# Simplicity of right-angled Hecke C∗-algebras

Mario Klisse  TU Delft, EWI/DIAM, P.O.Box 5031, 2600 GA Delft, The Netherlands
<EMAIL_ADDRESS>
(Date: . _MSC2010:_ 20C08, 20F55, 46L05, 46L65. The author is supported by the
NWO project “The structure of Hecke-von Neumann algebras”, 613.009.125.)

###### Abstract.

By exploiting properties of boundaries associated with Coxeter groups we
obtain a complete characterization of simple right-angled multi-parameter
Hecke C∗-algebras. This extends previous results by Caspers, Larsen and the
author. Based on work by Raum and Skalski, we further describe the center and
the character space of right-angled Hecke C∗-algebras.

## Introduction

(Iwahori) Hecke algebras are deformations of the group algebra of Coxeter
groups depending on a deformation (multi-)parameter $q$. They can be viewed as
an abstraction of certain endomorphism rings which naturally appear in the
representation theory of finite groups of Lie type [4] and are particularly
well studied in the case of spherical and affine Coxeter groups (see [21],
[24], [2], [25]). For other Coxeter groups they appear in the context of
buildings and Kac-Moody groups acting on them [33].

Hecke algebras of a given Coxeter system $(W,S)$ can naturally be represented
on the Hilbert space $\ell^{2}(W)$ of square-summable functions on $W$. They
complete to C∗-algebras $C_{r,q}^{\ast}(W)$ and von Neumann algebras
$\mathcal{N}_{q}(W)$. The study of these operator algebras gave insight in the
cohomology of associated buildings and its $\ell^{2}$-Betti numbers (see [2],
[15], [13]) and they are related to Dykema’s interpolated free group factors,
which play an important role in the treatment of the infamous free factor
problem (see [14], [30], [17]). Much earlier Hecke operator algebras of
spherical and affine type Coxeter systems have been studied in [29].

Despite their ubiquitousness, until now Hecke operator algebras are well
understood only in the case of spherical and affine type Coxeter groups. In
particular, the natural question for a characterization of the simplicity
(i.e. triviality of the ideal structure) of $C_{r,q}^{\ast}(W)$ and the
factoriality (i.e. triviality of the center) of $\mathcal{N}_{q}(W)$ is wide
open. Recently, the investigation of right-angled Hecke C∗-algebras and right-
angled Hecke-von Neumann algebras made some progress. In [17] Garncarek
characterized the factoriality of single-parameter Hecke-von Neumann algebras.
Complementing his ideas with a new combinatorial approach, the result was
later extended to the multi-parameter case by Raum and Skalski [31]. In [9]
Caspers, Larsen and the author studied the C∗-algebraic setting and proved,
using classical averaging techniques, simplicity and trace-uniqueness results
for right-angled Hecke C∗-algebras and certain ranges of deformation
parameters $q$. It is further known that the reduced group C∗-algebra of an
irreducible Coxeter system is simple if and only if the corresponding Coxeter
system is of non-affine type (see [16], [19], [11]). Other relevant references
treating non-affine Hecke operator algebras are [8], [10], [32].

A notion that was famously used by Kalantar and Kennedy [23] to solve the
longstanding question for the C∗-simplicity of a given (discrete) group is
that of Furstenberg’s boundary and (topological) boundary actions. The authors
established a link between dynamical properties of the Furstenberg boundary of
a given group and the structure of the corresponding group C∗-algebra, which
led to results on simplicity, uniqueness of trace and tightness of nuclear
embeddings of group C∗-algebras (see e.g. [18], [5], [26]) and inspired
various generalizations (see e.g. [1], [20], [22]). Inspired by the approach
in [18] our present work goes into a similar direction. In [27] the author
introduced and studied topological boundaries and compactifications associated
with connected rooted graphs. These are topological spaces that reflect
combinatorial properties of the underlying graph and which are particularly
tractable in the case of (Cayley graphs of) Coxeter groups. In the latter
context the spaces have been considered earlier by Caprace-Lécureux [7] and
Lam-Thomas [28]. The striking advantage of the construction is its close
connection to the Hecke operator algebras of the corresponding Coxeter system
(see Section 1 for more details), which has been utilized in [27, Section 4].
We will further exploit the implications of this connection by using it to
characterize the simplicity of right-angled Hecke C∗-algebras, thus extending
the results in [9] and (partially) answering [9, Question 5.13]. This leads to
a full classification of the simplicity in the right-angled case.

######  Theorem.

Let $(W,S)$ be a right-angled, irreducible Coxeter system and $q=(q_{s})_{s\in
S}\in\mathbb{R}_{>0}^{(W,S)}$ a multi-parameter. Let further $\mathcal{R}$ be
the region of convergence of the growth series $\sum_{\mathbf{w}\in
W}z_{\mathbf{w}}$, set $\mathcal{R}^{\prime}:=\\{(q_{s}^{\epsilon_{s}})_{s\in
S}\mid q\in\mathcal{R}\cap\mathbb{R}_{>0}^{(W,S)}\text{,
}\epsilon\in\left\\{-1,1\right\\}^{(W,S)}\\}$ and define
$\overline{\mathcal{R}^{\prime}}$ to be the closure of $\mathcal{R}^{\prime}$
in $\mathbb{R}_{>0}^{(W,S)}$. Then the following statements hold:

  * •

If $\\#S<\infty$, then the Hecke C∗-algebra $C_{r,q}^{\ast}(W)$ is simple if
and only if
$q\in\mathbb{R}_{>0}^{(W,S)}\setminus\overline{\mathcal{R}^{\prime}}$;

  * •

If $\\#S=\infty$, then the Hecke C∗-algebra $C_{r,q}^{\ast}(W)$ is simple for
all $q\in\mathbb{R}_{>0}^{(W,S)}$.

Using a Haagerup-type inequality from [9], we will also prove that the central
projections of right-angled Hecke-von Neumann algebras considered by Raum and
Skalski in [31] are already contained in the corresponding Hecke C∗-algebras.
This leads to a decomposition of $C_{r,q}^{\ast}(W)$ which is analogous to the
one of $\mathcal{N}_{q}(W)$ and can be used to characterize the space of
characters (i.e. unital, multiplicative, linear functionals) of
$C_{r,q}^{\ast}(W)$.

_Structure._ In Section 1 we recall general facts about Coxeter groups, the
boundaries introduced in [27] and multi-parameter Hecke algebras (resp. their
operator algebraic counterparts). In Section 2 a number of technical
statements related to certain C∗-algebras associated with Coxeter systems are
proved, which allow to translate group algebraic arguments into the
$q$-deformed setting. This will be used in Section 3 and 4 where the center,
the character space and the simplicity of right-angled Hecke C∗-algebras are
characterized.

### Acknowledgements

I am grateful to my supervisor Martijn Caspers for many enlightening
discussions and helpful feedback. It is also a pleasure to thank Sven Raum and
Adam Skalski for their comments on an earlier draft of this paper.

##  1\. Preliminaries and notation

###  1.1. General notation

We will write $\mathbb{N}:=\left\\{0,1,2,...\right\\}$ and $\mathbb{N}_{\geq
1}:=\left\\{1,2,...\right\\}$ for the natural numbers. Scalar products of
Hilbert spaces are linear in the first variable and we denote the bounded
operators on a Hilbert space $\mathcal{H}$ by $\mathcal{B}(\mathcal{H})$. For
a C∗-algebra $A$ we will write $\mathcal{S}(A)$ for the state space of $A$ and
endow it with the weak-$\ast$ topology. Further, if $(A,G,\alpha)$ is a
dynamical system we write $g.a:=\alpha_{g}(a)$ where $a\in A$, $g\in G$.
Similar notation is being used for (continuous) group actions on topological
spaces. The symbol $\odot$ denotes the algebraic tensor product of
$\ast$-algebras, $\otimes$ is the minimal tensor product of C∗-algebras,
$\overline{\otimes}$ denotes the tensor product of von Neumann algebras and we
write $\rtimes_{red}$ for reduced (C∗-algebraic) crossed products. Further,
the neutral element of a group is always denoted by $e$ and for a set $S$ we
write $\\#S$ for the number of elements in $S$ and $\chi_{S}$ for the
characteristic function on $S$.

###  1.2. Coxeter groups

A _Coxeter group_ $W$ is a group generated by a (possibly infinite) set $S$ of
the form

$\displaystyle W=\left\langle S\mid\forall s,t\in S\text{:
}(st)^{m_{st}}=e\right\rangle,$

where $m_{st}\in\left\\{1,2,...,\infty\right\\}$ with $m_{ss}=1$ and
$m_{st}\geq 2$ for all $s\neq t$. The pair $(W,S)$ is called a _Coxeter
system_. It is _right-angled_ if $m_{st}=2$ or $m_{st}=\infty$ for all $s\neq
t$. If the generating set $S$ is finite the system $(W,S)$ has _finite rank_.
The data of $(W,S)$ is usually encoded in its _Coxeter diagram_ whose vertex
set is $S$ and whose edge set is given by $\\{(s,t)\mid m_{st}\geq 3\\}$ where
every edge between two vertices $s,t\in S$ is labeled by the corresponding
exponent $m_{st}$.

For a subset $T\subseteq S$ the _special subgroup_ $W_{T}$ of $W$ generated by
$T$ is also a Coxeter group with the same exponents as $W$ (see [12, Theorem
4.1.6]). The system $(W,S)$ is _irreducible_ if its Coxeter diagram is
connected. This is the case if and only if $W$ does not decompose as a non-
trivial direct product of special subgroups.

Every element $\mathbf{w}\in W$ decomposes as a product
$\mathbf{w}=s_{1}...s_{n}$ of generators $s_{1},...,s_{n}\in S$. The
expression $s_{1}...s_{n}$ is called _reduced_ if it has minimal length. The
_word length_ of $\mathbf{w}$, denoted by $\left|\mathbf{w}\right|$, is then
defined to be the number of generators appearing in a reduced expression for
$\mathbf{w}$, where $\left|e\right|:=0$. One says that $\mathbf{w}$ _starts_
(resp. _ends_) with $\mathbf{v}\in W$ if
$|\mathbf{v}^{-1}\mathbf{w}|=|\mathbf{w}|-|\mathbf{v}|$ (resp.
$|\mathbf{w}\mathbf{v}^{-1}|=|\mathbf{w}|-|\mathbf{v}|$). In that case we
write $\mathbf{v}\leq_{R}\mathbf{w}$ (resp. $\mathbf{v}\leq_{L}\mathbf{w}$).
This defines a partial order on $W$ which is called the _weak right_ (resp.
_weak left_) _Bruhat order_ of $(W,S)$. It turns $W$ into a _complete meet-
semilattice_ (see [3, Theorem 3.2.1]).To simplify the notation we will usually
write $\leq$ instead of $\leq_{R}$.

In the right-angled case, cancellation of the form
$s_{1}...s_{n}=s_{1}...\widehat{s_{i}}...\widehat{s_{j}}...s_{n}$ for
$s_{1},...,s_{n}\in S$ implies that $s_{i}=s_{j}$ and that $s_{i}$ commutes
with $s_{i+1}...s_{j-1}$. Here
$s_{1}...s_{n}=s_{1}...\widehat{s_{i}}...\widehat{s_{j}}...s_{n}$ means that
$s_{i}$ and $s_{j}$ are removed from the expression $s_{1}...s_{n}$. We will
use this frequently without further mention. Another useful property is the
following.

######  Theorem 1.1 ( [3, Proposition 3.1.2 (vi)]).

Let $(W,S)$ be a Coxeter system, $\mathbf{v},\mathbf{w}\in W$ and $s\in S$
with $s\leq\mathbf{v}$, $s\leq\mathbf{w}$. Then, $\mathbf{v}\leq\mathbf{w}$ if
and only if $s\mathbf{v}\leq s\mathbf{w}$.

###  1.3. Topological boundaries of Coxeter groups

In [27] topological boundaries and compactifications associated with connected
rooted graphs were introduced and studied. These topological spaces are
particularly useful in the case of (Cayley graphs of) Coxeter groups. For
these, the spaces have been introduced earlier by Caprace and Lécureux in [7]
and by Lam and Thomas in [28] in a different setting, using different
formalisms. The construction that we will follow in this paper coincides with
the one in [27], but we will restrict to the case of Cayley graphs of Coxeter
groups. For more details and the general construction see [27].

Let $(W,S)$ be a finite rank Coxeter system and denote by $K:=\text{Cay}(W,S)$
the _Cayley graph_ of $W$ with respect to the generating set $S$, i.e. the
graph with vertex set $W$ and edge set $\\{(\mathbf{v},\mathbf{w})\in W\times
W\mid\mathbf{v}^{-1}\mathbf{w}\in S\\}$. The metric $d\text{: }W\times
W\rightarrow\mathbb{R}_{\geq 0}$ defined by
$d(\mathbf{v},\mathbf{w}):=|\mathbf{v}^{-1}\mathbf{w}|$ turns (the vertex set
of) $K$ into a metric space. A _geodesic path_ $\alpha$ in $K$ is a (possibly
infinite) sequence $\alpha_{0}\alpha_{1}...$ of vertices with
$d(\alpha_{i},\alpha_{j})=\left|i-j\right|$ for all $i,j$. For such a path
$\alpha$ and $\mathbf{w}\in W$ we write $\mathbf{w}\leq\alpha$ if
$\mathbf{w}\leq\alpha_{i}$ for all large enough $i$ and we write
$\mathbf{w}\nleq\alpha$ if $\mathbf{w}\nleq\alpha_{i}$ for all large enough
$i$. Now, define an equivalence relation $\sim$ on the set of all infinite
geodesic paths in $K$ via $\alpha\sim\beta$ if and only if for every
$\mathbf{\mathbf{w}}\in W$ the implications
$\mathbf{w}\leq\alpha\Leftrightarrow\mathbf{w}\leq\beta$ hold. Write
$\partial(W,S)$ for the set of corresponding equivalence classes. This set is
called the _boundary_ of $(W,S)$ and $\overline{(W,S)}:=W\cup\partial(W,S)$ is
called the _compactification_ of $(W,S)$. The weak right Bruhat order
naturally extends to a partial order $\leq$ on $\overline{(W,S)}$ (see [27,
Lemma 2.2]). We then equip $\overline{(W,S)}$ with the topology generated by
the subbase of sets of the form

$\displaystyle\mathcal{U}_{\mathbf{w}}:=\left\\{z\in\overline{(W,S)}\mid\mathbf{w}\leq
z\right\\}\text{\> and
\>}\mathcal{U}_{\mathbf{w}}^{c}:=\left\\{z\in\overline{(W,S)}\mid\mathbf{w}\nleq
z\right\\}\text{,}$

where $\mathbf{w}\in W$. This turns $\partial(W,S)$ and $\overline{(W,S)}$
into metrizable compact spaces and $W$ naturally embeds as a dense discrete
subset into $\overline{(W,S)}$. Further, the left action of $W$ on itself
induces a (continuous) action $W\curvearrowright\overline{(W,S)}$ with
$W.(\partial(W,S))=\partial(W,S)$. This action has some desirable properties,
one of which will play a role in the characterization of the simplicity of
right-angled Hecke C∗-algebras.

######  Theorem 1.2 ([27, Theorem 3.20 and Proposition 3.26]).

Let $(W,S)$ be a right-angled irreducible Coxeter system with
$3\leq\\#S<\infty$. Then the action $W\curvearrowright\partial(W,S)$ is a
_boundary action_ , meaning that the following statements hold:

  * •

_Minimality:_ For every $z\in\partial(W,S)$ the $W$-orbit
$W.z:=\left\\{\mathbf{w}.z\mid\mathbf{w}\in W\right\\}$ is dense in
$\partial(W,S)$;

  * •

_Strong proximality:_ For every probability measure
$\nu\in\text{Prob}\left(\partial(W,S)\right)$ the weak-$\ast$ closure of the
$W$-orbit $W.\nu$ contains a point mass
$\delta_{z}\in\text{Prob}(\partial(W,S))$ for some $z\in\partial(W,S)$.

Further, the action is _topologically free_ , i.e. for every $\mathbf{w}\in
W\setminus\left\\{e\right\\}$ the set
$(\partial(W,S))^{\mathbf{w}}:=\left\\{z\in\partial(W,S)\mid\mathbf{w}.z=z\right\\}$
has no inner points.

###  1.4. Multi-parameter Hecke algebras

For a Coxeter system $(W,S)$ define $\mathbb{R}_{>0}^{(W,S)}$ to be the set of
all multi-parameters $q=(q_{s})_{s\in S}\in\mathbb{R}^{S}$ for which
$q_{s}=q_{t}$ for all $s,t\in S$ which are conjugate to each other. The sets
$\mathbb{C}^{(W,S)}$ and $\left\\{-1,1\right\\}^{(W,S)}$ are defined in a
similar way. For a tuple $q=(q_{s})_{s\in S}\in\mathbb{R}_{>0}^{(W,S)}$, $s\in
S$ and a reduced expression $\mathbf{w}=s_{1}...s_{n}$ of $\mathbf{w}\in W$
write

$\displaystyle q_{\mathbf{w}}:=q_{s_{1}}...q_{s_{n}}\text{\> and
\>}p_{s}(q):=q_{s}^{-\frac{1}{2}}(q_{s}-1)\text{.}$

Then $q_{\mathbf{w}}$ does not depend on the choice of the reduced expression
for $\mathbf{w}$. Following the notation in [31] we also write
$q_{s,\epsilon}:=\epsilon_{s}q_{s}^{\epsilon_{s}}$ and
$q_{\mathbf{w},\epsilon}:=q_{s_{1},\epsilon}...q_{s_{n},\epsilon}$ for
$q\in\mathbb{R}_{>0}^{(W,S)}$, $\epsilon\in\left\\{-1,1\right\\}^{(W,S)}$.

By [12, Proposition 19.1.1] for $q\in\mathbb{R}_{>0}^{(W,S)}$ there exists a
unique (unital) $\ast$-algebra $\mathbb{C}_{q}\left[W\right]$ spanned by a
linear basis $\\{T_{\mathbf{w}}^{(q)}\mid\mathbf{w}\in W\\}$ such that for
$s\in S$, $\mathbf{w}\in W$ one has

$\displaystyle
T_{s}^{\left(q\right)}T_{\mathbf{w}}^{\left(q\right)}=\begin{cases}T_{s\mathbf{w}}^{\left(q\right)}&\text{,
if }s\nleq\mathbf{w}\\\
T_{s\mathbf{w}}^{\left(q\right)}+p_{s}(q)T_{\mathbf{w}}^{\left(q\right)}&\text{,
if }s\leq\mathbf{w}\end{cases}$ (1.1)

and

$\displaystyle(T_{\mathbf{w}}^{(q)})^{\ast}=T_{\mathbf{w}^{-1}}^{(q)}\text{.}$

This $\ast$-algebra is called the _(Iwahori) Hecke algebra_ of $(W,S)$ with
parameter $q$. Here we use a different normalization of the generators than in
[12], which coincides with the notation in [17], [8], [10], [9], [31] and
[27]. The equality (1.1) in particular implies that
$T_{\mathbf{w}}^{(q)}=T_{s_{1}}^{(q)}...T_{s_{n}}^{(q)}$ for a reduced
expression $\mathbf{w}=s_{1}...s_{n}$ of $\mathbf{w}\in W$. The $\ast$-algebra
$\mathbb{C}_{q}\left[W\right]$ can be represented on the bounded operators of
$\ell^{2}(W)$ via

$\displaystyle
T_{s}^{\left(q\right)}\delta_{\mathbf{w}}=\begin{cases}\delta_{s\mathbf{w}}&\text{,
if }s\nleq\mathbf{w}\\\
\delta_{s\mathbf{w}}+p_{s}(q)\delta_{\mathbf{w}}&\text{, if
}s\leq\mathbf{w}\end{cases},$

where $(\delta_{\mathbf{w}})_{\mathbf{w}\in W}$ denotes the canonical
orthonormal basis of $\ell^{2}(W)$. This defines a faithful
$\ast$-representation
$\mathbb{C}_{q}\left[W\right]\hookrightarrow\mathcal{B}(\ell^{2}(W))$,
therefore we will view $\mathbb{C}_{q}\left[W\right]$ as a $\ast$-subalgebra
of $\mathcal{B}(\ell^{2}(W))$. The norm closure
$C_{r,q}^{\ast}(W):=\overline{\mathbb{C}_{q}\left[W\right]}^{\left\|\cdot\right\|}$
is called the _(reduced) Hecke C ∗-algebra_ and the weak closure
$\mathcal{N}_{q}(W):=\overline{\mathbb{C}_{q}\left[W\right]}^{w.o.}$ is called
the _Hecke-von Neumann algebra_. Note that for $q_{s}=1$, $s\in S$,

$\displaystyle\mathbb{C}_{q}\left[W\right]=\mathbb{C}\left[W\right]\text{,
}C_{r,q}^{\ast}(W)=C_{r}^{\ast}(W)\text{, }\mathcal{N}_{q}(W)=\mathcal{L}(W)$

are the group algebra, reduced group C∗-algebra and group-von Neumann algebra
of $W$. Further, for every $q\in\mathbb{R}_{>0}^{(W,S)}$ the vector state
$x\mapsto\left\langle x\delta_{e},\delta_{e}\right\rangle$ restricts to a
faithful tracial state $\tau_{q}$ on $C_{r,q}^{\ast}(W)$ and
$\mathcal{N}_{q}(W)$ with $\tau_{q}(T_{\mathbf{w}}^{(q)})=0$ for all
$\mathbf{w}\in W$.

The following statement is well-known. A proof can be found in [9, Proposition
4.7].

######  Proposition 1.3.

Let $(W,S)$ be a Coxeter system, $q=(q_{s})_{s\in
S}\in\mathbb{R}_{>0}^{(W,S)}$ and $\epsilon=(\epsilon_{s})_{s\in
S}\in\left\\{-1,1\right\\}^{(W,S)}$. Set
$q^{\prime}:=(q_{s}^{\epsilon_{s}})_{s\in S}$. Then $C_{r,q}^{\ast}(W)\cong
C_{r,q^{\prime}}^{\ast}(W)$ via
$T_{s}^{(q)}\mapsto\epsilon_{s}T_{s}^{(q^{\prime})}$.

The following decomposition follows from the universal property of the Hecke
algebra and [9, Lemma 1].

######  Lemma 1.4.

Let $(W,S)$ be a Coxeter system which admits a non-trivial decomposition of
the form $(W,S)=(W_{T}\times W_{T^{\prime}},T\sqcup T^{\prime})$. Set
$q_{T}:=(q_{t})_{t\in T}$ and $q_{T^{\prime}}:=(q_{t})_{t\in T^{\prime}}$.
Then for every $q\in\mathbb{R}_{>0}^{(W,S)}$ the corresponding Hecke algebra
decomposes as an algebraic tensor product
$\mathbb{C}_{q}\left[W\right]\cong\mathbb{C}_{q_{T}}\left[W_{T}\right]\odot\mathbb{C}_{q_{T^{\prime}}}\left[W_{T^{\prime}}\right]$
via

$\displaystyle T_{t}^{(q)}\mapsto\begin{cases}T_{t}^{(q_{T})}\otimes 1&\text{,
if }t\in T\\\ 1\otimes T_{t}^{(q_{T^{\prime}})}&\text{, if }t\in
T^{\prime}\end{cases},$

This induces C∗-algebraic and von Neumann-algebraic isomorphisms
$C_{r,q}^{\ast}(W)\cong C_{r,q_{T}}^{\ast}(W_{T})\otimes
C_{r,q_{T^{\prime}}}^{\ast}(W_{T^{\prime}})$ and
$\mathcal{N}_{q}(W)\cong\mathcal{N}_{q_{T}}(W_{T})\overline{\otimes}\mathcal{N}_{q_{T^{\prime}}}(W_{T^{\prime}})$.

Proposition 1.3 and Lemma 1.4 allow to restrict in the treatment of the
question for the simplicity of Hecke C∗-algebras to irreducible Coxeter
systems and multi-parameters $q=(q_{s})_{s\in S}\in\mathbb{R}_{>0}^{(W,S)}$
with $0<q_{s}\leq 1$. Since simplicity of C∗-algebras is preserved by
inductive limits, it further suffices to consider finite rank Coxeter groups,
as the following lemma illustrates.

######  Lemma 1.5.

Let $(W,S)$ be a Coxeter system, $q=(q_{s})_{s\in
S}\in\mathbb{R}_{>0}^{(W,S)}$ and $\mathcal{S}:=\left\\{T\subseteq S\mid
T\text{ finite}\right\\}$. For $T\in\mathcal{S}$ set $q_{T}:=(q_{t})_{t\in
T}$. Then

$\displaystyle\left\\{(C_{r,q_{T}}^{\ast}(W_{T}),\phi_{T,T^{\prime}})\mid
T,T^{\prime}\in\mathcal{S}\text{ with }T\subseteq T^{\prime}\right\\}$

with $\phi_{T,T^{\prime}}(T_{t}^{(q_{T})}):=T_{t}^{(q_{T^{\prime}})}$ for
$t\in T$ defines an inductive system with
$C_{r,q}^{\ast}(W)\cong\underrightarrow{\lim}C_{r,q_{T}}^{\ast}(W_{T})$.

##  2\. The C∗-algebras $\mathcal{D}(W,S)$ and $\mathfrak{A}(W)$

The aim of this section is to recall the construction of the C∗-algebras
$\mathcal{D}(W,S)$ and $\mathfrak{A}(W)$ associated with a given Coxeter
system $(W,S)$ which appears in [27, Section 4]. We will further prove a
number of technical statements which will play a role in the later sections.

Let $(W,S)$ be a finite rank Coxeter system and define for $\mathbf{w}\in W$,
$P_{\mathbf{w}}\in\ell^{\infty}(W)\subseteq\mathcal{B}(\ell^{2}(W))$ to be the
orthogonal projection onto
$\overline{\text{Span}\left\\{\delta_{\mathbf{v}}\mid\mathbf{v}\in W\text{
with }\mathbf{w}\leq\mathbf{v}\right\\}}\subseteq\ell^{2}(W)$. Note that
$P_{e}=1$.

######  Remark 2.1.

Recall that $W$ equipped with the weak right Bruhat order defines a complete
meet-semilattice. If existent, denote the corresponding join of two elements
$\mathbf{v},\mathbf{w}\in W$ by $\mathbf{v}\vee\mathbf{w}$. We then have
$P_{\mathbf{v}}P_{\mathbf{w}}=P_{\mathbf{v}\vee\mathbf{w}}$ for all
$\mathbf{v},\mathbf{w}\in W$ where we assume that
$P_{\mathbf{v}\vee\mathbf{w}}=0$ if the join $\mathbf{v}\vee\mathbf{w}$ does
not exist. In particular the equalities $P_{s}P_{t}=0$ (i.e. $P_{s}$ and
$P_{t}$ are orthogonal to each other) if $m_{st}=\infty$ and
$P_{s}P_{t}=P_{st}$ if $m_{st}=2$ hold.

Denote the quotient map of $\mathcal{B}(\ell^{2}(W))$ onto
$\mathcal{B}(\ell^{2}(W))/\mathcal{K}$ by $\pi$ where
$\mathcal{K}:=\mathcal{K}(\ell^{2}(W))$ is the ideal of compact operators in
$\mathcal{B}(\ell^{2}(W))$ and write
$\tilde{P}_{\mathbf{w}}:=\pi(P_{\mathbf{w}})$ for $\mathbf{w}\in W$. It was
shown in [9, Proposition 2.8] that the commutative C∗-algebra
$\mathcal{D}(W,S)$ generated by all projections $P_{\mathbf{w}}$,
$\mathbf{w}\in W$ identifies with $C(\overline{(W,S)})$ via
$P_{\mathbf{w}}\mapsto\chi_{\mathcal{U}_{\mathbf{w}}}$. Similarly,
$\pi(\mathcal{D}(W,S))\cong C(\partial(W,S))$ via
$\tilde{P}_{\mathbf{w}}\mapsto\chi_{\mathcal{U}_{\mathbf{w}}\cap\partial(W,S)}$.
Further, let $\mathfrak{A}(W)$ be the C∗-subalgebra of
$\mathcal{B}(\ell^{2}(W))$ generated by the reduced group C∗-algebra
$C_{r}^{\ast}(W)$ and $\mathcal{D}(W,S)$. Since for $q=(q_{s})_{s\in
S}\in\mathbb{R}_{>0}^{(W,S)}$ and $s\in S$ the operator $T_{s}^{(q)}$
decomposes as $T_{s}^{(1)}+p_{s}(q)P_{s}$, we have an inclusion of the
corresponding Hecke C∗-algebra $C_{r,q}^{\ast}(W)\subseteq\mathfrak{A}(W)$. In
fact, $\mathfrak{A}(W)$ is the smallest C∗-subalgebra of
$\mathcal{B}(\ell^{2}(W))$ containing all Hecke C∗-algebras of $(W,S)$. It
naturally identifies with the reduced crossed product C∗-algebra of the action
$W\curvearrowright\overline{(W,S)}$ via

$\displaystyle\iota\text{: }\mathfrak{A}(W)\cong
C(\overline{(W,S)})\rtimes_{red}W\text{,
}P_{\mathbf{w}}\mapsto\chi_{\mathcal{U}_{\mathbf{w}}}\text{ and
}T_{\mathbf{w}}^{(1)}\mapsto\lambda_{\mathbf{w}}\text{,}$

where $\lambda$ denotes the left-regular representation of $W$. In a similar
way $\pi(\mathfrak{A}(W))$ identifies with $C(\partial(W,S))\rtimes_{red}W$
via

$\displaystyle\kappa\text{: }\pi(\mathfrak{A}(W))\cong
C(\partial(W,S))\rtimes_{red}W\text{,
}\tilde{P}_{\mathbf{w}}\mapsto\chi_{\mathcal{U}_{\mathbf{w}}\cap\partial(W,S)}\text{
and }\pi(T_{\mathbf{w}}^{(1)})\mapsto\lambda_{\mathbf{w}}\text{.}$

These maps are $W$-equivariant with respect to the action of $W$ on
$\mathfrak{A}(W)$ defined by
$\mathbf{w}.x:=T_{\mathbf{w}}^{(1)}xT_{\mathbf{w}^{-1}}^{(1)}$ for
$\mathbf{w}\in W$, $x\in\mathfrak{A}(W)$ and the action of $W$ on
$\pi(\mathfrak{A}(W))$ defined by
$\mathbf{w}.x:=T_{\mathbf{w}}^{(1)}xT_{\mathbf{w}^{-1}}^{(1)}$ for
$\mathbf{w}\in W$, $x\in\pi(\mathfrak{A}(W))$. Denote the _region of
convergence_ of the multi-variate growth series $W(z):=\sum_{\mathbf{w}\in
W}z_{\mathbf{w}}$ by

$\displaystyle\mathcal{R}:=\\{z\in\mathbb{C}^{(W,S)}\mid W(z)\text{
converges}\\},$

set

$\displaystyle\mathcal{R}^{\prime}:=\\{(q_{s}^{\epsilon_{s}})_{s\in S}\mid
q\in\mathcal{R}\cap\mathbb{R}_{>0}^{(W,S)}\text{,
}\epsilon\in\left\\{-1,1\right\\}^{(W,S)}\\}$

and let $\overline{\mathcal{R}^{\prime}}$ be the closure of
$\mathcal{R}^{\prime}$ in $\mathbb{R}_{>0}^{(W,S)}$. By [9, Corollary 4.4],
for $q\in\mathbb{R}_{>0}^{(W,S)}\setminus\mathcal{R}^{\prime}$ the restriction
of $\kappa\circ\pi$ to $C_{r,q}^{\ast}(W)$ factors to an embedding of
$C_{r,q}^{\ast}(W)$ into $C(\partial(W,S))\rtimes_{red}W$. We will therefore
often view $C_{r,q}^{\ast}(W)$ with
$q\in\mathbb{R}_{>0}^{(W,S)}\setminus\mathcal{R}^{\prime}$ as a C∗-subalgebra
of $C(\partial(W,S))\rtimes_{red}W$ and of $\pi(\mathfrak{A}(W))$.

###  2.1. Elementary properties of the action
$W\curvearrowright\mathcal{D}(W,S)$

Let us proceed with some technical statements which will play a role in the
following sections.

######  Proposition 2.2.

Let $(W,S)$ be a right-angled Coxeter system and $\mathbf{w}\in W$, $s\in S$.
Then the following equalities hold:

  1. (1)

$s.P_{\mathbf{w}}=P_{s\mathbf{w}}$ if $\mathbf{w}\notin C_{W}(s)$;

  2. (2)

$s.P_{\mathbf{w}}=P_{s\mathbf{w}}-P_{\mathbf{w}}$ if $\mathbf{w}\in C_{W}(s)$
and $s\leq\mathbf{w}$;

  3. (3)

$s.P_{\mathbf{w}}=P_{\mathbf{w}}$ if $\mathbf{w}\in C_{W}(s)$ and
$s\nleq\mathbf{w}$.

Here $C_{W}(s):=\left\\{\mathbf{v}\in W\mid s\mathbf{v}=\mathbf{v}s\right\\}$
denotes the centralizer of $s$ in $W$.

###### Proof.

First observe that by Proposition 1.1 for all $s\in S$ and
$\mathbf{v},\mathbf{w}\in W$ with $s\leq\mathbf{w}$, $s\nleq\mathbf{v}$ or
$s\nleq\mathbf{w}$, $s\leq\mathbf{v}$,

$\displaystyle(s.P_{\mathbf{w}})\delta_{\mathbf{v}}=T_{s}^{(1)}P_{\mathbf{w}}\delta_{s\mathbf{v}}=\begin{cases}\delta_{\mathbf{v}}&\text{,
if }\mathbf{w}\leq s\mathbf{v}\\\ 0&\text{, if }\mathbf{w}\nleq
s\mathbf{v}\end{cases}=\begin{cases}\delta_{\mathbf{v}}&\text{, if
}s\mathbf{w}\leq\mathbf{v}\\\ 0&\text{, if
}s\mathbf{w}\nleq\mathbf{v}\end{cases}=P_{s\mathbf{w}}\delta_{\mathbf{v}}\text{.}$
(2.1)

We will cover the remaining cases in the following.

_(1)_ : Assume that $\mathbf{w}\notin C_{W}(s)$. If $s\leq\mathbf{w}$ and
$s\leq\mathbf{v}$, then $\mathbf{w}\nleq s\mathbf{v}$ and
$s\mathbf{w}\nleq\mathbf{v}$. Indeed, if we assume that $\mathbf{w}\leq
s\mathbf{v}$, then $s\leq s\mathbf{v}$ in contradiction to $s\nleq
s\mathbf{v}$. Further, if we assume that $s\mathbf{w}\leq\mathbf{v}$, then
there exists $\mathbf{u}\in W$ with $\mathbf{v}=(s\mathbf{w})\mathbf{u}$ and
$\left|\mathbf{v}\right|=\left|s\mathbf{w}\right|+\left|\mathbf{u}\right|$.
Since $(W,S)$ is right-angled $s\leq\mathbf{v}$ implies that $s\leq\mathbf{u}$
and $s\mathbf{w}\in C_{W}(s)$. But then $\mathbf{w}\in C_{W}(s)$ in
contradiction to the assumption $\mathbf{w}\notin C_{W}(s)$. We get that
$(s.P_{\mathbf{w}})\delta_{\mathbf{v}}=0=P_{s\mathbf{w}}\delta_{\mathbf{v}}$.

If $s\nleq\mathbf{w}$ and $s\nleq\mathbf{v}$, then one obtains in the same way
$\mathbf{w}\nleq s\mathbf{v}$ and $s\mathbf{w}\nleq\mathbf{v}$ which implies
$(s.P_{\mathbf{w}})\delta_{\mathbf{v}}=0=P_{s\mathbf{w}}\delta_{\mathbf{v}}$.
With (2.1) this covers all possible cases. Hence,
$s.P_{\mathbf{w}}=P_{s\mathbf{w}}$.

_(2)_ : Assume that $\mathbf{w}\in C_{W}(s)$ and $s\leq\mathbf{w}$. If
$s\nleq\mathbf{v}$, then (2.1) implies
$(s.P_{\mathbf{w}})\delta_{\mathbf{v}}=P_{s\mathbf{w}}\delta_{\mathbf{v}}$ and
hence $(s.P_{\mathbf{w}})(1-P_{s})=P_{s\mathbf{w}}(1-P_{s})$. If
$s\leq\mathbf{v}$, then $\mathbf{w}\nleq s\mathbf{v}$ implies
$(s.P_{\mathbf{w}})\delta_{\mathbf{v}}=0$ and hence
$(s.P_{\mathbf{w}})P_{s}=0$. Combined this leads to

$\displaystyle
s.P_{\mathbf{w}}=(s.P_{\mathbf{w}})(1-P_{s})+(s.P_{\mathbf{w}})P_{s}=P_{s\mathbf{w}}(1-P_{s})=P_{s\mathbf{w}}-P_{s\mathbf{w}\vee
s}=P_{s\mathbf{w}}-P_{\mathbf{w}}\text{.}$

_(3)_ : Assume that $\mathbf{w}\in C_{W}(s)$ and $s\nleq\mathbf{w}$. If
$s\leq\mathbf{v}$, then
$(s.P_{\mathbf{w}})\delta_{\mathbf{v}}=P_{s\mathbf{w}}\delta_{\mathbf{v}}=P_{\mathbf{w}\vee
s}\delta_{\mathbf{v}}=P_{\mathbf{w}}P_{s}\delta_{\mathbf{v}}=P_{\mathbf{w}}\delta_{\mathbf{v}}$
by (2.1). So consider the case where $s\nleq\mathbf{v}$. If
$\mathbf{w}\leq\mathbf{v}$, then $\mathbf{v}=\mathbf{w}\mathbf{u}$ for some
$\mathbf{u}\in W$ with
$\left|\mathbf{v}\right|=\left|\mathbf{w}\right|+\left|\mathbf{u}\right|$ and
$s\nleq\mathbf{u}$. Hence,
$s\mathbf{v}=\mathbf{w}(s\mathbf{u})\geq\mathbf{w}$. Conversely, if
$\mathbf{w}\leq s\mathbf{v}$, then $s\mathbf{v}=\mathbf{w}\mathbf{u}$ for some
$\mathbf{u}\in W$ with
$\left|s\mathbf{v}\right|=\left|\mathbf{w}\right|+\left|\mathbf{u}\right|$ and
$s\leq\mathbf{u}$. We get that
$\mathbf{v}=s(s\mathbf{v})=\mathbf{w}(s\mathbf{u})\geq\mathbf{w}$. Together
this gives
$(s.P_{\mathbf{w}})\delta_{\mathbf{v}}=P_{\mathbf{w}}\delta_{\mathbf{v}}$,
i.e. $s.P_{\mathbf{w}}=P_{\mathbf{w}}$ as claimed. ∎

######  Remark 2.3.

Let $(W,S)$ be a right-angled Coxeter system, $q\in\mathbb{R}_{>0}^{(W,S)}$
and $s\in S$, $\mathbf{w}\in W$. Recall that
$T_{s}^{(q)}=T_{s}^{(1)}+p_{s}(q)P_{s}$ for $q\in\mathbb{R}_{>0}^{(W,S)}$. In
combination with Remark 2.1 the Proposition 2.2 leads to a description of the
conjugation of the generating projections in $\mathcal{D}(W,S)$ with the Hecke
operators $T_{s}^{(q)}$, $s\in S$. In particular, for $s\in S$, $\mathbf{w}\in
W$ with $\mathbf{w}\notin C_{W}(s)$ and $s\nleq\mathbf{w}$ the identities

$\displaystyle
T_{s}^{(q)}(1-P_{s})T_{s}^{(q)}=T_{s}^{(1)}(1-P_{s})T_{s}^{(1)}=P_{s}$

and

$\displaystyle
T_{s}^{(q)}P_{\mathbf{w}}T_{s}^{(q)}=T_{s}^{(1)}P_{\mathbf{w}}T_{s}^{(1)}=P_{s\mathbf{w}}$

hold.

###  2.2. Paths in the Coxeter diagram of right-angled Coxeter groups

To simplify the statements and proofs of later sections, we introduce the
following notion which already implicitly appears in the proof of [27, Theorem
3.20].

######  Definition 2.4.

Let $(W,S)$ be a right-angled finite rank Coxeter system. A _path
$s_{1}...s_{n}\in W$ in the Coxeter diagram of $(W,S)$_ is a product of
generators $s_{1},...,s_{n}\in S$ with $m_{s_{i}s_{i+1}}=\infty$ for
$i=1,...n-1$. We say that the path is _closed_ if $m_{s_{1}s_{n}}=\infty$ and
that the path _covers the whole graph_ if
$\left\\{s_{1},...,s_{n}\right\\}=S$.

######  Remark 2.5.

Let $(W,S)$ be a right-angled finite rank Coxeter system. For a closed path
$\mathbf{g}:=s_{1}...s_{n}\in W$ in the Coxeter diagram of $(W,S)$ that covers
the whole graph we have that
$\left|s\mathbf{g}\right|>\left|\mathbf{g}\right|$ for every $s\in
S\setminus\left\\{s_{1}\right\\}$ and
$C_{W}(\mathbf{g})=\left\\{\mathbf{g}^{i}\mid i\in\mathbb{Z}\right\\}$. In
particular,
$\left|\mathbf{g}^{n}\right|=\left|n\right|\left|\mathbf{g}\right|$ for every
$n\in\mathbb{Z}$. These facts have been crucial in the proof of [27, Theorem
3.20].

In the single-parameter case the following lemma appears in [8, Lemma 2.7].
The proof presented there translates verbatim to the multi-parameter case.
Therefore we omit it here.

######  Lemma 2.6 ([8, Lemma 2.7]).

Let $(W,S)$ be a right-angled Coxeter system. Denote the set of subsets of $S$
whose elements pairwise commute (including the empty set) by Cliq and write
$P_{\Gamma}:=\prod_{s\in\Gamma}P_{s}$ for $\Gamma\in\text{Cliq}$. Further, for
$\mathbf{w}\in W$ let $A_{\mathbf{w}}$ be the set of triples
$\left(\mathbf{w}^{\prime},\Gamma,\mathbf{w}^{\prime\prime}\right)$ with
$\mathbf{w}^{\prime},\mathbf{w}^{\prime\prime}\in W$ and
$\Gamma\in\text{Cliq}$ such that
$\mathbf{w}=\mathbf{w}^{\prime}\left(\prod_{s\in\Gamma}s\right)\mathbf{w}^{\prime\prime}$,
$|\mathbf{w}|=|\mathbf{w}^{\prime}|+|\prod_{s\in\Gamma}s|+|\mathbf{w}^{\prime\prime}|$
and $|\mathbf{w}^{\prime}t|>|\mathbf{w}^{\prime}|$ for all $t\in S$ with
$m_{st}=2$ for all $s\in\Gamma$. Then the operator $T_{\mathbf{w}}^{(q)}$
decomposes as

$\displaystyle
T_{\mathbf{w}}^{(q)}=\sum_{\left(\mathbf{w}^{\prime},\Gamma,\mathbf{w}^{\prime\prime}\right)\in
A_{\mathbf{w}}}\left(\prod_{s\in\Gamma}p_{s}(q)\right)T_{\mathbf{w}^{\prime}}^{(1)}P_{\Gamma}T_{\mathbf{w}^{\prime\prime}}^{(1)}\text{.}$

######  Corollary 2.7.

Let $(W,S)$ be a right-angled Coxeter system, $q=(q_{s})_{s\in
S}\in\mathbb{R}_{>0}^{(W,S)}$, $l\in\mathbb{N}$ and let
$\mathbf{g}:=s_{1}...s_{n}\in W$ be a closed path in the Coxeter diagram of
$(W,S)$. Then there exists an operator $x\in\mathfrak{A}(W)$ such that
$T_{\mathbf{g}^{l}}^{(q)}$ decomposes as
$T_{\mathbf{g}^{l}}^{(q)}=T_{\mathbf{g}^{l}}^{(1)}+P_{s_{1}}x$.

###### Proof.

Write $t_{1}...t_{m}$ for the reduced expression
$\mathbf{g}^{l}=(s_{1}...s_{n})(s_{1}...s_{n})...(s_{1}...s_{n})$ of
$\mathbf{g}^{l}$ where $m=nl$. By Lemma 2.6 the operator
$T_{\mathbf{g}^{l}}^{(q)}$ decomposes as

$\displaystyle
T_{\mathbf{g}^{l}}^{(q)}=T_{\mathbf{g}^{l}}^{(1)}+\sum_{i=1}^{m}p_{t_{i}}(q)T_{t_{1}...t_{i-1}}^{(1)}P_{t_{i}}T_{t_{i+1}...t_{m}}^{(1)}\text{.}$

Using the description in Proposition 2.2 we get that

$\displaystyle
T_{\mathbf{g}^{l}}^{(q)}=T_{\mathbf{g}^{l}}^{(1)}+\sum_{i=1}^{m}p_{t_{i}}(q)T_{t_{1}...t_{i-2}}^{(1)}P_{t_{i-1}t_{i}}T_{t_{i-1}t_{i+1}...t_{m}}^{(1)}=...=T_{\mathbf{g}^{l}}^{(1)}+\sum_{i=1}^{m}p_{t_{i}}(q)P_{t_{1}...t_{i}}T_{t_{1}...\widehat{t_{i}}...t_{m}}^{(1)}\text{,}$

so the claim follows by setting
$x:=\sum_{i=1}^{m}p_{t_{i}}(q)P_{t_{1}...t_{i}}T_{t_{1}...\widehat{t_{i}}...t_{n}}^{(1)}$.
∎

######  Lemma 2.8.

Let $(W,S)$ be an irreducible right-angled, finite rank Coxeter system, let
$\mathbf{g}:=s_{1}...s_{n}\in W$ be a path in the Coxeter diagram of $(W,S)$
that covers the whole graph and let $q=(q_{s})_{s\in
S}\in\mathbb{R}_{>0}^{(W,S)}$. Then the series $\sum_{\mathbf{w}\in
W}q_{\mathbf{w}}$ converges if and only if the series $\sum_{\mathbf{w}\in
W\text{: }\mathbf{g}\leq\mathbf{w}^{-1}}q_{\mathbf{w}}$ converges.

###### Proof.

Since all summands of the series are positive it is clear that the convergence
of $\sum_{\mathbf{w}\in W}q_{\mathbf{w}}$ implies the convergence of
$\sum_{\mathbf{w}\in W\text{: }\mathbf{g}\leq\mathbf{w}^{-1}}q_{\mathbf{w}}$.
So assume that $\sum_{\mathbf{w}\in W\text{:
}\mathbf{g}\leq\mathbf{w}^{-1}}q_{\mathbf{w}}$ converges. For every
$i,j\in\mathbb{N}$ with $i<j$ we have that

$\displaystyle\left|\sum_{\mathbf{w}\in W\text{: }\left|\mathbf{w}\right|\leq
i}q_{\mathbf{w}}-\sum_{\mathbf{w}\in W\text{: }\left|\mathbf{w}\right|\leq
j}q_{\mathbf{w}}\right|$ $\displaystyle=$ $\displaystyle\sum_{\mathbf{w}\in
W\text{: }i<\left|\mathbf{w}\right|\leq j}q_{\mathbf{w}}$ $\displaystyle=$
$\displaystyle\sum_{\mathbf{w}\in W\text{: }i<\left|\mathbf{w}\right|\leq
j\text{, }s_{n}\leq\mathbf{w}^{-1}}q_{\mathbf{w}}+\sum_{\mathbf{w}\in W\text{:
}i<\left|\mathbf{w}\right|\leq j\text{,
}s_{n}\nleq\mathbf{w}^{-1}}q_{\mathbf{w}}$ $\displaystyle=$
$\displaystyle\sum_{\mathbf{w}\in W\text{: }i-1<\left|\mathbf{w}\right|\leq
j-1\text{,
}s_{n}\nleq\mathbf{w}^{-1}}q_{s_{n}}q_{\mathbf{w}}+\sum_{\mathbf{w}\in
W\text{: }i<\left|\mathbf{w}\right|\leq j\text{,
}s_{n}\nleq\mathbf{w}^{-1}}q_{\mathbf{w}}$ $\displaystyle\leq$
$\displaystyle(1+q_{s_{n}})\sum_{\mathbf{w}\in W\text{:
}i-1<\left|\mathbf{w}\right|\leq j\text{,
}s_{n}\nleq\mathbf{w}^{-1}}q_{\mathbf{w}}$ $\displaystyle=$
$\displaystyle\frac{1+q_{s_{n}}}{q_{\mathbf{g}^{-1}}}\sum_{\mathbf{w}\in
W\text{: }i-1<\left|\mathbf{w}\right|\leq j\text{,
}s_{n}\nleq\mathbf{w}^{-1}}q_{\mathbf{g}^{-1}}q_{\mathbf{w}}$ $\displaystyle=$
$\displaystyle\frac{1+q_{s_{n}}}{q_{\mathbf{g}}}\sum_{\mathbf{w}\in W\text{:
}i-1<\left|\mathbf{g}^{-1}\mathbf{w}^{-1}\right|\leq j\text{,
}\mathbf{g}\leq\mathbf{w}^{-1}}q_{\mathbf{w}}$ $\displaystyle=$
$\displaystyle\frac{1+q_{s_{n}}}{q_{\mathbf{g}}}\sum_{\mathbf{w}\in W\text{:
}i-1+n<\left|\mathbf{w}\right|\leq j+n\text{,
}\mathbf{g}\leq\mathbf{w}^{-1}}q_{\mathbf{w}}\text{.}$

That implies that the sequence of partial sums of $\sum_{\mathbf{w}\in
W}q_{\mathbf{w}}$ is a Cauchy sequence and hence that the series converges. ∎

###  2.3. States on $\mathfrak{A}(W)$ and $\pi(\mathfrak{A}(W))$

Our solution of the simplicity question for right-angled Hecke C∗-algebras is
inspired by Haagerup’s approach to the unique trace property of group
C∗-algebras in [18]. The translation of the techniques into the deformed
setting requires the study of states on the C∗-algebras $\mathfrak{A}(W)$ and
$\pi(\mathfrak{A}(W))$.

######  Lemma 2.9.

Let $(W,S)$ be a right-angled, finite rank Coxeter system. For every
$\mathbf{u}\in W$ and $0<q<1$ the operator $\mathbf{Q}_{q}^{\mathbf{u}}$ on
$\ell^{2}(W)$ defined by

$\displaystyle\mathbf{Q}_{q}^{\mathbf{u}}:=\sum_{l=\left|\mathbf{u}\right|}^{\infty}\;\sum_{\mathbf{w}\in
W\text{:
}\left|\mathbf{w}\right|=l,\mathbf{u}\leq\mathbf{w}^{-1}}q^{l}P_{\mathbf{w}}$

exists (where the limit is taken with respect to the operator norm) and is
contained in the C∗-algebra
$\mathcal{D}(W,S)\subseteq\mathcal{B}(\ell^{2}(W))$.

###### Proof.

It suffices to show that the sequence
$(\mathbf{Q}_{q,i}^{\mathbf{u}})_{i\geq\left|\mathbf{u}\right|}$ with

$\displaystyle\mathbf{Q}_{q,i}^{\mathbf{u}}:=\sum_{l=\left|\mathbf{u}\right|}^{i}\;\sum_{\mathbf{w}\in
W\text{:
}\left|\mathbf{w}\right|=l,\mathbf{u}\leq\mathbf{w}^{-1}}q^{l}P_{\mathbf{w}}\in\mathcal{D}(W,S)$

is a Cauchy sequence. For $\mathbf{w}\in W$ and $l\in\mathbb{N}$ set
$\kappa_{\mathbf{w}}(l):=\\#\left\\{\mathbf{v}\in
W\mid\mathbf{v}\leq\mathbf{w}\text{ and }\left|\mathbf{v}\right|=l\right\\}$.
It has been shown in [8, Lemma 4.4] that $\kappa_{\mathbf{w}}(l)\leq
Cl^{\\#S-2}$ for some constant $C>0$. Using this in the third line of the
following inequalities, we get that for $i<j$ with
$i\geq\left|\mathbf{u}\right|$ and $\xi\in\ell^{2}(W)$,

$\displaystyle\left\|(\mathbf{Q}_{q,j}^{\mathbf{u}}-\mathbf{Q}_{q,i}^{\mathbf{u}})\xi\right\|_{2}$
$\displaystyle=$ $\displaystyle\left\|\sum_{\mathbf{v}\in
W}\left(\sum_{l=i+1}^{j}\;\sum_{\mathbf{w}\in W\text{:
}\left|\mathbf{w}\right|=l,\mathbf{w}\leq\mathbf{v},\mathbf{u}\leq\mathbf{w}^{-1}}q^{l}\right)\xi(\mathbf{v})\delta_{\mathbf{v}}\right\|_{2}$
$\displaystyle\leq$ $\displaystyle\sqrt{\sum_{\mathbf{v}\in
W}\left(\sum_{l=i+1}^{j}q^{l}\kappa_{\mathbf{v}}(l)\right)^{2}\left|\xi(\mathbf{v})\right|^{2}}$
$\displaystyle\leq$ $\displaystyle
C\left(\sum_{l=i+1}^{j}q^{l}l^{\\#S-2}\right)\left\|\xi\right\|_{2}\text{.}$

For $0<q<1$ the series $\sum_{l=1}^{\infty}q^{l}l^{\\#S-2}$ converges. This
implies the claim. ∎

The following proposition will play a crucial role in Subsection 3 and 4.
Recall that $\overline{\mathcal{R}^{\prime}}$ is the closure of
$\mathcal{R}^{\prime}:=\left\\{(q_{s}^{\epsilon_{s}})_{s\in S}\mid
q\in\mathcal{R}\cap\mathbb{R}_{>0}^{(W,S)}\text{,
}\epsilon\in\left\\{-1,1\right\\}^{(W,S)}\right\\}$ in
$\mathbb{R}_{>0}^{(W,S)}$, where $\mathcal{R}$ denotes the region of
convergence of the growth series of $(W,S)$.

######  Proposition 2.10.

Let $(W,S)$ be a right-angled, irreducible, finite rank Coxeter system,
$q=(q_{s})_{s\in
S}\in\mathbb{R}_{>0}^{(W,S)}\setminus\overline{\mathcal{R}^{\prime}}$ and let
$\mathbf{g}:=s_{1}...s_{n}\in W$ be a path in the Coxeter diagram of $(W,S)$
that covers the whole graph. Then, for every state $\phi$ on $\mathfrak{A}(W)$
there exists a sequence $(\mathbf{w}_{i})_{i\in\mathbb{N}}\subseteq W$ of
group elements with increasing word length,
$\mathbf{g}\leq\mathbf{w}_{i}^{-1}$ for all $i\in\mathbb{N}$ and
$q_{\mathbf{w}_{i}}^{-1}\phi(P_{\mathbf{w}_{i}})\rightarrow 0$.

The same statement holds, if one replaces $\mathfrak{A}(W)$ by
$\pi(\mathfrak{A}(W))$ and $P_{\mathbf{w}_{i}}$ by
$\tilde{P}_{\mathbf{w}_{i}}$.

###### Proof.

The set $\mathbb{R}_{>0}^{(W,S)}\setminus\overline{\mathcal{R}^{\prime}}$ is
open in $\mathbb{R}_{>0}^{(W,S)}$, so there exist positive real numbers
$q^{\prime},\lambda\in\left(0,1\right)$ such that
$q^{\prime}q:=(q^{\prime}q_{s})_{s\in
S}\in\mathbb{R}_{>0}^{(W,S)}\setminus\overline{\mathcal{R}^{\prime}}$ and
$\lambda q^{\prime}q:=(\lambda q^{\prime}q_{s})_{s\in
S}\in\mathbb{R}_{>0}^{(W,S)}\setminus\overline{\mathcal{R}^{\prime}}$. In
particular, Lemma 2.8 implies that the series
$\sum_{\mathbf{g}\leq\mathbf{w}^{-1}}\lambda^{\left|\mathbf{w}\right|}(q^{\prime}q)_{\mathbf{w}}$
diverges. By the root test criterium for convergence,

$\displaystyle\limsup_{l}\left(\sum_{\mathbf{w}\in W\text{:
}\left|\mathbf{w}\right|=l,\mathbf{g}\leq\mathbf{w}^{-1}}\lambda^{l}(q^{\prime}q)_{\mathbf{w}}\right)^{1/l}\geq
1$

and hence

$\displaystyle\limsup_{l}\left(\sum_{\mathbf{w}\in W\text{:
}\left|\mathbf{w}\right|=l,\mathbf{g}\leq\mathbf{w}^{-1}}(q^{\prime}q)_{\mathbf{w}}\right)^{1/l}>1\text{.}$

One can thus find a strictly increasing sequence
$\left(l_{i}\right)_{i\in\mathbb{N}}\subseteq\mathbb{N}$ and a constant $C>1$
such that for all $i\in\mathbb{N}$,

$\displaystyle\left(\sum_{\mathbf{w}\in W\text{:
}\left|\mathbf{w}\right|=l_{i},\mathbf{g}\leq\mathbf{w}^{-1}}(q^{\prime}q)_{\mathbf{w}}\right)^{1/l_{i}}\geq
C\text{.}$ (2.2)

For $\mathbf{w}\in W$ define the set

$\displaystyle C_{\mathbf{w}}:=\left\\{\mathbf{v}\in
W\mid\mathbf{g}\leq\mathbf{v}^{-1}\text{ and
}z_{\mathbf{v}}=z_{\mathbf{w}}\text{ for all }z=(z_{s})_{s\in
S}\in\mathbb{C}^{(W,S)}\right\\}\text{,}$

and choose for every $i\in\mathbb{N}$ an element $\mathbf{w}_{i}\in W$ with
$\\#C_{\mathbf{w}_{i}}(q^{\prime}q)_{\mathbf{w}_{i}}=\max_{\left|\mathbf{v}\right|=l_{i},\mathbf{g}\leq\mathbf{v}^{-1}}\\#C_{\mathbf{v}}(q^{\prime}q)_{\mathbf{v}}$
that has length $l_{i}$ and satisfies $\mathbf{g}\leq\mathbf{w}_{i}^{-1}$.
Since by the definition of $C_{\mathbf{w}_{i}}$ the equality
$\\#C_{\mathbf{v}}(q^{\prime}q)_{\mathbf{v}}=\\#C_{\mathbf{w}_{i}}(q^{\prime}q)_{\mathbf{w}_{i}}$
holds for all $\mathbf{v}\in C_{\mathbf{w}_{i}}$, this element can be chosen
in such a way that $\phi(P_{\mathbf{w}_{i}})\leq\phi(P_{\mathbf{v}})$ for all
$\mathbf{v}\in C_{\mathbf{w}_{i}}$. Now, by picking a suitable subset
$\mathcal{M}\subseteq W$ of elements $\mathbf{w}\in W$ with length $l_{i}$ and
$\mathbf{g}\leq\mathbf{w}^{-1}$, the sum $\sum_{\mathbf{w}\in W\text{:
}\left|\mathbf{w}\right|=l_{i},\mathbf{g}\leq\mathbf{w}^{-1}}(q^{\prime}q)_{\mathbf{w}}$
can be written as
$\sum_{\mathbf{w}\in\mathcal{M}}\\#C_{\mathbf{w}}(q^{\prime}q)_{\mathbf{w}}$.
By the choice of $\mathbf{w}_{i}$ we hence have

$\displaystyle\sum_{\mathbf{w}\in W\text{:
}\left|\mathbf{w}\right|=l_{i},\mathbf{g}\leq\mathbf{w}^{-1}}(q^{\prime}q)_{\mathbf{w}}\leq\left(l_{i}+1\right)^{\left|S\right|}\\#C_{\mathbf{w}_{i}}(q^{\prime}q)_{\mathbf{w}_{i}}$

which implies in combination with (2.2) that for all $i\in\mathbb{N}$,

$\displaystyle\\#C_{\mathbf{w}_{i}}(q^{\prime})^{l_{i}}q_{\mathbf{w}_{i}}\geq\frac{C^{l_{i}}}{\left(l_{i}+1\right)^{\left|S\right|}}\text{.}$
(2.3)

It follows from Lemma 2.9 that the series $\sum_{\mathbf{w}\in W\text{:
}\mathbf{g}\leq\mathbf{w}^{-1}}(q^{\prime})^{\left|\mathbf{w}\right|}\phi(P_{\mathbf{w}})$
converges. By the same argument as above we hence have that

$\displaystyle\limsup_{l}\left(\sum_{\mathbf{w}\in W\text{:
}\left|\mathbf{w}\right|=l,\mathbf{g}\leq\mathbf{w}^{-1}}(q^{\prime})^{l}\phi(P_{\mathbf{w}})\right)^{1/l}<1\text{.}$

One can therefore assume that $\left(\sum_{\mathbf{w}\in W\text{:
}\left|\mathbf{w}\right|=l_{i}\text{,
}\mathbf{g}\leq\mathbf{w}^{-1}}(q^{\prime})^{l_{i}}\phi(P_{\mathbf{w}})\right)^{1/l_{i}}\leq
L$ for all $i\in\mathbb{N}$ where $0<L<1$ . But then, by the choice of
$\mathbf{w}_{i}$,

$\displaystyle\\#C_{\mathbf{w}_{i}}(q^{\prime})^{l_{i}}\phi(P_{\mathbf{w}_{i}})\leq(q^{\prime})^{l_{i}}\sum_{\mathbf{w}\in
C_{\mathbf{w}_{i}}}\phi(P_{\mathbf{w}})\leq\sum_{\mathbf{w}\in W\text{:
}\left|\mathbf{w}\right|=l_{i},\mathbf{g}\leq\mathbf{w}^{-1}}(q^{\prime})^{l_{i}}\phi(P_{\mathbf{w}})\leq
L^{l_{i}}$

and thus with (2.3)

$\displaystyle 0\leq
q_{\mathbf{w}_{i}}^{-1}\phi(P_{\mathbf{w}_{i}})<\frac{L^{l_{i}}}{\\#C_{\mathbf{w}_{i}}(q^{\prime})^{l_{i}}q_{\mathbf{w}_{i}}}\leq\left(l_{i}+1\right)^{\left|S\right|}\left(\frac{L}{C}\right)^{l_{i}}\rightarrow
0\text{.}$

This implies the first part of the statement. The second part is an immediate
consequence, since $\pi(\mathfrak{A}(W))$ is a quotient of $\mathfrak{A}(W)$.
That finishes the proof. ∎

######  Remark 2.11.

The proof of Proposition 2.10 significantly simplifies in the case of single-
parameters $q$. Indeed, if we follow the notation of Proposition 2.10 and
assume that $q_{s}=q_{t}$ for all $s,t\in S$, Lemma 2.9 implies that for
$i\in\mathbb{N}$ and $0<q^{\prime}<1$,

$\displaystyle\sum_{\mathbf{w}\in W\text{:
}\left|\mathbf{w}\right|=i,\mathbf{g}\leq\mathbf{w}^{-1}}(q^{\prime})^{i}\phi(P_{\mathbf{w}})\leq\phi(\mathbf{Q}_{q^{\prime}}^{\mathbf{g}})\text{.}$

One can thus find an element $\mathbf{w}_{i}$ of length $i$ with
$\mathbf{g}\leq\mathbf{w}_{i}^{-1}$ such that
$\phi(P_{\mathbf{w}_{i}})\leq\left(\\#L_{i}^{\mathbf{g}}(q^{\prime})^{i}\right)^{-1}\phi({\mathbf{Q}}_{q^{\prime}}^{\mathbf{g}})$
where $L_{i}^{\mathbf{g}}:=\left\\{\mathbf{w}\in
W\mid\left|\mathbf{w}\right|=i\text{,
}\mathbf{g}\leq\mathbf{w}_{i}^{-1}\right\\}$. We get that

$\displaystyle
q_{\mathbf{w}_{i}}^{-1}\phi(P_{\mathbf{w}_{i}})\leq\frac{\phi(\mathbf{Q}_{q^{\prime}}^{\mathbf{g}})}{\\#L_{i}^{\mathbf{g}}(q^{\prime}q)_{\mathbf{w}_{i}}}\text{.}$

The Cauchy-Hadamard formula (for radii of convergence of power series) implies
that for increasing $i$, if $q^{\prime}$ is close enough to $1$, the
expression on the right approaches $0$.

##  3\. Central projections in Hecke C∗-algebras

In [31] Raum and Skalski, generalizing the single-parameter results by
Garncarek [17], proved that for a right-angled, irreducible, finite rank
Coxeter system $(W,S)$ with at least three generators and $q=(q_{s})_{s\in
S}\in\mathbb{R}_{>0}^{(W,S)}$ the corresponding Hecke-von Neumann algebra
$\mathcal{N}_{q}(W)$ decomposes as $\mathcal{N}_{q}(W)\cong
M\oplus\bigoplus_{\epsilon\in\left\\{-1,1\right\\}^{(W,S)}\text{:
}\left|q_{\epsilon}\right|\in\mathcal{R}^{\prime}}\mathbb{C}$ where
$q_{\epsilon}:=(\epsilon_{s}q_{s}^{\epsilon_{s}})_{s\in
S},\left|q_{\epsilon}\right|:=(q_{s}^{\epsilon_{s}})_{s\in S}$ and where $M$
is a factor. In particular, $\mathcal{N}_{q}(W)$ is a factor if and only if
$q\in\mathbb{R}_{>0}^{(W,S)}\setminus\mathcal{R}^{\prime}$. It is a natural
question whether for $q\in\mathcal{R}^{\prime}$ the central projections in
$\mathcal{N}_{q}(W)$ are already contained in the corresponding Hecke
C∗-algebra $C_{r,q}^{\ast}(W)$. We will prove this by using a Haagerup-type
inequality from [9]. We will further characterize the characters (i.e. unital,
linear, multiplicative functionals) of right-angled Hecke C∗-algebras.

###  3.1. The center of right-angled Hecke C∗-algebras

######  Theorem 3.1 ([9, Theorem 3.4]).

Let $(W,S)$ be a right-angled, finite rank Coxeter system and let
$q=(q_{s})_{s\in S}\in\mathbb{R}_{>0}^{(W,S)}$. Then there exists a constant
$C>0$ such that for every $l\in\mathbb{N}_{\geq 1}$ and $x\in
C_{r,q}^{\ast}(W)$ of the form $x:=\sum_{\mathbf{w}\in W\text{:
}\left|\mathbf{w}\right|=l}c_{\mathbf{w}}T_{\mathbf{w}}^{(q)}$ with
coefficients $c_{\mathbf{w}}\in\mathbb{C}$ we have $\left\|x\right\|\leq
Cl\left\|x\right\|_{2}$.

We will further need the following easy lemma.

######  Lemma 3.2.

Let $(W,S)$ be a finite rank Coxeter system. Then the intersection
$\mathcal{R}\cap\mathbb{R}_{>0}^{(W,S)}$ of the region of convergence
$\mathcal{R}$ of the growth series $W(z)=\sum_{\mathbf{w}\in W}z_{\mathbf{w}}$
with $\mathbb{R}_{>0}^{(W,S)}$ is open in $\mathbb{R}_{>0}^{(W,S)}$.

###### Proof.

Assume that the set $\mathcal{R}\cap\mathbb{R}_{>0}^{(W,S)}$ is not open in
$\mathbb{R}_{>0}^{(W,S)}$ and let $q\in\mathcal{R}\cap\mathbb{R}_{>0}^{(W,S)}$
be a point on its boundary. Since $\sum_{\mathbf{w}\in W}q_{\mathbf{w}}$
converges, the power series $f(z):=\sum_{\mathbf{w}\in
W}q_{\mathbf{w}}z^{\left|\mathbf{w}\right|}$ absolutely converges for all
$z\in\mathbb{C}$ with $\left|z\right|\leq 1$. But the radius of convergence of
$f$ coincides with the distance of the origin to the closest pole of $f$,
hence there exists $\lambda>1$ such that $\sum_{\mathbf{w}\in
W}q_{\mathbf{w}}z^{\left|\mathbf{w}\right|}$ absolutely converges for all
$z\in\mathbb{C}$ with $\left|z\right|<\lambda$. This implies that
$(2^{-1}(1+\lambda)q_{s})_{s\in S}\in\mathcal{R}\cap\mathbb{R}_{>0}^{(W,S)}$
which contradicts the choice of $q$. ∎

######  Proposition 3.3.

Let $(W,S)$ be a right-angled, finite rank Coxeter system and let
$q=(q_{s})_{s\in S}\in\mathbb{R}_{>0}^{(W,S)}$ with $0<q_{s}\leq 1$ for all
$s\in S$. Further, let $\epsilon\in\left\\{-1,1\right\\}^{(W,S)}$,
$q_{\epsilon}:=\left(\epsilon_{s}q_{s}^{\epsilon_{s}}\right)_{s\in S}$ and
assume that
$\left|q_{\epsilon}\right|:=\left(q_{s}^{\epsilon_{s}}\right)_{s\in
S}\in\mathcal{R}$. Then the operator

$\displaystyle
E_{q,\epsilon}=\frac{1}{W(\left|q_{\epsilon}\right|)}\sum_{i=0}^{\infty}\;\sum_{\mathbf{w}\text{:
}\left|\mathbf{w}\right|=i}\left(\sqrt{q}\right)_{\mathbf{w},\epsilon}T_{\mathbf{w}}^{(q)}$

exists (where the limit is taken with respect to the operator norm), it is a
projection in $C_{r,q}^{\ast}(W)$ and it satisfies
$T_{s}^{(q)}E_{q,\epsilon}=E_{q,\epsilon}T_{s}^{(q)}=\epsilon_{s}q_{s}^{\frac{\epsilon_{s}}{2}}E_{q,\epsilon}$
for all $s\in S$. For distinct
$\epsilon,\epsilon^{\prime}\in\left\\{-1,1\right\\}^{(W,S)}$ with
$\left|q_{\epsilon}\right|,\left|q_{\epsilon^{\prime}}\right|\in\mathcal{R}$
the projections $E_{q,\epsilon}$ and $E_{q,\epsilon^{\prime}}$ are orthogonal
to each other.

###### Proof.

By assumption $\left|q_{\epsilon}\right|\in\mathcal{R}$, so Lemma 3.2 implies
that there exists $\lambda>1$ such that still $\left|\lambda
q_{\epsilon}\right|:=\left(\lambda q_{s}^{\epsilon_{s}}\right)_{s\in
S}\in\mathcal{R}$. Using the root test criterium for convergence,

$\displaystyle\limsup_{l}\left(\sum_{\mathbf{w}\in W\text{:
}\left|\mathbf{w}\right|=l}\lambda^{l}\left|q_{\mathbf{w},\epsilon}\right|\right)^{1/l}\leq
1$

and hence

$\displaystyle\limsup_{l}\left(\sum_{\mathbf{w}\in W\text{:
}\left|\mathbf{w}\right|=l}\left|q_{\mathbf{w},\epsilon}\right|\right)^{1/l}<1\text{.}$

One can therefor find $l_{0}\in\mathbb{N}$ and $0<L<1$ such that for all
$l\geq l_{0}$,

$\displaystyle\sum_{\mathbf{w}\in W\text{:
}\left|\mathbf{w}\right|=l}\left|q_{\mathbf{w},\epsilon}\right|<L^{l}\text{.}$

Now set
$E_{q,\epsilon}^{(i)}:=\left(W(\left|q_{\epsilon}\right|)\right)^{-1}\sum_{l=0}^{i}\sum_{\mathbf{w}\text{:
}\left|\mathbf{w}\right|=l}\left(\sqrt{q}\right)_{\mathbf{w},\epsilon}T_{\mathbf{w}}^{(q)}$.
For $i,j\in\mathbb{N}$ with $i<j$ and $i\geq l_{0}$ we have by Theorem 3.1 and
the inequality (3.1) that

$\displaystyle\left\|E_{q,\epsilon}^{(j)}-E_{q,\epsilon}^{(i)}\right\|$
$\displaystyle\leq$
$\displaystyle\frac{1}{W(\left|q_{\epsilon}\right|)}\sum_{l=i+1}^{j}\left\|\sum_{\mathbf{w}\text{:
}\left|\mathbf{w}\right|=l}\left(\sqrt{q}\right)_{\mathbf{w},\epsilon}T_{\mathbf{w}}^{(q)}\right\|$
$\displaystyle\leq$
$\displaystyle\frac{1}{W(\left|q_{\epsilon}\right|)}\sum_{l=i+1}^{j}Cl\sqrt{\sum_{\mathbf{w}\text{:
}\left|\mathbf{w}\right|=l}\left|q_{\mathbf{w},\epsilon}\right|}$
$\displaystyle<$
$\displaystyle\frac{1}{W(\left|q_{\epsilon}\right|)}\sum_{l=i+1}^{j}ClL^{\frac{l}{2}}\text{.}$

The series $\sum_{l=0}^{\infty}lL^{\frac{l}{2}}$ converges, so
$(E_{q,\epsilon}^{(i)})_{i\in\mathbb{N}}\subseteq C_{r,q}^{\ast}(W)$ converges
to $E_{q,\epsilon}\in C_{r,q}^{\ast}(W)$. The remaining statements follow from
short calculations (compare also with [12, Lemma 19.2.5], [17, Theorem 5.3]
and [31, Proposition 2.2]) that we omit here. ∎

######  Remark 3.4.

In [31] Raum and Skalski introduced the notion of _Hecke eigenvectors_ (for
the parameter $q$). These are non-zero elements $\eta\in\ell^{2}(W)$
satisfying $T_{s}^{(q)}\eta\in\mathbb{C}\eta$ for all $s\in S$. The central
projections considered in Proposition 3.3 are exactly the orthogonal
projections onto the Hecke eigenspaces.

The following corollary follows from [31, Theorem A] and Proposition 1.3.

######  Corollary 3.5.

Let $(W,S)$ be a right-angled, finite rank Coxeter system with $\\#S\geq 3$
and let $q\in\mathbb{R}_{>0}^{(W,S)}$. Then the center of the Hecke C∗-algebra
$C_{r,q}^{\ast}(W)$ coincides with the center of the Hecke-von Neumann algebra
$\mathcal{N}_{q}(W)$.

One other immediate consequence is that right-angled Hecke C∗-algebras admit a
decomposition which is analogous to the one of their von Neumann-algebraic
counterparts.

######  Corollary 3.6.

Let $(W,S)$ be a right-angled, finite rank Coxeter system with $\\#S\geq 3$
and let $q=(q_{s})_{s\in S}\in\mathbb{R}_{>0}^{(W,S)}$. Then the corresponding
Hecke C∗-algebra $C_{r,q}^{\ast}(W)$ decomposes as

$\displaystyle
C_{r,q}^{\ast}(W)\cong\pi(C_{r,q}^{\ast}(W))\oplus\bigoplus_{\epsilon\in\left\\{-1,1\right\\}^{(W,S)}\text{:
}\left|q_{\epsilon}\right|\in\mathcal{R}^{\prime}}\mathbb{C},$

where $\pi$ denotes the quotient map of $\mathcal{B}(\ell^{2}(W))$ onto
$\mathcal{B}(\ell^{2}(W))/\mathcal{K}(\ell^{2}(W))$.

###### Proof.

By Proposition 3.3 $C_{r,q}^{\ast}(W)$ decomposes as
$A\oplus\bigoplus_{\epsilon\in\left\\{-1,1\right\\}^{(W,S)}\text{:
}\left|q_{\epsilon}\right|\in\mathcal{R}^{\prime}}\mathbb{C}$ where

$\displaystyle
A=C_{r,q}^{\ast}(W)\prod_{\epsilon\in\left\\{-1,1\right\\}^{(W,S)}\text{:
}\left|q_{\epsilon}\right|\in\mathcal{R}^{\prime}}(1-E_{q,\epsilon})\subseteq\mathcal{B}(\ell^{2}(W)).$

By [31, Theorem A] the von Neumann algebra
$A^{\prime\prime}\subseteq\mathcal{B}(\ell^{2}(W))$ is a factor, necessarily
of type $\text{II}_{1}$, so $A$ contains no compact operators. This implies
that $A\cong\pi(C_{r,q}^{\ast}(W))$ from which the claim follows. ∎

###  3.2. Characters on Hecke C∗-algebras

The operators appearing in Propositon 3.3 are projections onto one-dimensional
subspaces of $\ell^{2}(W)$ and thus induce characters on the right-angled
Hecke C∗-algebras. Let us prove that all characters on right-angled Hecke
C∗-algebras arise in such a manner.

######  Proposition 3.7.

Let $(W,S)$ be a right-angled, irreducible, finite rank Coxeter system and
$q=(q_{s})_{s\in S}\in\mathbb{R}_{>0}^{(W,S)}$. Then the set of characters of
the corresponding Hecke C∗-algebra $C_{r,q}^{\ast}(W)$ is given by

$\displaystyle\\{\chi_{q_{\epsilon}}\mid\epsilon\in\left\\{-1,1\right\\}^{(W,S)}\text{
with }\left|q_{\epsilon}\right|\in\overline{\mathcal{R}^{\prime}}\\}\text{,}$

where $\left|q_{\epsilon}\right|:=(q_{s}^{\epsilon_{s}})_{s\in
S}\in\mathbb{R}_{>0}^{(W,S)}$ and $\chi_{q_{\epsilon}}$ satisfies
$\chi_{q_{\epsilon}}(T_{s}^{(q)}):=\epsilon_{s}q_{s}^{\frac{\epsilon_{s}}{2}}$
for all $s\in S$.

###### Proof.

By Proposition 1.3 we can assume that $0<q_{s}\leq 1$ for all $s\in S$.
Arguing exactly as in the proof of [9, Lemma 5.3] it follows from Proposition
3.3 (or also from [31, Proposition 2.2]) that for every
$\epsilon\in\left\\{-1,1\right\\}^{(W,S)}$ with
$\left|q_{\epsilon}\right|\in\overline{\mathcal{R}^{\prime}}$ the character
$\chi_{q_{\epsilon}}$ exists. Conversely, let $\chi$ be a state on
$\mathfrak{A}(W)$ which restricts to a character on $C_{r,q}^{\ast}(W)$. For
$s\in S$ the Hecke relation $(T_{s}^{(q)})^{2}=1+p_{s}(q)T_{s}^{(q)}$ implies
$(\chi(T_{s}^{(q)}))^{2}-p_{s}(q)\chi(T_{s}^{(q)})=1$ and hence
$\chi(T_{s}^{(q)})\in\\{q_{s}^{\frac{1}{2}},-q_{s}^{-\frac{1}{2}}\\}$. One can
thus find $\epsilon\in\left\\{-1,1\right\\}^{(W,S)}$ with
$\chi=\chi_{q_{\epsilon}}$. Now assume that
$\left|q_{\epsilon}\right|\notin\overline{\mathcal{R}^{\prime}}$, fix $s\in S$
and choose a path $\mathbf{g}:=s_{1}...s_{n}\in W$ in the Coxeter diagram of
$(W,S)$ that covers the whole graph and for which $m_{ss_{1}}=\infty$. By
Proposition 2.10 there exists a sequence
$(\mathbf{w}_{i})_{i\in\mathbb{N}}\subseteq W$ of increasing word length with
$\mathbf{g}\leq\mathbf{w}_{i}^{-1}$ for all $i\in\mathbb{N}$ and
$\left|q_{\mathbf{w}_{i},\epsilon}^{-1}\chi(P_{\mathbf{w}_{i}})\right|\rightarrow
0$. We further have that $P_{\mathbf{w}_{i}s}\leq P_{\mathbf{w}_{i}}$, so
$\chi(P_{\mathbf{w}_{i}s})\leq\chi(P_{\mathbf{w}_{i}})$. Using that
$T_{\mathbf{w}_{i}}^{(q)}$ and $T_{\mathbf{w}_{i}^{-1}}^{(q)}$ lie in the
multiplicative domain of $\chi$ (see for instance [6, Proposition 1.5.7]) in
combination with Proposition 2.2 (as well as Remark 2.3) one concludes

$\displaystyle\left|\chi(T_{s}^{(q)})\right|$ $\displaystyle=$
$\displaystyle\left|q_{\mathbf{w}_{i},\epsilon}^{-1}\right|\left|\chi(T_{\mathbf{w}_{i}}^{(q)}(1-P_{s})T_{s}^{(q)}T_{\mathbf{w}_{i}^{-1}}^{(q)})+\chi(T_{\mathbf{w}_{i}}^{(q)}P_{s}T_{s}^{(q)}T_{\mathbf{w}_{i}^{-1}}^{(q)})\right|$
$\displaystyle\leq$
$\displaystyle\left|q_{\mathbf{w}_{i},\epsilon}^{-1}\chi(T_{\mathbf{w}_{i}}^{(q)}T_{s}^{(1)}T_{\mathbf{w}_{i}^{-1}}^{(1)}P_{\mathbf{w}_{i}s})\right|+\left|q_{\mathbf{w}_{i},\epsilon}^{-1}\chi(P_{\mathbf{w}_{i}s}T_{\mathbf{w}_{i}}^{(1)}T_{s}^{(q)}T_{\mathbf{w}_{i}^{-1}}^{(q)})\right|$
$\displaystyle=$
$\displaystyle\left|q_{\mathbf{w}_{i},\epsilon}^{-1/2}\chi(T_{s\mathbf{w}_{i}^{-1}}^{(1)}P_{\mathbf{w}_{i}s})\right|+\left|q_{s,\epsilon}q_{\mathbf{w}_{i},\epsilon}^{-1/2}\chi(P_{\mathbf{w}_{i}s}T_{\mathbf{w}_{i}}^{(1)})\right|\text{.}$

The Cauchy-Schwarz inequality then implies

$\displaystyle\left|\chi(T_{s}^{(q)})\right|$ $\displaystyle\leq$
$\displaystyle(1+q_{s}^{-1/2})\sqrt{\left|q_{\mathbf{w}_{i},\epsilon}^{-1}\chi\left(P_{\mathbf{w}_{i}s}\right)\right|}\rightarrow
0\text{.}$

This contradicts
$\chi(T_{s}^{(q)})\in\\{q_{s}^{\frac{1}{2}},-q_{s}^{-\frac{1}{2}}\\}$. ∎

##  4\. Simplicity of right-angled Hecke C∗-algebras

In this last section we study the simplicity of right-angled Hecke C∗-algebras
(recall that a C∗-algebra is _simple_ if it contains no non-trivial, two-
sided, closed ideal). Our approach is inspired by [18]. It requires the
following lemma which immediately follows from [27, Lemma 4.2].

######  Lemma 4.1.

Let $(W,S)$ be a Coxeter system, $q=(q_{s})_{s\in
S}\in\mathbb{R}_{>0}^{(W,S)}$ and $\mathbf{w}\in W$. Let further
$\mathbf{w}=s_{1}...s_{n}$ with $s_{1},...,s_{n}\in S$ be a reduced expression
for $\mathbf{w}$. Then,

$\displaystyle\prod_{i=1}^{n}\min\left\\{q_{s_{i}}^{\pm
1}\right\\}\leq(T_{\mathbf{w}}^{(q)})^{\ast}T_{\mathbf{w}}^{(q)}\leq\prod_{i=1}^{n}\max\left\\{q_{s_{i}}^{\pm}\right\\}\text{.}$

Recall that for a finite rank Coxeter system $(W,S)$ and
$q\in\mathbb{R}_{>0}^{(W,S)}\setminus\mathcal{R}^{\prime}$ the Hecke
C∗-algebra $C_{r,q}^{\ast}(W)$ can be viewed as a C∗-subalgebra of
$\pi(\mathfrak{A}(W))$ (see Section 2). We will use this observation
frequently.

######  Proposition 4.2.

Let $(W,S)$ be a right-angled, irreducible, finite rank Coxeter system,
$q=(q_{s})_{s\in
S}\in\mathbb{R}_{>0}^{(W,S)}\setminus\overline{\mathcal{R}^{\prime}}$ with
$0<q_{s}\leq 1$ for all $s\in S$ and let $I\neq C_{r,q}^{\ast}(W)$ be an ideal
where we view $C_{r,q}^{\ast}(W)$ as a C∗-subalgebra of
$\pi(\mathfrak{A}(W))$. Then, for every two elements $s,t\in S$ with
$m_{st}=\infty$ there exists a state $\phi$ on $\pi(\mathfrak{A}(W))$ that
vanishes on $I$ and which satisfies $\phi(P_{s})=1$, $\phi(P_{t})=0$.

###### Proof.

Choose a state on $C_{r,q}^{\ast}(W)$ that vanishes on $I$. We can extend it
to a state $\psi$ on $\pi(\mathfrak{A}(W))$. Further let
$\mathbf{g}:=s_{1}...s_{n}\in W$ with $s_{1}:=s$, $s_{2}:=t$ be a path in the
Coxeter diagram of $(W,S)$ that covers the whole graph and let
$(\mathbf{w}_{i})_{i\in\mathbb{N}}\subseteq W$ be a sequence as in Corollary
2.10, i.e. the $\mathbf{w}_{i}$ have increasing word length,
$\mathbf{g}\leq\mathbf{w}_{i}^{-1}$ for all $i\in\mathbb{N}$ and
$q_{\mathbf{w}_{i}}^{-1}\psi(\tilde{P}_{\mathbf{w}_{i}})\rightarrow 0$. Note
that $\psi(T_{\mathbf{w}_{i}}^{(q)}T_{\mathbf{w}_{i}^{-1}}^{(q)})=0$ is not
possible since then Lemma 4.1 and the Cauchy-Schwarz inequality would imply

$\displaystyle
0=\psi(T_{\mathbf{w}_{i}}^{(q)}T_{\mathbf{w}_{i}^{-1}}^{(q)})\geq
q_{\mathbf{w}_{i}s_{1}}\psi((T_{s_{1}}^{(q)})^{2})\geq
q_{\mathbf{w}_{i}s_{1}}|\psi(T_{s_{1}}^{(q)})|^{2}$

and thus $\psi((T_{s_{1}}^{(q)})^{2})=\psi(T_{s_{1}}^{(q)})=0$. This
contradicts the identity
$(T_{s_{1}}^{(q)})^{2}=1+p_{s}\left(q\right)T_{s_{1}}^{(q)}$. With Proposition
2.2 (as well as Remark 2.3) and Lemma 4.1 we get that for $i\in\mathbb{N}$,

$\displaystyle\left|\frac{\psi(T_{\mathbf{w}_{i}}^{(q)}\tilde{P}_{s}T_{\mathbf{w}_{i}^{-1}}^{(q)})}{\psi(T_{\mathbf{w}_{i}}^{(q)}T_{\mathbf{w}_{i}^{-1}}^{(q)})}-1\right|=\left|\frac{\psi(T_{\mathbf{w}_{i}}^{(q)}(\tilde{P}_{s}-1)T_{\mathbf{w}_{i}^{-1}}^{(q)})}{\psi(T_{\mathbf{w}_{i}}^{(q)}T_{\mathbf{w}_{i}^{-1}}^{(q)})}\right|=\left|\frac{\psi(\tilde{P}_{\mathbf{w}_{i}})}{\psi(T_{\mathbf{w}_{i}}^{(q)}T_{\mathbf{w}_{i}^{-1}}^{(q)})}\right|\leq
q_{\mathbf{w}_{i}}^{-1}\psi(\tilde{P}_{\mathbf{w}_{i}})\rightarrow 0\text{.}$

The weak-$\ast$ compactness of the state space
$\mathcal{S}(\pi(\mathfrak{A}(W)))$ implies that we can find a subsequence of

$\displaystyle\left((\psi(T_{\mathbf{w}_{i}}^{(q)}T_{\mathbf{w}_{i}^{-1}}^{(q)}))^{-1}\psi(T_{\mathbf{w}_{i}}^{(q)}(\cdot)T_{\mathbf{w}_{i}^{-1}}^{(q)})\right)_{i\in\mathbb{N}}\subseteq\mathcal{S}(\pi(\mathfrak{A}(W)))$

that weak-$\ast$ converges to a state $\phi$. By construction, this state
vanishes on the ideal $I$, we have $\phi(\tilde{P}_{s})=1$ and hence also
$\phi(\tilde{P}_{t})=0$ since $0\leq\tilde{P}_{t}\leq 1-\tilde{P}_{s}$. ∎

Recall that the inner action of the group $W$ on $\pi(\mathfrak{A}(W))$
defined by $\mathbf{w}.x:=T_{\mathbf{w}}^{(1)}xT_{\mathbf{w}^{-1}}^{(1)}$ for
$\mathbf{w}\in W$, $x\in\pi(\mathfrak{A}(W))$ induces an action of $W$ on the
state space of $\pi(\mathfrak{A}(W))$ via
$(\mathbf{w}.\phi)(x):=\phi(T_{\mathbf{w}}^{(1)}xT_{\mathbf{w}^{-1}}^{(1)})$
for $\phi\in\mathcal{S}(\pi(\mathfrak{A}(W)))$, $\mathbf{w}\in W$ and
$x\in\pi(\mathfrak{A}(W))$.

We are now ready to characterize the simplicity of right-angled Hecke
C∗-algebras.

######  Theorem 4.3.

Let $(W,S)$ be a right-angled, irreducible Coxeter system and let
$q=(q_{s})_{s\in S}\in\mathbb{R}_{>0}^{(W,S)}$ be a multi-parameter. Then the
following statements hold:

  * •

If $\\#S<\infty$, then the Hecke C∗-algebra $C_{r,q}^{\ast}(W)$ is simple if
and only if
$q\in\mathbb{R}_{>0}^{(W,S)}\setminus\overline{\mathcal{R}^{\prime}}$;

  * •

If $\\#S=\infty$, then the Hecke C∗-algebra $C_{r,q}^{\ast}(W)$ is simple for
all $q\in\mathbb{R}_{>0}^{(W,S)}$.

###### Proof.

Let us first consider the finitely generated case. By Proposition 3.7 the
Hecke C∗-algebra $C_{r,q}^{\ast}(W)$ is not simple for
$q\in\overline{\mathcal{R}^{\prime}}$. For the treatment of the case where
$q\in\mathbb{R}_{>0}^{(W,S)}\setminus\overline{\mathcal{R}^{\prime}}$ by
Proposition 1.3 it suffices to consider multi-parameters with $0<q_{s}\leq 1$
for all $s\in S$. View $C_{r,q}^{\ast}(W)$ as a C∗-subalgebra of
$\pi(\mathfrak{A}(W))$ and assume that $I\neq C_{r,q}^{\ast}(W)$ is an ideal
in $C_{r,q}^{\ast}(W)$. Further choose a closed path
$\mathbf{g}:=s_{1}...s_{n}$ in the Coxeter diagram of $(W,S)$ that covers the
whole graph. Proposition 4.2 implies that we can find a state $\phi$ on
$\pi(\mathfrak{A}(W))$ that vanishes on $I$ and for which
$\phi(\tilde{P}_{s_{1}})=1$, $\phi(\tilde{P}_{s_{n}})=0$ holds. In particular
the projections $\tilde{P}_{s_{1}}$, $\tilde{P}_{s_{n}}$ are contained in the
multiplicative domain of $\phi$ (see for instance [6, Proposition 1.5.7]).

By the identification $\pi(\mathcal{D}(W,S))\cong C(\partial(W,S))$ and the
equality $\phi(\tilde{P}_{s_{1}})=1$ the restriction of $\phi$ to
$\pi(\mathcal{D}(W,S))$ corresponds to a probability measure $\mu$ on the
boundary $\partial(W,S)$ whose support is contained in the set of all
$z\in\partial(W,S)$ with $s_{1}\leq z$. The sequence
$(\mathbf{g}^{i}.\mu)_{i\in\mathbb{N}}$ hence weak-$\ast$ converges to the
point mass $\delta_{\mathbf{g}^{\infty}}\in\text{Prob}(\partial(W,S))$ where
$g^{\infty}:=\lim_{l}\mathbf{g}^{l}\in\partial(W,S)$ and where
$\text{Prob}(\partial(W,S))$ denotes the space of all probability measures on
$\partial(W,S)$ (compare also with the proof of [27, Theorem 3.20]). This
implies that there exists an increasing sequence
$(i_{k})_{k\in\mathbb{N}}\subseteq\mathbb{N}$ for which
$\left(\mathbf{g}^{-i_{k}}.\phi\right)_{k\in\mathbb{N}}$ weak-$\ast$ converges
to a state $\psi$ whose restriction to $\mathcal{D}(W,S)$ is multiplicative.
The product $s_{n}...s_{1}$ also defines a path in the Coxeter diagram of
$(W,S)$. Using Lemma 2.7 and $\phi(\tilde{P}_{s_{n}})=0$ one deduces that for
$a\in I$

$\displaystyle\psi(a)=\lim_{k}\phi(T_{\mathbf{g}^{-i_{k}}}^{(1)}aT_{\mathbf{g}^{i_{k}}}^{(1)})=\lim_{k}\phi(T_{\mathbf{g}^{-i_{k}}}^{(q)}aT_{\mathbf{g}^{i_{k}}}^{(q)})=0\text{,}$

so $\psi$ vanishes on the ideal $I$.

Now, let $J$ be the ideal in $\pi(\mathfrak{A}(W))$ generated by $I$. Since
$\pi(\mathfrak{A}(W))$ identifies with the crossed product C∗-algebra
$C(\partial(W,S))\rtimes_{red}W$, every element in $\pi(\mathfrak{A}(W))$ can
be approximated by finite sums of the form $\sum_{\mathbf{w}\in
W}f_{\mathbf{w}}T_{\mathbf{w}}^{(1)}$ where
$f_{\mathbf{w}}\in\pi(\mathcal{D}(W,S))$. Using
$T_{s}^{(1)}=T_{s}^{(q)}-p_{s}(q)P_{s}$ for $s\in S$, one concludes via
induction that every such operator can be written as a finite sum the form
$\sum_{\mathbf{w}\in W}g_{\mathbf{w}}T_{\mathbf{w}}^{(q)}$ for suitable
$g_{\mathbf{w}}\in\pi(\mathcal{D}(W,S))$. But for all $a\in I$,
$g,h\in\mathcal{D}(W,S)$ and $\mathbf{v},\mathbf{w}\in W$ we have that

$\displaystyle\psi((gT_{\mathbf{w}}^{(q)})a(T_{\mathbf{v}}^{(q)}h))=\psi(g)\psi(T_{\mathbf{w}}^{(q)}aT_{\mathbf{v}}^{(q)})\psi(h)=0$

since $T_{\mathbf{w}}^{(q)}aT_{\mathbf{v}}^{(q)}\in I$, so the state $\psi$
vanishes on $J$. In particular, since $\psi\neq 0$, $J$ can not coincide with
the whole C∗-algebra $\pi(\mathfrak{A}(W))$. But $\pi(\mathfrak{A}(W))$ is
simple by [27, Corollary 4.11], so $J=0$. We get that $C_{r,q}^{\ast}(W)$ must
be simple as well. This completes the finitely generated case.

For infinitely generated Coxeter groups and $q\in\mathbb{R}_{>0}^{(W,S)}$ the
simplicity of $C_{r,q}^{\ast}(W)$ follows from the previous part and Lemma
1.5. ∎

## References

  * [1] A. Bearden, M. Kalantar, _Topological boundaries of unitary representations_ , to appear in Int. Math. Res. Not. (IMRN).
  * [2] J. N. Bernstein, _Le “centre” de Bernstein_ , Representations of reductive groups over a local field, 1–32, Travaux en Cours, Hermann, Paris, 1984.
  * [3] A. Björner, F. Brenti, _Combinatorics of Coxeter groups_ , Graduate Texts in Mathematics, 231. Springer, New York, 2005.
  * [4] N. Bourbaki, _Lie groups and Lie algebras. Chapters 4-6_ , Translated from the 1968 French original by Andrew Pressley, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. xii+300 pp.
  * [5] E. Breuillard, M. Kalantar, M. Kennedy, N. Ozawa, _C ∗-simplicity and the unique trace property for discrete groups_, Publ. Math. Inst. Hautes Études Sci. 126 (2017), 35–71.
  * [6] N. Brown, N. Ozawa, _C ∗-algebras and finite-dimensional approximations_, Graduate Studies in Mathematics, 88. American Mathematical Society, Providence, RI, 2008. xvi+509 pp.
  * [7] P. E. Caprace, J. Lécureux, _Combinatorial and group-theoretic compactifications of buildings_ , Ann. Inst. Fourier (Grenoble) 61 (2011), no. 2, 619–672.
  * [8] M. Caspers, _Absence of Cartan subalgebras for right-angled Hecke von Neumann algebras_ , Anal. PDE 13 (2020), no. 1, 1-28.
  * [9] M. Caspers, M. Klisse, N.S. Larsen, _Graph product Khintchine inequalities and Hecke C ∗-algebras: Haagerup inequalities, (non)simplicity, nuclearity and exactness_, J. Funct. Anal. 280 (2021), no. 1, 108795, 41 pp.
  * [10] M. Caspers, A. Skalski, M. Wasilewski, _On masas in $q$-deformed von Neumann algebras_, Pacific J. Math. 302 (2019), no. 1, 1–21.
  * [11] Y. De Cornulier, _Semisimple Zariski closure of Coxeter groups_ , arXiv preprint arXiv:1211.5635 (2012).
  * [12] M. W. Davis, _The Geometry and Topology of Coxeter groups_ , London Mathematical Society Monographs Series, 32. Princeton University Press, Princeton, NJ, 2008.
  * [13] M. W. Davis, J. Dymara, T. Januszkiewicz, B. Okun,_Weighted $L^{2}$-cohomology of Coxeter groups_, Geom. Topol. 11 (2007), 47–138.
  * [14] K. Dykema, _Free products of hyperfinite von Neumann algebras and free dimension_ , Duke Math. J. 69 (1993), no. 1, 97–119.
  * [15] J. Dymara, _Thin buildings_ , Geom. Topol. 10 (2006), 667–694.
  * [16] G. Fendler, _Simplicity of the reduced C ∗-algebras of certain Coxeter groups_, Illinois J. Math. 47 (2003), no. 3, 883–897.
  * [17] L. Garncarek, _Factoriality of Hecke-von Neumann algebras of right-angled Coxeter groups_ , J. Funct. Anal. 270 (2016), no. 3, 1202–1219.
  * [18] U. Haagerup, _A new look at C ∗-simplicity and the unique trace property of a group_, Operator algebras and applications - the Abel Symposium 2015, 167-176, Abel Symp., 12, Springer, 2017.
  * [19] P. de La Harpe, _On simplicity of reduced C ∗‐algebras of groups_, Bull. London Math. Soc. 39 (2007), no. 1, 1–26.
  * [20] Y. Hartman, M. Kalantar, _Stationary C ∗-dynamical systems_, to appear in the Journal of the European Mathematical Society (JEMS).
  * [21] N. Iwahori, H. Matsumoto, _On some Bruhat decomposition and the structure of Hecke rings of $\mathfrak{p}$-adic Chevalley groups_, Inst. Hautes Études Sci. Publ. Math. No. 25 (1965), 5–48.
  * [22] M. Kalantar, P. Kasprzak, A. Skalski, R. Vergnioux, _Noncommutative Furstenberg boundary_ , to appear in Analysis and PDE.
  * [23] M. Kalantar, M. Kennedy, _Boundaries of reduced C ∗-algebras of discrete groups_, J. Reine Angew. Math. 727 (2017), 247–267.
  * [24] D. Kazhdan, G. Lusztig, _Representations of Coxeter groups and Hecke algebras_ , Invent. Math. 53 (1979), no. 2, 165–184.
  * [25] D. Kazhdan, G. Lusztig, _Proof of the Deligne-Langlands conjecture for Hecke algebras_ , Invent. Math. 87 (1987), no. 1, 153–215.
  * [26] M. Kennedy, _An intrinsic characterization of C ∗-simplicity_, Ann. Sci. Éc. Norm. Supér. (4) 53 (2020), no. 5, 1105-1119.
  * [27] M. Klisse, _Topological boundaries of connected graphs and Coxeter groups_ , arXiv preprint arXiv:2010.03414 (2020).
  * [28] T. Lam, A. Thomas, _Infinite reduced words and the Tits boundary of a Coxeter group_ , Int. Math. Res. Not. IMRN 2015, no. 17, 7690–7733.
  * [29] H. Matsumoto, _Analyse harmonique dans les systèmes de Tits bornologiques de type affine_ , Lecture Notes in Mathematics, Vol. 590, Springer-Verlag, Berlin-New York, 1977. i+219 pp.
  * [30] F. Radulescu, _Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, of noninteger index_ , Invent. Math. 115 (1994), no. 2, 347–389.
  * [31] S. Raum, A. Skalski, _Factorial multiparameter Hecke von Neumann algebras and representations of groups acting on right-angled buildings_ , arXiv preprint arXiv: 2008.00919 (2020).
  * [32] S. Raum, A. Skalski, _K-theory of right-angled Hecke C ∗-algebras_, arXiv preprint arXiv:2103.02962 (2021).
  * [33] B. Rémy, _Buildings and Kac-Moody groups_ , Buildings, finite geometries and groups, 231–250, Springer Proc. Math., 10, Springer, New York, 2012.