\xi\in
L^{2}(G,H)$. The reduced crossed product $C(X)\rtimes_{r}G$ of $C(X)$ by $G$
is defined to be $\overline{\Pi(C_{c}(G,C(X)))}^{\|\cdot\|}\subset
B(L^{2}(G)\otimes H)$. This definition does not depend on the choice of a
faithful representation $\pi\colon C(X)\hookrightarrow B(H)$.

For a compact open subgroup $K$ of $G$, $p_{K}$ denotes a projection
$\displaystyle\frac{1}{\mu(K)}\chi_{K}$ in $C(X)\rtimes_{r}G$. A net
$(p_{K})_{K<G:\text{compact open subgroup}}$ of projections forms an
approximate unit for $C(X)\rtimes_{r}G$. Note that for a compact open subgroup
$K<G$, there is a canonical embedding $C(X)\rtimes_{r}K\subset
C(X)\rtimes_{r}G$. For $K$-invariant function $f$ on $X$, $[f]_{K}\in
C(X)\rtimes_{r}K$ denotes a constant function on $K$ whose value is $f$. There
is a faithful conditional expectation $E_{K}\colon
p_{K}(C(X)\rtimes_{r}G)p_{K}\to[C(K\backslash X)]_{K}$ such that
$E_{K}(p_{K}fp_{K})=(p_{K}fp_{K})|_{K}$ for $f\in C_{c}(G,C(X))$ (see [19,
Lemma 2.1.]).

##  3\. Main theorems

Our two main theorems are inspired by the result of [10] and [2].

######  Theorem 3.1.

Let $G$ be a totally disconnected locally compact group. If $G$ admits a
topologically free boundary, then $C(\partial_{F}G)\rtimes_{r}G$ is simple.

By the result of Le Boudec and Tsankov ([12, Corollary 1.3.]), the condition
that $G$ admits a topologically free boundary is equivalent to the freeness of
the action $G\curvearrowright\partial_{F}G$.

For a subset $A$ of a topological group $G$, $\langle A\rangle$ denotes the
closed subgroup of $G$ generated by $A$.

######  Lemma 3.2.

Let $C\subset G$ be a compact subset of $G$ and $K$ be a compact open subgroup
of $G$. Then there exists a compact open subgroup $L$ of $G$ such that for
every $g\in C$, $\langle L\cup gLg^{-1}\rangle$ is contained in $K$.

###### Proof.

Let $K_{g}:=K\cap g^{-1}Kg$ and $U_{g}:=\\{h\in G\mid\langle K_{g}\cup
hK_{g}h^{-1}\rangle\subset K\\}$ for $g\in C$. Then $U_{g}$ are open subsets
and, by compactness of $C$, there exists a finite subset $F$ of $C$ such that
$C\subset\bigcup_{g\in F}U_{g}$. A compact open subgroup $L:=\bigcap_{g\in
F}K_{g}$ has the desired property. ∎

######  Proposition 3.3.

Let $K$ be a totally disconnected compact group and $X$ be a compact Hausdorff
space on which $K$ acts freely. Then every closed ideal of $C(X)\rtimes_{r}K$
is of the form $C_{0}(U)\rtimes_{r}K$ where $U$ is a $K$-invariant open subset
of $X$.

###### Proof.

Let $I$ be a closed ideal of $C(X)\rtimes_{r}K$. For each compact open normal
subgroup $L\triangleleft K$, $p_{L}(C(X)\rtimes_{r}K)p_{L}\cong C(L\backslash
X)\rtimes_{r}K/L$. Since $K/L$ is finite and the action $K/L\curvearrowright
L\backslash X$ is free, by theorems by Kawamura–Tomiyama ([10, Theorem 4.1.])
and Sierakowski ([17, Proposition 1.3. and Proposition 1.6.]), $p_{L}Ip_{L}$
is of the form $C_{0}(U_{L})\rtimes_{r}K/L$ where $U_{L}\subset L\backslash X$
is a $K/L$-invariant open subset. Let $\tilde{U_{L}}$ be an inverse image of
$U_{L}$ under the canonical quotient map $X\to L\backslash X$. Then it is easy
to see that $\tilde{U_{L}}$ does not depend on $L$ and
$I=C_{0}(\tilde{U_{L}})\rtimes_{r}K$. ∎

###### Proof of Theorem 3.1.

Assume that the action $G\curvearrowright\partial_{F}G$ is free. Suppose that
there exists a non-trivial ideal $I\triangleleft
C(\partial_{F}G)\rtimes_{r}G$. Let $a\in I$ be a non-zero positive element.
Take a compact open subgroup $K<G$ such that $p_{K}ap_{K}\neq 0$.

First, we will show that $I\cap C(\partial_{F}G)\rtimes_{r}K=0$. Suppose that
$I\cap C(\partial_{F}G)\rtimes_{r}K\neq 0$. Then, by Proposition 3.3, there
exists a $K$-invariant open subset $U\subset\partial_{F}G$ such that $I\cap
C(\partial_{F}G)\rtimes_{r}K=C_{0}(U)\rtimes_{r}K$. Suppose that
$U\neq\partial_{F}G$. Take $x\in\partial_{F}G\setminus U$ and $g\in G$ such
that $g^{-1}.x\in U$. Let $L<G$ be a compact open subgroup such that $L<K$,
$g^{-1}Lg<K$, and $g^{-1}L.x\subset U$ and $f\in C_{0}(U)$ be a positive
function which equals to $1$ on $g^{-1}L.x$. Define $\tilde{f}\colon
g^{-1}Lg\to C_{0}(U)$ as a constant function whose value is $f$. Note that
$\tilde{f}\in C_{0}(U)\rtimes_{r}K=I\cap C(\partial_{F}G)\rtimes_{r}K$. One
can see that $\chi_{Lg}\cdot\tilde{f}\cdot\chi_{g^{-1}L}$, which is also in
$I\cap C(\partial_{F}G)\rtimes_{r}K=C_{0}(U)\rtimes_{r}K$, is
$\int_{L}\alpha_{\ell g}(f)\,d\ell$ on $L$ and $0$ on $K\setminus L$ where
$\alpha\colon G\curvearrowright C(\partial_{F}G)$ denotes the action induced
by the action $G\curvearrowright\partial_{F}G$. Therefore,
$\int_{L}\alpha_{\ell g}(f)\,d\ell$ must be in $C_{0}(U)$, but
$\int_{L}\alpha_{\ell g}(f)(x)\,d\ell=\int_{L}f(g^{-1}\ell^{-1}x)\,d\ell=1$,
which is contradiction. Hence $U=\partial_{F}G$, i.e.,
$C(\partial_{F}G)\rtimes_{r}K\subset I$, but in this case, an approximate unit
$(p_{M})_{M<K:\text{compact open subgroup}}$ for
$C(\partial_{F}G)\rtimes_{r}G$ is containd in $I$, which contradicts to non-
triviality of $I$. Therefore, $I\cap C(\partial_{F}G)\rtimes_{r}K=0$.

Let $\varepsilon>0$ be arbitrary and take $b\in C_{c}(G,C(\partial_{F}G))_{+}$
such that $\|a-b\|<\varepsilon$. Put
$\tilde{b}:=p_{K}bp_{K}-E_{K}(p_{K}bp_{K})$. Then
$\operatorname{supp}{\tilde{b}}\subset G\setminus K$. Define a compact set
$C:=K\cdot\operatorname{supp}{\tilde{b}}\cdot K\subset G\setminus K$. By Lemma
3.2, there exists a compact open subgroup $L<G$ such that $\langle L\cup
gLg^{-1}\rangle\subset K$ for every $g\in C$. Since the action
$G\curvearrowright\partial_{F}G$ is free, $Kg.x\neq K.x$ for every $g\in C$
and $x\in\partial_{F}G$. In particular, $\langle L\cup gLg^{-1}\rangle
g.x\neq\langle L\cup gLg^{-1}\rangle.x$ for every $g\in C$ and
$x\in\partial_{F}G$.

Let $A:=p_{L}(C(\partial_{F}G)\rtimes_{r}K)p_{L}$. Then
$A,\,p_{L}Ip_{L}\subset p_{L}(C(\partial_{F}G)\rtimes_{r}G)p_{L}$ and $A\cap
p_{L}Ip_{L}=0$. Define a u.c.p. map $\pi$ as a composition of

$\displaystyle A+p_{L}Ip_{L}\twoheadrightarrow A+p_{L}Ip_{L}/p_{L}Ip_{L}\cong
A\xrightarrow{E_{L}}[C(L\backslash\partial_{F}G)]_{L}.$

Since $[C(L\backslash\partial_{F}G)]_{L}\cong C(L\backslash\partial_{F}G)$ is
an injective $\mathrm{C}^{*}$-algebra, there is a u.c.p. extension $\Phi\colon
p_{L}(C(\partial_{F}G)\rtimes_{r}G)p_{L}\to[C(L\backslash\partial_{F}G)]_{L}$
of $\pi$. Note that $[C(L\backslash\partial_{F}G)]_{L}$ is in a multiplicative
domain of $\Phi$ since $\pi|_{[C(L\backslash\partial_{F}G)]_{L}}$ is the
identity map (see [4, Proposition 1.5.7. and Definition 1.5.8.] for a
multiplicative domain).

Since $\operatorname{supp}{\tilde{b}}$ is compact, there is a finite subset
$F$ of $G$ such that $\operatorname{supp}{\tilde{b}}\subset\bigcup_{g\in
F}LgL$, $\tilde{b}|_{LgL}\neq 0$ for every $g\in F$, and a map $F\ni g\mapsto
LgL\in L\backslash G/L$ is injective. Then $F\subset C$ and
$\tilde{b}=\sum_{g\in F}\tilde{b}|_{LgL}$. We will show that
$\Phi(\tilde{b}|_{LgL})=0$ for every $g\in F$. We identify
$\Phi(\tilde{b}|_{LgL})$ with its value in $C(L\backslash\partial_{F}G)$.
Suppose that there exists $x\in\partial_{F}G$ such that
$\Phi(\tilde{b}|_{LgL})(L.x)\neq 0$. Take a non-trivial clopen subset $U$ of
$\langle L\cup gLg^{-1}\rangle\backslash\partial_{F}G$ such that $\langle
L\cup gLg^{-1}\rangle.x\in U$ and $\langle L\cup gLg^{-1}\rangle g.x\not\in
U$. Let $\tilde{U}$ be an inverse image of $U$ under the canonical quotient
$\partial_{F}G\twoheadrightarrow\langle L\cup
gLg^{-1}\rangle\backslash\partial_{F}G$. Note that $\tilde{U}$ is
$L$-invariant and $g^{-1}\tilde{U}$ is also $L$-invariant. Define
$f:=\chi_{\tilde{U}}\in C(L\backslash\partial_{F}G)$. Then one can check that
$[f]_{L}\cdot\tilde{b}|_{LgL}=\tilde{b}|_{LgL}\cdot[\alpha_{g^{-1}}(f)]_{L}$
and so,
$f\cdot\Phi(\tilde{b}|_{LgL})=\Phi(\tilde{b}|_{LgL})\cdot\alpha_{g^{-1}}(f)$.
Evaluating this equation at $L.x\in L\backslash\partial_{F}G$ leads to a
contradiction. Therefore, $\Phi(\tilde{b})=0$ and we have

$\displaystyle\|E_{K}(p_{K}ap_{K})\|$
$\displaystyle=\|\Phi(E_{K}(p_{K}ap_{K}))\|$
$\displaystyle\leq\|\Phi(E_{K}(p_{K}bp_{K}))\|+\varepsilon$
$\displaystyle=\|\Phi(p_{K}bp_{K})\|+\varepsilon$
$\displaystyle\leq\|\Phi(p_{K}ap_{K})\|+2\varepsilon$
$\displaystyle=2\epsilon.$

This is a contradiction since $E_{K}$ is faithful and $\varepsilon$ is
arbitrary. ∎

A partial converse of Theorem 3.1 holds.

For a locally compact group $G$, $\operatorname{Sub}{(G)}$ denotes the set of
closed subgroups of $G$ equipped with the Chabauty topology; topology defined
by

$\\{H\in\operatorname{Sub}{(G)}\mid H\cap K=\emptyset,H\cap
U_{1}\neq\emptyset,\ldots,H\cap U_{n}\neq\emptyset\\}$

where $K\subset G$ is a compact subset and $U_{1},\ldots U_{n}\subset G$ are
open subsets. Given a compact $G$-space $X$, one can obtain the stabilizer map
$X\ni x\mapsto G_{x}\in\operatorname{Sub}{(G)}$. This map is not necessarily
continuous. Note that if the action $G\curvearrowright X$ is topologically
free, then the continuity of the stabilizer map is equivalent to the freeness
of the action.

See [1] for the definition and properties of amenable actions.

######  Theorem 3.4.

Let $G$ be a totally disconnected locally compact group and $X$ be a compact
$G$-space. Assume that the action $G\curvearrowright X$ is amenable and the
stabilizer map $X\ni x\mapsto G_{x}\in\operatorname{Sub}{(G)}$ is continuous.
If $C(X)\rtimes_{r}G$ is simple, then the action $G\curvearrowright X$ is
free.

The following lemma follows from [9, Appendix].

######  Lemma 3.5.

Let $G$ be a locally compact group and $X$ be a compact $G$-space. Assume that
the stabilizer map $X\ni x\mapsto G_{x}\in\operatorname{Sub}{(G)}$ is
continuous. Then there is a family of measures $\\{\mu_{x}\\}_{x\in X}$ such
that

  * •

for each $x\in X$, $\mu_{x}$ is a left Haar measure on $G_{x}$,

  * •

for every $f\in C_{c}(G)$, a map $x\mapsto\int_{G_{x}}f(a)\,d\mu_{x}(a)$ is
continuous.

###### Proof of Theorem 3.4.

Suppose that the action $G\curvearrowright X$ is not topologically free. In
this case, there is a non-empty open subset $U\subset X$ such that $G_{x}$ is
non-trivial for all $x\in U$. Fix some $x\in U$. Take a family of measures
$\\{\mu_{y}\\}_{y\in X}$ as Lemma 3.5. Let $\rho$ be a rho-function for
$(G,G_{x})$ and $\nu$ be the associated quasi-invariant regular Borel measure
on $G/G_{x}$; they satisfy
$\displaystyle\frac{dg.\nu}{d\nu}(aG_{x})=\frac{\rho(g^{-1}a)}{\rho(a)}$.
Define a $*$-homomorphism $\pi\colon C(X)\to B(L^{2}(G/G_{x}))$ and a group
homomorphism $u\colon G\to\mathcal{U}(L^{2}(G/G_{x}))$ by the relations

$\displaystyle[\pi(f)\xi](aG_{x}):=f(a.x)\xi(aG_{x}),$
$\displaystyle[u_{g}\xi](aG_{x}):=\left(\displaystyle\frac{dg.\nu}{d\nu}(aG_{x})\right)^{\frac{1}{2}}\xi(g^{-1}aG_{x}).$

Then we obtain a $*$-homomorphism $\tilde{\pi}\colon C(X)\rtimes_{\lambda}G\to
B(L^{2}(G/G_{x}))$ from $\pi$ and $u$ since the action $G\curvearrowright X$
is amenable.

To prove non-faithfulness of $\tilde{\pi}$, first, we construct a non-zero
function $f_{0}\in C_{c}(G,C(X))$ such that
$\displaystyle\int_{G_{y}}[f_{0}(sh)](y)(\Delta_{G}(h)\Delta_{G_{y}}(h))^{-\frac{1}{2}}\,d\mu_{y}(h)=0$
for every $y\in X$ and $s\in G$. Take two distinct points $g_{1},g_{2}\in
G_{x}$ and two positive functions $f_{1},f_{2}\in C_{c}(G)$ such that
$\operatorname{supp}{f_{1}}\cap\operatorname{supp}{f_{2}}=\emptyset$ and
$f_{i}(g_{i})>0$ for $i=1,2$. Define $\eta_{0}$ by

$[\eta_{0}(s)](y)=f_{1}(s)\int_{G_{y}}f_{2}(h)\,d\mu_{y}(h)-f_{2}(s)\int_{G_{y}}f_{1}(h)\,d\mu_{y}(h).$

Then $\eta_{0}\in C_{c}(G,C(X))$, $[\eta_{0}(\cdot)](x)|_{G_{x}}\neq 0$, and
$\int_{G_{y}}[\eta_{0}(h)](y)\,d\mu_{y}(h)=0$ for every $y\in X$. Define a map
$\tau\colon\\{(g,y)\in G\times X\mid g.y=y\\}\to\mathbb{R}$ by
$\tau(g,y):=\Delta_{G_{y}}(g)^{-\frac{1}{2}}$. This map is continuous. Indeed,
fix some compact open subgroup $K$ of $G$ and assume $(g_{i},y_{i})\to(g,y)$
in $\\{(g,y)\in G\times X\mid g.y=y\\}$. Then

$\displaystyle\mu_{y_{i}}(K\cap
G_{y_{i}})=\int_{G_{y_{i}}}\chi_{K}(h)\,d\mu_{y_{i}}(h)\to\int_{G_{y}}\chi_{K}(h)\,d\mu_{y}(h)=\mu_{y}(K\cap
G_{y})$

and

$\displaystyle|\mu_{y_{i}}(Kg_{i}\cap G_{y_{i}})-\mu_{y}(Kg\cap G_{y})|$
$\displaystyle=\left|\int_{G_{y_{i}}}\chi_{K{g_{i}}}(h)\,d\mu_{y_{i}}(h)-\int_{G_{y}}\chi_{Kg}(h)\,d\mu_{y}(h)\right|$
$\displaystyle\leq\left|\int_{G_{y_{i}}}\chi_{K{g_{i}}}(h)\,d\mu_{y_{i}}(h)-\int_{G_{y_{i}}}\chi_{Kg}(h)\,d\mu_{y_{i}}(h)\right|$
$\displaystyle\
+\left|\int_{G_{y_{i}}}\chi_{Kg}(h)\,d\mu_{y_{i}}(h)-\int_{G_{y}}\chi_{Kg}(h)\,d\mu_{y}(h)\right|$

hold. There exists a compact subset $C\subset G$ such that $Kg_{i}\subset C$
eventually. Note that $\\{\mu_{y}(C\cap G_{y})\\}_{y\in X}$ is bounded by the
construction in Lemma 3.5. Hence one has

$\displaystyle\left|\int_{G_{y_{i}}}\chi_{K{g_{i}}}(h)\,d\mu_{y_{i}}(h)-\int_{G_{y_{i}}}\chi_{Kg}(h)\,d\mu_{y_{i}}(h)\right|$
$\displaystyle\leq\mu_{y_{i}}(C\cap
G_{y_{i}})\|\chi_{Kg_{i}}-\chi_{Kg}\|_{\infty}$ $\displaystyle\to 0.$

Moreover, by the definition of $\\{\mu_{y}\\}_{y\in X}$, one has

$\left|\int_{G_{y_{i}}}\chi_{Kg}(h)\,d\mu_{y_{i}}(h)-\int_{G_{y}}\chi_{Kg}(h)\,d\mu_{y}(h)\right|\to
0.$

Therefore, we have

$\tau(g_{i},y_{i})=\left(\frac{\mu_{y_{i}}(K\cap
G_{y_{i}})}{\mu_{y_{i}}(Kg_{i}\cap
G_{y_{i}})}\right)^{\frac{1}{2}}\to\left(\frac{\mu_{y}(K\cap
G_{y})}{\mu_{y}(Kg\cap G_{y})}\right)^{\frac{1}{2}}=\tau(g,y).$

Let $\tilde{\tau}\in C(G\times X)$ be an extension of $\tau$ (the existence of
an extension is ensured by the Tietze extension theorem). Define $\eta$ by
$[\eta(h)](y):=\tilde{\tau}(h,y)\Delta_{G}(h)^{\frac{1}{2}}[\eta_{0}(h)](y)$.
Then $\eta\in C_{c}(G,C(X))$ and $[\eta(\cdot)](x)|_{G_{x}}\neq 0$. By the
Tietze extension theorem, one can find a non-zero function $\zeta\in C_{c}(G)$
such that

$\int_{G_{x}}\zeta(k^{-1})[\eta(k)](x)\,d\mu_{x}(k)\neq 0.$

Define $f_{0}$ by
$\displaystyle[f_{0}(h)](y):=\int_{G_{y}}\zeta(hk^{-1})[\eta(k)](y)\,d\mu_{y}(k)$.
Then $f_{0}\in C_{c}(G,C(X))$ and $f_{0}\neq 0$ since $[f_{0}(e)](x)\neq 0$.
For $y\in X$ and $s\in G$,

$\displaystyle\int_{G_{y}}[f_{0}(sh)](y)(\Delta_{G}(h)\Delta_{G_{y}}(h))^{-\frac{1}{2}}\,d\mu_{y}(h)$
$\displaystyle=\int_{G_{y}}\int_{G_{y}}\zeta(shk^{-1})[\eta(k)](y)(\Delta_{G}(h)\Delta_{G_{y}}(h))^{-\frac{1}{2}}\,d\mu_{y}(k)\,d\mu_{y}(h)$
$\displaystyle=\int_{G_{y}}\left(\int_{G_{y}}[\eta_{0}(kh)](y)\,d\mu_{y}(h)\right)\zeta(sk^{-1})\Delta_{G}(k)^{\frac{1}{2}}\Delta_{G_{y}}(k)^{-\frac{1}{2}}\,d\mu_{y}(k)$
$\displaystyle=0$

holds and this shows that this $f_{0}$ has the desired property.

Let $[f(s)](y)=f_{0}(s,s^{-1}.y)$ and we will show that $\tilde{\pi}(f)=0$.
Take $\xi\in C_{c}(G/G_{x})\subset L^{2}(G/G_{x})$ arbitrarily. Then for
$aG_{x}\in G/G_{x}$, one has

$\displaystyle[\tilde{\pi}(f)\xi](aG_{x})$
$\displaystyle=\int_{G}[f(g)](a.x)\left(\displaystyle\frac{dg.\nu}{d\nu}(aG_{x})\right)^{\frac{1}{2}}\xi(g^{-1}aG_{x})\,d\mu(g)$
$\displaystyle=\int_{G}[f(ag^{-1})](a.x)\left(\frac{\rho(g)}{\rho(a)}\right)^{\frac{1}{2}}\Delta_{G}(g)^{-1}\xi(gG_{x})\,d\mu(g)$
$\displaystyle=\int_{G/G_{x}}\int_{G_{x}}[f(ah^{-1}b^{-1})](a.x)\left(\frac{\rho(bh)}{\rho(a)}\right)^{\frac{1}{2}}\Delta_{G}(bh)^{-1}\rho(bh)^{-1}\xi(bG_{x})\,d\mu_{x}(h)\,d\nu(bG_{x})$
$\displaystyle=\int_{G/G_{x}}\left(\int_{G_{x}}[f(ah^{-1}b^{-1})](a.x)(\Delta_{G}(h)\Delta_{G_{x}}(h))^{-\frac{1}{2}}\,d\mu_{x}(h)\right)(\rho(a)\rho(b))^{-\frac{1}{2}}\Delta_{G}(b)^{-1}\xi(bG_{x})\,d\nu(bG_{x})$

where the third equality uses the following formula; for $\varphi\in
C_{c}(G)$,

$\displaystyle\int_{G}\varphi(g)\,dg=\int_{G/G_{x}}\int_{G_{x}}\varphi(bh)\rho(bh)^{-1}\,d\mu_{x}(h)\,d\nu(bG_{x}).$

For each $b\in G$,

$\displaystyle\int_{G_{x}}[f(ah^{-1}b^{-1})](a.x)(\Delta_{G}(h)\Delta_{G_{x}}(h))^{-\frac{1}{2}}\,d\mu_{x}(h)$
$\displaystyle=c_{b}\int_{G_{b.x}}[f(ab^{-1}k)](a.x)(\Delta_{G}(k)\Delta_{G_{b.x}}(k))^{-\frac{1}{2}}\,d\mu_{b.x}(k)$
$\displaystyle=c_{b}\int_{G_{b.x}}[f_{0}(ab^{-1}k)](b.x)(\Delta_{G}(k)\Delta_{G_{b.x}}(k))^{-\frac{1}{2}}\,d\mu_{b.x}(k)$
$\displaystyle=0$

holds where $c_{b}$ is defined as follows; for $b\in G$,
$(\operatorname{Ad}{b})_{*}\mu_{x}$ is a left Haar measure on $G_{b.x}$ where
$\operatorname{Ad}{b}\colon G_{x}\to G_{b.x}$ is a conjugation. A positive
number $c_{b}>0$ is defined as satisfying
$(\operatorname{Ad}{b})_{*}\mu_{x}=c_{b}\mu_{b.x}$. Therefore,
$\tilde{\pi}(f)=0$ and $C(X)\rtimes_{r}G$ is not simple. ∎

##  4\. Examples

###  4.1. Groups with a piecewise minimal-strongly-proximal action

In [5, Theorem 1.3.], Caprace, Le Boudec, and Matte Bon proved that any
totally disconnected locally compact group with a piecewise minimal-strongly-
proximal action on a totally disconnected compact space admits a topologically
free boundary. Therefore, a group with this assumption satisfies our condition
in Theorem 3.1. In particular, our theorem applies to the Neretin groups (see
[5, Section 4] for the Neretin groups).

######  Corollary 4.1.

Let $\mathcal{N}_{d,k}$ be the Neretin group and
$\partial_{F}\mathcal{N}_{d,k}$ be its Furstenberg boundary. Then
$C(\partial_{F}\mathcal{N}_{d,k})\rtimes_{r}\mathcal{N}_{d,k}$ is simple.

###  4.2. HNN extensions

We will recall and fix notations of HNN extensions and its Bass–Serre trees.
Let $H$ be a locally compact group, $H_{0}$ be its open subgroup, and
$\theta\colon H_{0}\hookrightarrow H$ be an open, injective, continuous
homomorphism. Then the HNN extension $G=\textrm{HNN}(H,H_{0},\theta)$
corresponding $H,H_{0},\text{and }\theta$ is defined to be $\langle H,t\mid
th_{0}t^{-1}=\theta(h_{0})\text{ for }h_{0}\in H_{0}\rangle$. If systems of
representatives $S_{-1}$ and $S_{1}$ of $H/H_{0}$ and $H/\theta(H_{0})$
respectively such that $e\in S_{1}$ and $e\in S_{-1}$ are fixed, then any
element $g$ of $G$ is represented as the unique normal form
$g=g_{1}t^{\varepsilon_{1}}g_{2}t^{\varepsilon_{2}}\cdots
g_{n}t^{\varepsilon_{n}}g_{n+1}$ where

  * •

$\varepsilon_{i}\in\\{\pm 1\\}$ and $g_{n+1}\in H$,

  * •

$g_{i}\in S_{\varepsilon_{i}}$,

  * •

If $g_{i}=1$, then $\varepsilon_{i-1}=\varepsilon_{i}$.

This ensures that the canonical homomorphism $H\to G$ is an embedding. There
is a unique topology on $G$ such that the embedding $H\hookrightarrow G$ is
continuous and open ([6, Proposition 8.B.10.]). A Bass–Serre tree associated
with $G$ is as follows; the vertex set is $G/H$, the edge set is $G/H_{0}$,
and an edge $gH_{0}\in G/H_{0}$ connects $gH$ and $gt^{-1}H$. The HNN
extension acts on its Bass–Serre tree by left translation.

######  Example 4.2.

Let $m,n$ be integers with $2\leq|m|\leq n$. The Baumslag–Solitar group
$\mathrm{BS}(m,n)$ is defined as $\langle a,t\mid ta^{m}t^{-1}=a^{n}\rangle$;
an HNN extension corresponding to $\mathbb{Z}=\langle
a\rangle,m\mathbb{Z}=\langle a^{m}\rangle,\text{ and }\theta(a^{m})=a^{n}$.
Its Bass–Serre tree is $(m+n)$-regular tree $T_{m+n}$. Let $\mathrm{G}(m,n)$
be a closure of $\mathrm{BS}(m,n)$ in $\operatorname{Aut}{(T_{m+n})}$. This
group is also an HNN extension, corresponding to $\overline{\langle
a\rangle},\overline{\langle a^{m}\rangle},\text{ and
}\overline{\theta}(h_{0})=th_{0}t^{-1}\text{ for }h_{0}\in\overline{\langle
a^{m}\rangle}$. Note that it is shown by Raum that this group is
$\mathrm{C}^{*}$-simple ([16]).

Let $G=\textrm{HNN}(H,H_{0},\theta)$ be an HNN extension. If
$1<[H:H_{0}]<\infty$ and $1<[H:\theta(H_{0})]<\infty$, then $\partial T$ is
closed in $T\sqcup\partial T$ with the shadow topology and the action of $G$
on its Bass–Serre tree $T$ is minimal, i.e., there is no non-trivial
$G$-invariant subtree, and of general type, i.e., there exist two hyperbolic
elements in $G$ without common fixed points in $\partial T$ (see [7,
Proposition 19.]). Therefore, the induced action of $G$ on the boundary of the
Bass–Serre tree $\partial T$ is a boundary action.

######  Proposition 4.3.

Let $G$ be an HNN extension corresponding to a totally disconnected locally
compact group $H$, its open subgroup $H_{0}$, and an open, injective,
continuous homomorphism $\theta\colon H_{0}\hookrightarrow H$, and $T$ be its
Bass–Serre tree. Assume that $1<[H:H_{0}]<\infty$,
$1<[H:\theta(H_{0})]<\infty$, there exist $a\in H\setminus H_{0}$ and $b\in
H\setminus\theta(H_{0})$ such that $a$ centralizes $H_{0}$ and $b$ centralizes
$\theta(H_{0})$, and $\bigcap_{n\geq 0}\theta^{-n}(H_{0}^{(n)})$ is trivial
where $H_{0}^{(0)}=H_{0}$ and $H_{0}^{(n+1)}=H_{0}\cap\theta(H_{0}^{(n)})$.
Then the action of $G$ on the boundary of the Bass–Serre tree $\partial T$ is
topologically free.

In particular, the action $\mathrm{G}(m,n)\curvearrowright\partial T_{m+n}$ is
topologically free.

###### Proof.

Consider an element $\xi$ of $\partial T$ represented by a ray

$btH,btat^{-1}H,btat^{-2}H,btat^{-3}H,btat^{-3}btH,btat^{-1}H,btat^{-2}H,\ldots$

(this element is represented by
$\prod_{i=1}^{\infty}(btat^{-1})(t^{-2})^{i}$). We will show that $G_{\xi}$ is
trivial. Suppose that there is a non-trivial element $g\in G_{\xi}$. Then $g$
fixes some ray representing $\xi$ or acts on some ray representing $\xi$ by
translation. First, assume that $g$ acts on some ray representing $\xi$ by
translation of degree $d\geq 1$. Then for sufficiently large $N$, wiriting
$\prod_{i=1}^{N}(btat^{-1})(t^{-2})^{i}$ as $x$, $g$ sends the three vertices
$xH,xbtH,xbtat^{-1}H$ to $xbtat^{-(d-1)}H,xbtat^{-d}H,xbtat^{-(d+1)}H$.
However, such an element $g$ does not exist because $G$-action on the tree $T$
preserves directions of edges, which is defined as follows; the source of an
edge $gH_{0}$ is $gH$ and the range of $gH_{0}$ is $gt^{-1}H$. Next, assume
that $g$ fixes some ray representing $\xi$. In this case, for large $N$,
wiriting $\prod_{i=1}^{N}(btat^{-1})(t^{-2})^{i}$ as $x$,

$g\in x\left(H\cap\bigcap_{i\geq
N+1}\left(\prod_{j=N+1}^{i}(btat^{-1})(t^{-2})^{j}\right)H\left(\prod_{j=N+1}^{i}(btat^{-1})(t^{-2})^{j}\right)^{-1}\right)x^{-1}.$

However, it is easy to see that

$H\cap\left(\prod_{j=N+1}^{i}(btat^{-1})(t^{-2})^{j}\right)H\left(\prod_{j=N+1}^{i}(btat^{-1})(t^{-2})^{j}\right)^{-1}\subset\theta^{-M(i)}(H_{0}^{(M(i)+1)})$

for $i\geq N+1$ where $M(i)$ is a some positive integer with $M(i)\to\infty$
as $i\to\infty$. This leads to a contradiction to a non-triviality of $g$. ∎

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