Sven Raum and Adam Skalski

††footnotetext:  MSC classification: Primary 46L80, Secondary 19K99, 20C08,
46L09 ††footnotetext:  Keywords: right-angled Coxeter group; Hecke algebra;
K-theory; UCT; amalgamated free product; classification

Abstract. Exploiting the graph product structure and results concerning
amalgamated free products of $\mathrm{C}^{*}$-algebras we provide an explicit
computation of the K-theoretic invariants of right-angled Hecke
$\mathrm{C}^{*}$-algebras, including concrete algebraic representants of a
basis in $\mathrm{K}$-theory. On the way, we show that these Hecke algebras
are KK-equivalent with their undeformed counterparts and satisfy the UCT. Our
results are applied to study the isomorphism problem for Hecke
$\mathrm{C}^{*}$-algebras, highlighting the limits of K-theoretic
classification, both for varying Coxeter type as well as for fixed Coxeter
type.

##  1 Introduction

The notion of Coxeter groups originates in the study of reflection groups and
Weyl groups. One of the most important tools when understanding combinatorial
and representation theoretic questions associated with these groups are Hecke
algebras as studied for example by Kazhdan and Lusztig [kazhdanlusztig79].
While traditionally focused on affine Coxeter types, by the time, important
results on Hecke algebras could also be obtained for arbitrary Coxeter types
[eliaswilliamson14]. Operator algebraic versions of Hecke algebras have been
for the first time studied in [davisdymarajanusykiewiczokun07]. These have
recently experienced a sharp rise in attention. Originally introduced as a
mean to study cohomology of buildings [davis08, Section 11], Garncarek
established them as an interesting class of operator algebras in their own
right, by proving a clear parameter dependence of factoriality for single
parameter Hecke von Neumann algebras of right-angled type. Since then, Hecke
operator algebras have been studied from various angles, informed by typical
questions asked about group operator algebras. This comes as no surprise,
since first of all Hecke operator algebras can be considered deformations of
the reduced group $\mathrm{C}^{*}$-algebra and the group von Neumann algebra,
and second for certain rational deformation parameters, Hecke algebras arise
as corners of group algebras associated with groups acting on buildings, see
for example [raumskalski20, Section 2]. On the other hand a constant source of
inspiration for operator algebraists is the comparison of results known for
reduced group $\mathrm{C}^{*}$-algebras of free groups with possible results
on Hecke $\mathrm{C}^{*}$-algebras. In [capsersskalskiwasilewski19], Caspers,
Skalski and Wasilewski studied MASAs of Hecke von Neumann algebras of free
product type in analogy with questions about the free group factors
[bocaradulescu92]. In [caspers20], Caspers obtained results on the absence of
Cartan subalgebras and strong solidity for Hecke von Neumann algebras of
right-angled type, which is analogue to key questions about the structure of
group von Neumann algebras addressed by Ozawa [ozawa04-solid] and Chifan and
Sinclair [chifansinclair13]. The structure of Hecke von Neumann algebras also
motivated work of Klisse [klisse2020], which offered and exploited a new
perspective on certain boundaries of Coxeter complexes, earlier introduced in
[capracelecureux11] and [lamthomas15]. Hecke $\mathrm{C}^{*}$-algebras have
been put in the spotlight by the work of Caspers, Klisse and Larsen
[caspersklisselarsen19], which obtains a $\mathrm{C}^{*}$-simplicity result
for right-angled Hecke algebras with certain deformation parameter (a complete
characterization of the relevant parameter range was recently obtained in
[klisse2021]). The investigation of $\mathrm{C}^{*}$-simplicity of group
$\mathrm{C}^{*}$-algebras has developed a long tradition since Powers showed
that reduced group $\mathrm{C}^{*}$-algebras of free groups are simple
[powers75]. Recently it has experienced a breakthrough thanks to the work of
Kalantar and Kennedy [kalantarkennedy14-boundaries] and Breuillard, Kalantar,
Kennedy and Ozawa [breuillardkalantarkennedyozawa14]. See also
[raum19-bourbaki] for a survey of more recent developments of the field. It is
no surprise that in the study of Hecke operator algebras, the right-angled
case has received the most attention. This is because a natural description as
graph products directly relates them to amalgamated free products via a
dévissage procedure. Free products and amalgamated free products are of major
importance in the world of operator algebras since the advent of Voiculescu’s
free probability theory [voiculescu85]. Operator algebraic graph products have
recently been introduced by Caspers and Fima [caspersfima17], laying the basis
for applications of work on amalgamated free products to right-angled Hecke
operator algebras. Moreover, the particular case of Hecke operator algebras of
free product type lies directly in the scope of work of Dykema
[dykema93-hyperfinite, dykema99] and Dykema and Rørdam [dykemarordam1998] on
the structure and K-theory of free product $\mathrm{C}^{*}$-algebras and von
Neumann algebras, guiding the way to potential future results on Hecke
operator algebras of general right-angled type and beyond. The present article
investigates the classification of right-angled Hecke
$\mathrm{C}^{*}$-algebras using K-theoretic tools. K-theory of free product
type algebras has been an object of interest for a long time. In
[pimsnervoiculescu82], Pimsner and Voiculescu provided the first calculation
of $\mathrm{K}$-theory of the reduced group $\mathrm{C}^{*}$-algebras of free
groups, and could infer absence of non-trivial projections as well as non-
isomorphism for different finite numbers of generators. Shortly after, Cuntz
established a foundational perspective on $\mathrm{K}$-theory of free product
$\mathrm{C}^{*}$-algebras in general, introducing the notion of
$\mathrm{K}$-amenability [cuntz83-k-amenability] together with a conceptually
simple calculation of the $\mathrm{K}$-theory of universal free products
[cuntz82-internal-structure, cuntz82-free-products]. Germain was first to
establish unconditional results on the K-theory of free product
$\mathrm{C}^{*}$-algebras in [germain96, germain97]. An extension of this work
to amalgamated free products was only recently obtained by Fima and Germain in
[fimagermain15]. The K-theory of Hecke algebras has been mainly considered in
the affine case so far, where Solleveld proved a longstanding conjecture of
Higson and Plymen that describes parameter independence of K-theory
[solleveld18]. For deformed Hecke $\mathrm{C}^{*}$-algebras of non-affine
type, no results are stated in the literature, although the special case of
Coxeter groups of free product type is covered by known results on K-theory
for free product $\mathrm{C}^{*}$-algebras. In the undeformed case the
K-theory groups in the even case (including the right-angled one) have been
computed by Sánchez-Garcia in [sanchezgarcia07]. This was achieved via
computing the Bredon K-homology of the underlying Coxeter group via the Davis
complex, and then appealing to the Baum-Connes conjecture, valid for the class
of groups in question via a combination of the main results of
[bozejkojanuszkiewiczspatzier88] and [higsonkasparov97]. It should be noted
that as often when appealing to the Baum-Connes isomorphism the arguments of
[sanchezgarcia07] do not lead to the explicit identification of generators of
K-theory. The main results of this paper provide an explicit computation of
K-theory in both deformed and undeformed case, including concrete algebraic
representants of K-theory classes, described in Notation 4.1. These are
projections $p_{C}\in\mathrm{C}^{*}_{q}(W)$ associated with a clique $C$ in
the commutation graph of a right-angled Coxeter group $W$.

######  Theorem A (Corollary LABEL:cor:k-theory-hecke-algebras).

Let $(W,S)$ be a right-angled Coxeter system. Denote by $\Gamma$ the
associated commutation graph and by $\mathcal{C}$ the set of all, possibly
empty, cliques of $\Gamma$. Then the following statements hold true.

  * •

The map $C\mapsto[p_{C}]$ induces an isomorphism
$\mathbb{Z}^{\mathcal{C}}\cong\mathrm{K}_{0}(\mathrm{C}^{*}_{\mathbf{q}}(W))$.

  * •

$\mathrm{K}_{1}(\mathrm{C}^{*}_{\mathbf{q}}(W))=0$.

In particular, $\mathrm{K}_{0}(\mathrm{C}^{*}_{q}(W))$ is a free abelian group
whose rank is the number of cliques in the commutation graph of $W$. It is
generated by the projections from the copies
$\mathrm{C}^{*}(\mathbb{Z}/2\mathbb{Z})\subset\mathrm{C}^{*}(W)$ associated
with the Coxeter generators $s\in S$.

Our proof of this result reduces the problem of K-theory computations to the
full group $\mathrm{C}^{*}$-algebra of the Coxeter group in question, by means
of a KK-equivalence, analogue to K-amenability results. Our key ingredients to
establish this KK-equivalence are Fima’s and Germain’s work on amalgamated
free products [fimagermain15], combined with the dévissage procedure for graph
product $\mathrm{C}^{*}$-algebras obtained by Caspers and Fima
[caspersfima17]. The K-theory computations for full group
$\mathrm{C}^{*}$-algebras of right-angled Coxeter groups then make use of six-
term exact sequences established in [fimagermain15]; this in particular gives
an alternative approach to [sanchezgarcia07, Corollary 7.3]. Classifying
$\mathrm{C}^{*}$-algebras by their K-theory invariants has been one of central
topics of the theory of operator algebras for almost fifty years. To wit,
Elliott’s classification programme for nuclear $\mathrm{C}^{*}$-algebras
[elliott76] saw recent structural breakthroughs, see e.g. [gonglinniu15,
tikuisiswhitewinter17, cgstw19]. But even in the early days K-theory proved to
be a useful invariant beyond the nuclear world too. In particular the
aforementioned result of Pimsner and Voiculescu [pimsnervoiculescu82]
distinguished the reduced $\mathrm{C}^{*}$-algebras of different free groups
by computing their $\mathrm{K}$-theory. Generally speaking, non-isomorphism
problems regarding group $\mathrm{C}^{*}$-algebras, even in the amenable
realm, are highly non-trivial, and only recent years brought the first
rigidity results regarding algebras of this type; see [eckhardtraum18] and
references therein. Note that the group $\mathrm{C}^{*}$-algebra of a right-
angled Coxeter group depends only on the associated Coxeter graph, and the
fact that the correspondence between (isomorphism classes of) Coxeter graphs
and (isomorphism classes of) Coxeter groups is bijective in the right-angled
case was established only recently (see [radcliffe03] and references therein),
whereas for general Coxeter groups the bijectivity is known to fail. Theorem A
implies that there is no hope to use K-theory groups in themselves to
distinguish Hecke $\mathrm{C}^{*}$-algebras associated with different right-
angled Coxeter groups, as is shown in Example LABEL:ex:identical-k-theory.
Adding further data, the explicit nature of our K-theory calculations allow us
to identify the trace pairing in K-theory in Proposition LABEL:prop:trace-
pairing, but also this additional information does not suffices to identify
the Coxeter type, as Example LABEL:ex:identical-trace-pairing shows. We point
out that classification results for formally similar structures such as
$\mathrm{C}^{*}$-algebras of right-angled Artin monoids achieved by Eilers, Li
and Ruiz [eilersliruiz16] are based on amenability of groupoids naturally
arising when studying semigroup operator algebras. Similar groupoid models
cannot be expected for operator algebras arising from right-angled Coxeter or
Artin groups. Fixing a Coxeter type one might hope that K-theoretic
information, and in particular the trace pairing suffices to recover the
deformation parameter of a Hecke $\mathrm{C}^{*}$-algebra. The discussion of
K-theoretic invariants for solving this classification problem is the second
main contribution of this article. For the purposes of our paper it is
convenient to consider the following version of the Elliott invariant of a
unital $\mathrm{C}^{*}$-algebra $A$.

$\displaystyle\mathrm{Ell_{unord}}(A)=(\mathrm{K}_{0}(A),\mathrm{K}_{1}(A),[1]_{\mathrm{K}_{0}(A)},\mathrm{T}(A),\rho_{A})\,\text{,}$

where $\mathrm{T}(A)$ denotes the trace simplex of $A$ and $\rho_{A}$ is the
trace pairing of $\mathrm{K}_{0}(A)$ and $\mathrm{T}(A)$. Combining our
K-theory calculations with Dykema’s work on the structure of free product
$\mathrm{C}^{*}$-algebras, which classified in particular their traces, allows
us to determine this unordered Elliott invariant for Hecke
$\mathrm{C}^{*}$-algebras of free product type. Focusing on the single
parameter case, we obtain the following result, which illustrates both the
power and limitations of K-theoretic invariants when it comes to classifying
the algebras in our class.

######  Theorem B (Theorem LABEL:thm:classification).

Suppose $n\in\mathbb{N}_{\geq 3}$ and let $q_{1},q_{2}\in(0,1]$. Then the
following statements hold true.

  * •

If
$\mathrm{C}^{*}_{q_{1}}(\mathbb{Z}/2\mathbb{Z}^{*n})\cong\mathrm{C}^{*}_{q_{2}}(\mathbb{Z}/2\mathbb{Z}^{*n})$
and one of $q_{1}$ or $q_{2}$ lies in either of the sets $(0,\frac{1}{n-1})$,
$\\{\frac{1}{n-1}\\}$, $(\frac{1}{n-1}\,1]$, then so does the other.

  * •

Assume
$\mathrm{C}^{*}_{q_{1}}(\mathbb{Z}/2\mathbb{Z}^{*n})\cong\mathrm{C}^{*}_{q_{2}}(\mathbb{Z}/2\mathbb{Z}^{*n})$
and $q_{1},q_{2}\in(\frac{1}{n-1},1]$.

    * –

If $q_{1}$ or $q_{2}$ is irrational, then $q_{1}=q_{2}$.

    * –

If $q_{1}$ or $q_{2}$ is rational, then so is the other. Further,
$\frac{1}{1+q_{1}}$ and $\frac{1}{1+q_{2}}$ have the same order in
$\mathbb{Q}/\mathbb{Z}$.

    * –

If $q_{1},q_{2}$ are rational and $\frac{1}{1+q_{1}}$, $\frac{1}{1+q_{2}}$
have the same order in $\mathbb{Q}/\mathbb{Z}$, then the unordered Elliott
invariants of $\mathrm{C}^{*}_{q_{1}}(\mathbb{Z}/2\mathbb{Z}^{*n})$ and
$\mathrm{C}^{*}_{q_{2}}(\mathbb{Z}/2\mathbb{Z}^{*n})$ are isomorphic.

  * •

If $q_{1},q_{2}\in(0,\frac{1}{n-1})$, then the unordered Elliott invariants of
$\mathrm{C}^{*}_{q_{1}}(\mathbb{Z}/2\mathbb{Z}^{*n})$ and
$\mathrm{C}^{*}_{q_{2}}(\mathbb{Z}/2\mathbb{Z}^{*n})$ are isomorphic.

It is natural to wonder why the version of the Elliott invariant used above
does not involve the positive part of $\mathrm{K}_{0}$. One can expect that
the order in $\mathrm{K}_{0}$ is determined by the pairing with traces. This
indeed is known to hold in the undeformed case of
$\mathrm{C}^{*}_{\mathrm{red}}((\mathbb{Z}/2\mathbb{Z})^{*n})$, $n\geq 3$, as
follows from the main result of [dykemarordam1998]. That theorem, which as far
as we know is currently the best available statement regarding the
relationship between the order of $\mathrm{K}_{0}$ and traces for free
products, assumes the so-called _Avitzour conditions_. These are known to be
satisfied only in the undeformed case and we have not been able to determine
the order structure of $\mathrm{K}_{0}$ in the general case. The calculation
of ordered K-theory for free products not satisfying the Avitzour conditions
remains an interesting open problem. Finally we should note that it is natural
to expect that the deformed algebras mentioned in Theorem B are non-isomorphic
whenever $q_{1}\neq q_{2}$. The opposite statement would solve the free group
factor conjecture by showing that all the free group factors must be
isomorphic.

### Organisation of the article

After this introduction, in Section 2, we recall basic facts concerning the
graph products of groups and algebras, right-angled Coxeter groups, Hecke
algebras and K-theory of amalgamated free products. In Section 3 we establish
the KK-equivalence result which implies that the K-theory groups of Hecke
$\mathrm{C}^{*}$-algebras do not depend on the deformation parameter. Section
4 contains our first main result, Theorem A. We compute the K-theoretic
invariants of the Hecke-$\mathrm{C}^{*}$-algebras of right-angled type, and
discuss limitations of K-theoretic recovery of the Coxeter type. In Section
LABEL:sec:classification, we study the scope and limitations of K-theoretic
classification of Hecke $\mathrm{C}^{*}$-algebras of free product type in
detail and prove Theorem B.

### Acknowledgements

S.R. was supported by the Swedish Research Council through grant number
2018-04243 and by the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (grant agreement no.
677120-INDEX). A.S. was partially supported by the National Science Center
(NCN) grant no. 2014/14/E/ST1/00525. The major part of our work on this
article was completed while S.R. was visiting IM PAN in autumn 2020. We would
like to thank Piotr Nowak for making this visit possible. We are also grateful
to him for helpful conversations on the structure and K-theory of Hecke
$\mathrm{C}^{*}$-algebras. We thank Julian Kranz for clarifying comments
regarding the UCT results and Klaus Thomsen and Jamie Gabe for pointing out an
imprecision in an earlier version of this article. Finally we are very
grateful to the referee for several useful comments, and in particular for
pointing to us the reference [sanchezgarcia07].

##  2 Preliminaries

In this section we recall basic facts concerning the graph products of groups
and algebras, right-angled Coxeter groups, Hecke algebras and K-theory of
amalgamated free products. We also show how in the case of algebras with free
K-theory one can deduce from six-term exact sequences stability of the UCT
under the amalgamated free products.

######  Notation 2.1.

Multi-indices will be denoted by a bold font throughout this article. We
assume throughout that the $\mathrm{C}^{*}$-algebras we consider are
separable.

###  2.1 K-theory and the UCT for amalgamated free products

For information regarding $\mathrm{K}$-theory and $\mathrm{KK}$-theory we
refer to [blackadar98]. A detailed introduction to reduced and universal
amalgamated free products can be found in [fimagermain15, Section 2]. The
identification of the $\mathrm{K}$-theory groups of amalgamated free products
has been achieved by Fima and Germain in [fimagermain15]. Concretely speaking,
[fimagermain15, Corollary 4.12] implies that if $A_{1},B,A_{2}$ are unital
separable $\mathrm{C}^{*}$-algebras, with unital inclusions $\alpha_{1}:B\to
A_{1}$ and $\alpha_{2}:B\to A_{2}$, then for every separable
$\mathrm{C}^{*}$-algebra $C$ there are the following six-term short exact
sequences in $\mathrm{K}$-theory and $\mathrm{KK}$-theory:

$\textstyle{\mathrm{K}_{0}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{K}_{0}(A_{1})\oplus\mathrm{K}_{0}(A_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{K}_{0}(A_{1}*_{B}A_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{K}_{1}(A_{1}*_{B}A_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{K}_{1}(A_{1})\oplus\mathrm{K}_{1}(A_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{K}_{1}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{\mathrm{KK}^{0}(B,C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{KK}^{0}(A_{1},C)\oplus\mathrm{KK}^{0}(A_{2},C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{KK}^{0}(A_{1}*_{B}A_{2},C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{KK}^{1}(A_{1}*_{B}A_{2},C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{KK}^{1}(A_{1},C)\oplus\mathrm{KK}^{1}(A_{2},C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{KK}^{1}(B,C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Recall from [blackadar98, Chapter 23] that a separable
$\mathrm{C}^{*}$-algebra $A$ satisfies the UCT (the Universal Coefficient
Theorem) if the natural short exact sequence

$\displaystyle
0\to\mathrm{Ext}_{\mathbb{Z}}^{1}(\mathrm{K}_{*}(A),\mathrm{K}_{*}(C))\longrightarrow\mathrm{KK}^{*}(A,C)\stackrel{{\scriptstyle\gamma}}{{\longrightarrow}}\mathop{\mathrm{Hom}}(\mathrm{K}_{*}(A),\mathrm{K}_{*}(C))\to
0$

is exact for every separable $\mathrm{C}^{*}$-algebra $C$. Here the map
$\gamma$ is the dual of the $\mathrm{KK}$-pairing
$\mathrm{K}_{*}(A)\otimes\mathrm{KK}^{*}(A,C)\to\mathrm{K}_{*}(C)$, and as
such is natural in $A$ and $C$. We shall now record a consequence of the
results of [fimagermain15].

######  Proposition 2.2.

Let $A_{1},A_{2},B$ be unital separable $\mathrm{C}^{*}$-algebras with unital
inclusions with conditional expectations $\alpha_{1}:B\to A_{1}$ and
$\alpha_{2}:B\to A_{2}$. If $A_{1},A_{2},B$ satisfy the UCT,then
$A_{1}*_{B}A_{2}$ also satisfies the UCT.

###### Proof.

Under the assumptions above [fimagermain15, Theorem 4.1] says that the
suspension of $A_{1}*_{B}A_{2}$ is KK-equivalent to the mapping cone
determined by the direct sum of the inclusion maps $B\subset A_{1}$ and
$B\subset A_{2}$ (see the proof of Proposition 3.1 for more details). As
mapping cones of the algebras satisfying the UCT also satisfy the UCT, and the
property of satisfying the UCT is stable under taking direct products and also
under passing to suspensions, the result follows.

If in the context of the above proposition one assumes in addition that the
$\mathrm{K}_{*}$-groups of $A_{1}$, $A_{2}$, $B$ and $A_{1}*_{B}A_{2}$ are
free (as will be the case in our application), one can give a direct proof,
using only the six-term exact sequences.

###  2.2 Graph products of groups and algebras

Throughout the paper, by a graph $\Gamma=(\mathrm{V},\mathrm{E})$ we will
understand a finite, and possibly empty, simplicial graph. Formally,
$\mathrm{V}$ denotes a finite set and
$\mathrm{E}\subset\mathrm{V}\times\mathrm{V}$ is a symmetric subset that does
not intersect the diagonal. We will sometimes write $\mathrm{V}(\Gamma)$ for
$V$ and $\mathrm{E}(\Gamma)$ for $E$. We denote the _induced subgraph_ on a
subset $V^{\prime}\subset\mathrm{V}(\Gamma)$ by $\Gamma|_{V^{\prime}}$, so
that
$\mathrm{E}(\Gamma|_{V^{\prime}}):=\mathrm{E}\cap(\mathrm{V}^{\prime}\times\mathrm{V}^{\prime})$.
Given a vertex $v\in\mathrm{V}$, we denote by $\Gamma\setminus v$ the induced
subgraph of $\Gamma$ with vertex set $\mathrm{V}\setminus\\{v\\}$. A _clique_
in a graph $\Gamma$ is a complete subgraph, i.e. a subgraph $\Gamma^{\prime}$
of $\Gamma$ such that
$\mathrm{E}(\Gamma^{\prime})=\Gamma^{\prime}\times\Gamma^{\prime}\setminus\\{(\gamma,\gamma):\gamma\in\Gamma^{\prime}\\}$.
For an element $w\in\mathrm{V}$ we consider the graphs
$\mathrm{Link}(w)=\Gamma|_{\\{v\in\mathrm{V}:(v,w)\in\mathrm{E}\\}}$ and
$\mathrm{Star}(w)=\Gamma|_{\\{v\in\mathrm{V}:(v,w)\in\mathrm{E}\\}\cup\\{w\\}}$.
We will need in Section 4 a simple lemma counting cliques in a given graph.
Recall that the empty set is a clique in every graph.

######  Lemma 2.3.

Denote by $N(\Gamma)$ the number of cliques of arbitrary size in a graph
$\Gamma$. Then for all $v\in\mathrm{V}(\Gamma)$

$\displaystyle N(\Gamma)=N(\mathrm{Link}(v))+N(\Gamma\setminus v)\,\text{.}$

###### Proof.

For a clique $C$ in $\Gamma$, there are precisely two options. Either
$v\in\mathrm{V}(C)$ and $C$ is contained in $\mathrm{Star}(v)$ or
$v\notin\mathrm{V}(C)$ and $C$ is contained in $\Gamma\setminus v$. In the
former case, $C\setminus v$ is a clique in $\mathrm{Link}(v)$, and every
clique in $\mathrm{Link}(v)$ is of this form.

Suppose that $\Gamma=(\mathrm{V},\mathrm{E})$ is a graph and that for every
$v\in\mathrm{V}$ we have an associated group $G_{v}$. Based on this data,
Green introduced in [green-thesis] a construction of the graph product of
groups, leading to a group $G_{\Gamma}$, generated by copies of all $G_{v}$,
$v\in\mathrm{V}$, such that $G_{v}$ commutes with $G_{w}$ if and only if
$(v,w)\in\mathrm{E}$. In [caspersfima17] Caspers and Fima transferred this
construction to the operator algebra world, taking into account various
possible variants. For us the key notions will be the _universal_ graph
product of unital $\mathrm{C}^{*}$-algebras, and the _reduced_ graph product
of unital $\mathrm{C}^{*}$-algebras equipped with faithful states. Thus if for
every $v\in\mathrm{V}$ we are given a unital $\mathrm{C}^{*}$-algebra $A_{v}$
(with a faithful state $\varphi_{v}$), we have the corresponding universal
graph product $\prod_{\Gamma}A_{v}$ (and the reduced graph product
$\prod_{\Gamma}(A_{v},\varphi_{v})$). The constructions are closely related to
the amalgamated free product of $\mathrm{C}^{*}$-algebras, and in particular
[caspersfima17, Section 2] contains the following identifications, inspired by
Serre’s dévissage procedure for groups acting on trees.

######  Theorem 2.4 (See Theorem 2.15 and Remark 2.16 of [caspersfima17]).

Let $\Gamma=(\mathrm{V},E)$ be a graph and $(A_{v})_{v\in\mathrm{V}}$ a family
of unital $\mathrm{C}^{*}$-algebras. Let $w\in\mathrm{V}$.

  * •

There is an isomorphism of $\mathrm{C}^{*}$-algebras
$\prod_{\Gamma}A_{v}\to\left(\prod_{\mathrm{Star}(w)}A_{v}\right)*_{\prod_{\mathrm{Link}(w)}A_{v}}\left(\prod_{\Gamma\setminus
w}A_{v}\right)$, commuting with the natural inclusions of
$(A_{v})_{v\in\mathrm{V}}$ in each side.

  * •

If $\varphi_{v}$ is a faithful state on $A_{v}$ for all $v\in\mathrm{V}$, then
there is an isomorphism of $\mathrm{C}^{*}$-algebras
$\prod_{\Gamma}(A_{v},\varphi_{v})\to\left(\prod_{\mathrm{Star}(w)}(A_{v},\varphi_{v})\right)*_{\prod_{\mathrm{Link}(w)}(A_{v},\varphi_{v})}\left(\prod_{\Gamma\setminus
w}(A_{v},\varphi_{v})\right)$, commuting with the natural inclusions of
$(A_{v})_{v\in\mathrm{V}}$ in each side.

###  2.3 Coxeter systems and Hecke algebras

A good introduction to Coxeter groups can be found for example in
[humphreys90]. Here we will consider only the finitely generated right-angled
case. A _Coxeter matrix of right-angled type_ over a non-empty finite set $S$
is a symmetric $S\times S$ matrix whose diagonal entries equal $1$ and whose
off-diagonal entries lie in the set $\\{2,\infty\\}$. Let $M=(m_{st})_{s,t\in
S}$ be a Coxeter matrix. The _Coxeter system_ $(W,S)$ associated with $M$ is
given by the group

$\displaystyle W=\langle S\mid\forall s,t\in S:\,s^{2}=e\textup{ and
}st=ts\textup{ whenever }m_{st}=2\rangle$

together with its generating set $S$. Given $w\in W$ we denote by $|w|$ its
_word length_ with respect to the generating set $S$. The data contained in
the Coxeter matrix can be equivalently encoded in the _commutation graph_
$\Gamma_{(W,S)}$, with the vertex set $S$ and the edge set $\\{(s,t)\in
S\times S:m_{st}=2\\}$. Note that the commutation graph is the complement of
the standard (unlabelled) _Coxeter graph_ of $(W,S)$. In
[caspersklisselarsen19] the commutation graph was called Coxeter diagram,
which is unfortunately in conflict with the standard terminology employed in
the theory of Coxeter groups. It is easy to see, and was in fact the original
inspiration to study graph products of groups, that $W$ is isomorphic to the
graph product of groups of order $2$; the relevant isomorphism
$W\cong\prod_{\Gamma_{(W,S)}}\mathbb{Z}/2\mathbb{Z}$ preserves generating
sets. The Coxeter matrix and its Coxeter system are called _irreducible_ if
the Coxeter matrix is not the Kronecker tensor product of two non-trivial
Coxeter matrices. Equivalently, there is no partition $S=S_{1}\sqcup S_{2}$
such that $W=\langle S_{1}\rangle\oplus\langle S_{2}\rangle$. Given $W$ as
above we will write $\mathrm{C}^{*}(W)$ for the universal group
$\mathrm{C}^{*}$-algebra of $W$ and $\mathrm{C}^{*}_{r}(W)$ for its reduced
group $\mathrm{C}^{*}$-algebra. The central object of study in this paper is
the family of unital $\mathrm{C}^{*}$-algebras arising as deformations of
$\mathrm{C}^{*}_{r}(W)$, the so-called Hecke $\mathrm{C}^{*}$-algebras of
right-angled Coxeter groups. For the origins of the construction and its
connections to geometry we refer to [davis08] and [raumskalski20, Section
2.4]; many properties of the algebras in the class were recently established
in [caspersklisselarsen19]. Here we will take a direct way of introducing the
Hecke operator algebras as generated by certain unitary representations of
$W$. Fix $(W,S)$ as above and a multiparameter $\mathbf{q}\in(0,1]^{S}$.
Consider $\lambda_{\mathbf{q}}:W\to B(\ell^{2}(W))$, the unitary
representation given by the formulas ($s\in S$, $w\in W$)

(2.1)
$\displaystyle\lambda_{\mathbf{q}}(s)\delta_{w}=\begin{cases}\phantom{-}\frac{1-q_{s}}{1+q_{s}}\delta_{w}+\frac{2q^{1/2}}{1+q_{s}}\delta_{sw}&\text{if
}|sw|>|w|,\\\
-\frac{1-q_{s}}{1+q_{s}}\delta_{w}+\frac{2q^{1/2}}{1+q_{s}}\delta_{sw}&\text{if
}|sw|<|w|\,\text{.}\end{cases}$

We then denote by $\mathrm{C}^{*}_{\mathbf{q}}(W)$ the
$\mathrm{C}^{*}$-algebra generated by $\lambda_{\mathbf{q}}(W)$. Note that for
$\mathbf{q}={\bf{1}}$ the representation $\lambda_{\mathbf{q}}$ is the left
regular representation, so that we have
$\mathrm{C}^{*}_{\textbf{1}}(W)=\mathrm{C}^{*}_{r}(W)$. The representations
$\lambda_{\mathbf{q}}$ can be also viewed as graph product representations,
which was the point of view exploited in [caspers20, caspersklisselarsen19]
and systematically investigated in [raumskalski20, Section 4]. Each of them is
equipped with a canonical trace
$\tau:\mathrm{C}^{*}_{\mathbf{q}}(W)\to\mathbb{C}$, given as the vector state
associated with $\delta_{e}\in\ell^{2}(W)$. Finally if $q\in(0,1]$ we will
write simply $\mathrm{C}^{*}_{q}(W)$ for $\mathrm{C}^{*}_{\mathbf{q}}(W)$ with
$\mathbf{q}=(q,\ldots,q)\in(0,1]^{S}$. The group isomorphism mentioned earlier
and the general properties of the graph products of groups and operator
algebras, [caspersfima17, Example 2.6] yield the natural isomorphism

$\displaystyle\mathrm{C}^{*}(W)\cong\prod_{\Gamma_{(W,S)}}\mathrm{C}^{*}(\mathbb{Z}/2\mathbb{Z}).$

Further by [caspers20, Corollary 3.4], see also [caspersklisselarsen19,
Section 1.10], if we view
$\mathrm{C}^{*}(\mathbb{Z}/2\mathbb{Z})\cong\mathbb{C}\chi_{1}\oplus\mathbb{C}\chi_{-1}$
via the Fourier transform and for each $s\in S$ we denote by
$\varphi_{s}\in\mathcal{S}(\mathrm{C}^{*}(\mathbb{Z}/2\mathbb{Z}))$ the state
given by

$\displaystyle\varphi_{s}(\chi_{1})$
$\displaystyle=\frac{1}{2}\langle(1+\lambda_{\mathbf{q}}(s))\delta_{e},\delta_{e}\rangle=\frac{1}{1+q_{s}},$
$\displaystyle\varphi_{s}(\chi_{-1})$
$\displaystyle=1-\varphi_{s}(\chi_{1})=\frac{q_{s}}{1+q_{s}},$

then we also have the trace preserving isomorphism

$\displaystyle\mathrm{C}^{*}_{\mathbf{q}}(W)\cong\prod_{\Gamma_{(W,S)}}(\mathrm{C}^{*}(\mathbb{Z}/2\mathbb{Z}),\varphi_{s}),$

extending the isomorphism of groups
$W\to\prod_{\Gamma(W,S)}\mathbb{Z}/2\mathbb{Z}$.

##  3 The reduction map is a KK-equivalence

In this section we will prove that the unitary representations
$\lambda_{\mathbf{q}}:W\to\mathrm{C}^{*}_{\mathbf{q}}(W)$ described in Section
2.3 induce a KK-equivalence of the Hecke-$\mathrm{C}^{*}$-algebra with
$\mathrm{C}^{*}(W)$, in analogy with K-amenability results
[cuntz83-k-amenability]. This implies that the K-theory of Hecke
$\mathrm{C}^{*}$-algebras associated to right-angled Coxeter groups does not
depend on the deformation parameter. In particular, it coincides with the
K-theory of the reduced group $\mathrm{C}^{*}$-algebra of the underlying
group. We will derive this result from a KK-equivalence between universal
graph products and reduced graph products of unital $\mathrm{C}^{*}$-algebras.
Once again we refer to [fimagermain15, Section 2] for information on
amalgamated free products of $\mathrm{C}^{*}$-algebras. In the next
proposition we will use the approach to $\mathbf{KK}$-category via the
language of triangulated categories, as developed by Meyer and Nest in
[meyernest06]. We refer the reader to [neeman01] for a treatment of
triangulated categories.

######  Proposition 3.1.

For $i\in\\{1,2\\}$, let $A_{i},B_{i},C_{i}$ be unital separable
$\mathrm{C}^{*}$-algebras with unital inclusions with conditional expectations
$\alpha_{i}:C_{i}\to A_{i}$ and $\beta_{i}:C_{i}\to B_{i}$. If there is a
commutative diagram of unital *-homomorphisms

$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{A}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
C_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{C}}$$\textstyle{B_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{B}}$$\textstyle{A_{2}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
C_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{2}}$

such that $\varphi_{A},\varphi_{B},\varphi_{C}$ are
$\mathrm{K}\mathrm{K}$-equivalences, then the induced *-homomorphism
$\varphi:A_{1}*_{C_{1}}B_{1}\to A_{2}*_{C_{2}}B_{2}$ is a
$\mathrm{K}\mathrm{K}$-equivalence too.

###### Proof.

Throughout the proof we denote by $\mathrm{S}$ the suspension functor for
$\mathrm{C}^{*}$-algebras, realised by tensoring with
$\mathrm{C}_{0}((-1,1))$. Recall that given a unital ∗-homomorphism
$\alpha:A\to B$ between two unital $\mathrm{C}^{*}$-algebras, its cone is
defined as

$\mathrm{cone}(\alpha)=\\{(f,a)\in\mathrm{C}_{0}([0,1),B)\oplus A\mid
f(0)=\alpha(a)\\}.$

We shall consider the Kasparov category $\mathbf{KK}$ whose objects are
$\mathrm{C}^{*}$-algebras, and whose Hom-spaces are KK-groups with the
Kasparov product defining the composition. Slightly extending the category of
objects and adapting the morphisms leads to a triangulated category
$\boldsymbol{\widetilde{KK}}$, as described in [meyernest06, Section 2 and
Appendix A]. The only aspect to take care of when trying to view $\mathbf{KK}$
as a triangulated category, which neccessitates the extension mentioned above,
is that the suspension functor on $\mathbf{KK}$ is not an isomorphism, but
merely an equivalence of categories. As in [meyernest06], we will not make the
formal distinction between $\boldsymbol{\widetilde{KK}}$ and $\mathbf{KK}$. By
[meyernest06, Section 2.1] the mapping cone of a *-homomorphism $\alpha:A\to
B$ gives rise to a distinguished triangle

$\displaystyle\mathrm{S}B\to\mathrm{cone}(\alpha)\to
A\stackrel{{\scriptstyle\psi}}{{\to}}B\,\text{.}$

Let $i\in\\{1,2\\}$ and consider the $\mathrm{C}^{*}$-subalgebra

$\displaystyle D_{i}=\\{f\in\mathrm{S}(A_{i}*_{C_{i}}B_{i})\mid
f((-1,0))\subset A_{i},f((0,1))\subset B_{i},\text{ and }f(0)\in
C_{i}\\}\subset\mathrm{S}(A_{i}*_{C_{i}}B_{i})\,\text{,}$

which is *-isomorphic with the mapping cone of
$\alpha_{i}\oplus\beta_{i}:C_{i}\to A_{i}\oplus B_{i}$, via the map that takes
$(f,c)\in\mathrm{cone}(\alpha_{i}\oplus\beta_{i})$ to the function

$\displaystyle t\mapsto\begin{cases}\pi_{A}(f(-t))&\text{ if }t<0\\\ c&\text{
if }t=0\\\ \pi_{B}(f(t))&\text{ if }t>0\,\text{.}\end{cases}$

Then [fimagermain15, Theorem 4.1] says that the inclusion
$D_{i}\hookrightarrow\mathrm{S}(A_{i}*_{C_{i}}B_{i})$ is a KK-equivalence.
Applying the construction mentioned in the beginning of the proof to the
*-homomorphism $\alpha_{i}\oplus\beta_{i}:C_{i}\to A_{i}\oplus B_{i}$, we
obtain a distinguished triangle

$\displaystyle\mathrm{S}(A_{i}\oplus
B_{i})\to\mathrm{cone}(\alpha_{i}\oplus\beta_{i})\to C_{i}\to A_{i}\oplus
B_{i}\,\text{.}$

By the axiom TR2 of a triangulated category (see e.g. [neeman01, p.29f]) also
the shifted triangle

$\displaystyle\mathrm{S}C_{i}\to\mathrm{S}(A_{i}\oplus
B_{i})\to\mathrm{cone}(\alpha_{i}\oplus\beta_{i})\to C_{i}$

is distinguished. Denoting by
$\varphi_{\mathrm{cone}}:\mathrm{cone}(\alpha_{1}\oplus\beta_{1})\to\mathrm{cone}(\alpha_{2}\oplus\beta_{2})$
the *-homomorphism induced by $\varphi_{A}\oplus\varphi_{B}$, we observe that
the diagram

$\textstyle{\mathrm{S}C_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{S}\varphi_{C}}$$\textstyle{\mathrm{S}(A_{1}\oplus
B_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{S}(\varphi_{A}\oplus\varphi_{B})}$$\textstyle{\mathrm{cone}(\alpha_{1}\oplus\beta_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\mathrm{cone}}}$$\textstyle{C_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{C}}$$\textstyle{\mathrm{S}C_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{S}(A_{2}\oplus
B_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{cone}(\alpha_{2}\oplus\beta_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{2}}$

commutes. Since $\mathrm{S}\varphi_{C}$ and
$\mathrm{S}(\varphi_{A}\oplus\varphi_{B})$ are KK-equivalences, [neeman01,
Proposition 1.1.20] says that $\varphi_{\mathrm{cone}}$ is a KK-equivalence
too. Consider now the commuting square

$\textstyle{\mathrm{cone}(\alpha_{1}\oplus\beta_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\mathrm{cone}}}$$\scriptstyle{\cong_{\mathbf{KK}}}$$\textstyle{\mathrm{S}(A_{1}*_{\mathrm{C}_{1}}B_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{S}\varphi}$$\textstyle{\mathrm{cone}(\alpha_{2}\oplus\beta_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong_{\mathbf{KK}}}$$\textstyle{\mathrm{S}(A_{2}*_{\mathrm{C}_{2}}B_{2})\,\text{.}}$

Since three out of four morphisms are KK-equivalences, it follows that also
$\mathrm{S}\varphi$ is a KK-equivalence, and so is $\varphi$.

We can now combine the last result with the dévissage process for graph
product $\mathrm{C}^{*}$-algebras described in Theorem 2.4 in order to
establish a general theorem on the KK-equivalence of reduced and universal
graph products of $\mathrm{C}^{*}$-algebras. We use Skandalis’ notion of
$\mathrm{K}$-nuclearity [skandalis88].

######  Theorem 3.2.

Let $\Gamma=(\mathrm{V},E)$ be a graph and let
$(A_{v},\varphi_{v})_{v\in\mathrm{V}}$ be a family of unital, separable and
K-nuclear $\mathrm{C}^{*}$-algebras with faithful tracial states. Then the
reduction map $\prod_{\Gamma}A_{v}\to\prod_{\Gamma}(A_{v},\varphi_{v})$ is a
KK-equivalence. If moreover each of the algebras $A_{v}$ satisfies the UCT, so
do the graph products $\prod_{\Gamma}A_{v}$ and
$\prod_{\Gamma}(A_{v},\varphi_{v})$.

###### Proof.

We prove the first part of the theorem by induction on the number of vertices
$|\mathrm{V}|$. When $\mathrm{V}$ is a complete graph, and in particular in
the trivial base case of $|V|=1$, the statement follows from the K-nuclearity
assumption. Indeed, in this case the reduced graph product is just the minimal
tensor product of $\mathrm{C}^{*}$-algebras, and the universal graph product
is the corresponding maximal tensor product. The reduction map is then a KK-
equivalence by the inductive application of [skandalis88, Proposition 3.5
(a)]. Fix then $\Gamma=(\mathrm{V},\mathrm{E})$ and
$(A_{v},\varphi_{v})_{v\in\mathrm{V}}$ as in the statement of the theorem and
assume that the theorem is established for all graphs whose number of vertices
is strictly lower than that of $\mathrm{V}$. By our initial remark, we may
assume that $\Gamma$ is not a complete graph, so that we can choose $w\in V$
such that $\mathrm{Star}(w)$ has fewer vertices than $\mathrm{V}$. By Theorem
2.4, there are natural isomorphisms

$\displaystyle\prod_{\Gamma}A_{v}$
$\displaystyle\cong\left(\prod_{\mathrm{Star}(w)}A_{v}\right)*_{\mathrm{max},\prod_{\mathrm{Link}(w)}A_{v}}\left(\prod_{\Gamma\setminus
w}A_{v}\right)$ $\displaystyle\prod_{\Gamma}(A_{v},\varphi_{v})$
$\displaystyle\cong\left(\prod_{\mathrm{Star}(w)}(A_{v},\varphi_{v})\right)*_{\mathrm{red},\prod_{\mathrm{Link}(w)}(A_{v},\varphi_{v})}\left(\prod_{\Gamma\setminus
w}(A_{v},\varphi_{v})\right)\,\text{.}$

We distinguish explicitly between maximal and reduced amalgamated free
product, since we will also apply the former version to
$\mathrm{C}^{*}$-algebras with a prescribed state. So we obtain a description
of the reduction map $\prod_{\Gamma}A_{v}\to\prod_{\Gamma}(A_{v},\varphi_{v})$
by the following diagram.

$\textstyle{\prod_{\Gamma}A_{v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\prod_{\mathrm{Star}(w)}A_{v}*_{\mathrm{max},\prod_{\mathrm{Link}(w)}A_{v}}\prod_{\Gamma\setminus
w}A_{v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\left(\prod_{\mathrm{Star}(w)}(A_{v},\varphi_{v})\right)*_{\mathrm{max},\prod_{\mathrm{Link}(w)}(A_{v},\varphi_{v})}\left(\prod_{\Gamma\setminus
w}(A_{v},\varphi_{v})\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{\prod_{\Gamma}(A_{v},\varphi_{v})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\left(\prod_{\mathrm{Star}(w)}(A_{v},\varphi_{v})\right)*_{\mathrm{red},\prod_{\mathrm{Link}(w)}(A_{v},\varphi_{v})}\left(\prod_{\Gamma\setminus
w}(A_{v},\varphi_{v})\right)}$

We now use the fact that the traces on $A_{v}$ provide conditional
expectations allowing us to apply Proposition 3.1 and results from
[fimagermain15]. The map $\beta$ is a KK-equivalence by [fimagermain15,
Corollary 4.12]. The map $\alpha$ is a KK-equivalence by Proposition 3.1 and
the $\mathrm{K}$-nuclearity assumption on $A_{w}$. So also the reduction map
of the graph product is a KK-equivalence. The second statement can be shown
along the same lines, using Proposition 2.2 and the fact – observed by
Skandalis in [skandalis88, Section 5] – that minimal (as well as maximal)
tensor products of $\mathrm{C}^{*}$-algebras satisfying the UCT also satisfy
the UCT. Note also that a $\mathrm{C}^{*}$-algebra satisfying the UCT is
neccessarily K-nuclear.

For the formulation of the next corollary, recall the representation
$\lambda_{\mathbf{q}}$ of a right-angled Coxeter group $W$ defined in (2.1)
and denote its canonical extension to a unital ∗-homomorphism from
$\mathrm{C}^{*}(W)$ to $\mathrm{C}^{*}_{\mathbf{q}}(W)$ by the same symbol.

######  Corollary 3.3.

Let $(W,S)$ be a right-angled Coxeter system and
$\mathbf{q}\in\mathbb{R}_{>0}^{S}$. Then
$\lambda_{\mathbf{q}}:\mathrm{C}^{*}(W)\to\mathrm{C}^{*}_{\mathbf{q}}(W)$ is a
KK-equivalence. Moreover the algebra $\mathrm{C}^{*}_{\mathbf{q}}(W)$
satisfies the UCT.

###### Proof.

The identifications introduced in Subsection 2.3 imply that we obtain a
commutative diagram

$\textstyle{\mathrm{C}^{*}(W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\prod_{\Gamma_{(W,S)}}\mathrm{C}^{*}(\mathbb{Z}/2\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{C}^{*}_{q}(W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\prod_{\Gamma_{(W,S)}}(\mathrm{C}^{*}(\mathbb{Z}/2\mathbb{Z}),\varphi_{s})\,\text{.}}$

By Theorem 3.2, the right vertical morphism is a KK-equivalence. So also the
left vertical morphism is a KK-equivalence. The last statement follows from
the above identification and the second part of Theorem 3.2.

One could alternatively deduce the UCT for $\mathrm{C}^{*}(W)$ from the
results of [tu99], as by [bozejkojanuszkiewiczspatzier88] all Coxeter groups
have the Haagerup property.

##  4 Computing K-theoretic invariants of Hecke $\mathrm{C}^{*}$-algebras

In this section we first provide an alternative (to that of [sanchezgarcia07])
method of computing the K-theory for the full group $\mathrm{C}^{*}$-algebra
of a right-angled Coxeter system $(W,S)$ , allowing us to describe explicit
generators in terms of the commutation graph of $(W,S)$. Combined with the KK-
equivalence established in the previous section, this also provides a very
concrete description of the K-theory of the corresponding Hecke
$\mathrm{C}^{*}$-algebra. We then exploit the explicit nature of our
calculations in order to describe the trace pairing. In order to formulate our
$\mathrm{K}$-theory calculations of universal group $\mathrm{C}^{*}$-algebras,
let us introduce the following notation.

######  Notation 4.1.

Let $(W,S)$ be a right-angled Coxeter system with commutation graph $\Gamma$.
Recall from the last part of Section 2.3 the minimal projection
$\chi_{1}\in\mathrm{C}^{*}(\mathbb{Z}/2\mathbb{Z})$ associated with the non-
trivial irreducible representation of $\mathbb{Z}/2\mathbb{Z}$. We associate
with each non-empty clique $C$ of $\Gamma$ the projection

$\displaystyle p_{C}=\otimes_{s\in
C}\chi_{1}^{(s)}\in\prod_{\Gamma}\mathrm{C}^{*}(\mathbb{Z}/2\mathbb{Z})\cong\mathrm{C}^{*}(W)\,\text{,}$

where by $\chi_{1}^{(s)}$ we denote the minimal projection in the copy of
$\mathrm{C}^{*}(\mathbb{Z}/2\mathbb{Z})$ generated by $s\in C\subset S$, and
set $p_{\emptyset}=1_{\mathrm{C}^{*}(W)}$.

######  Theorem 4.2.

Let $(W,S)$ be a right-angled Coxeter system. Denote by $\Gamma$ the
associated commutation graph and by $\mathcal{C}$ the set of all cliques of
$\Gamma$ (including the empty clique). Then the following statements about
$\mathrm{K}$-theory of the universal group $\mathrm{C}^{*}$-algebra hold true.

  * •

The map $C\mapsto[p_{C}]$ induces an isomorphism
$\mathbb{Z}^{\mathcal{C}}\cong\mathrm{K}_{0}(\mathrm{C}^{*}(W))$.

  * •

$\mathrm{K}_{1}(\mathrm{C}^{*}(W))=0$.

In particular, $\mathrm{K}_{0}(\mathrm{C}^{*}(W))$ is a free abelian group
whose rank is the numbers of cliques in the commutation graph of $(W,S)$ and
it is generated by the projections from the copies
$\mathrm{C}^{*}(\mathbb{Z}/2\mathbb{Z})\subset\mathrm{C}^{*}(W)$ associated
with the Coxeter generators $s\in S$.

###### Proof.

We will make use of the natural identification of $\mathrm{C}^{*}(W)$ with the
graph product $\prod_{\Gamma}\mathrm{C}^{*}(\mathbb{Z}/2\mathbb{Z})$ described
in the last part of Section 2.3. In particular, this identification preserves
the inclusion of the copies of $\mathrm{C}^{*}(\mathbb{Z}/2\mathbb{Z})$
associated with the Coxeter generators in $S$. Let us first observe that the
statement of the theorem follows from direct K-theory computations if $W$ is
abelian, and in particular if $W$ has rank 1. Let $(W,S)$ be non-abelian of
rank at least 2 and assume that the result is proven for all right-angled
Coxeter groups whose rank is strictly lower than the rank of $W$. We first
show that $\mathrm{K}_{1}(\mathrm{C}^{*}(W))=0$. Since $W$ is not abelian,
there is $s\in S$ such that $\mathrm{Star}(s)\neq\Gamma$. Then we have an
amalgamated free product decomposition

$\displaystyle\prod_{\Gamma}\mathrm{C}^{*}(\mathbb{Z}/2\mathbb{Z})\to\left(\prod_{\mathrm{Star}(s)}\mathrm{C}^{*}(\mathbb{Z}/2\mathbb{Z})\right)*_{\prod_{\mathrm{Link}(s)}\mathrm{C}^{*}(\mathbb{Z}/2\mathbb{Z})}\left(\prod_{\Gamma\setminus
s}\mathrm{C}^{*}(\mathbb{Z}/2\mathbb{Z})\right)\,\text{,}$

by Theorem 2.4. Note that the isomorphism
$\prod_{\mathrm{Star}(s)}\mathrm{C}^{*}(\mathbb{Z}/2\mathbb{Z})\cong\mathrm{C}^{*}(\mathbb{Z}/2\mathbb{Z})\otimes\prod_{\mathrm{Link}(s)}\mathrm{C}^{*}(\mathbb{Z}/2\mathbb{Z})$
preserves the natural inclusions of
$\prod_{\mathrm{Link}(s)}\mathrm{C}^{*}(\mathbb{Z}/2\mathbb{Z})$. This implies
that the inclusion
$\prod_{\mathrm{Link}(s)}\mathrm{C}^{*}(\mathbb{Z}/2\mathbb{Z})\to\prod_{\mathrm{Star}(s)}\mathrm{C}^{*}(\mathbb{Z}/2\mathbb{Z})$
has a right inverse and thus induces an injection in K-theory. Using the
6-term exact sequence for amalgamated free products from [fimagermain15,
Corollary 4.12], which was recalled in Section 2.1, we obtain the following
6-term exact sequence.