Graph product Khintchine inequalities and Hecke C*-algebras: Haagerup inequalities, (non)simplicity, nuclearity and exactness

Graph products of groups were introduced by Green in her thesis. They have an operator algebraic counterpart introduced and explored by Fima and the first-named author. In this paper we prove Khintchine type inequalities for general C$^{\ast}$-algebraic graph products which generalize results by Ricard and Xu on free products of C$^{\ast}$-algebras. We apply these inequalities in the context of (right-angled) Hecke C$^{\ast}$-algebras, which are deformations of the group algebra of Coxeter groups. For these we deduce a Haagerup inequality. We further use this to study the simplicity and trace uniqueness of (right-angled) Hecke C$^{\ast}$-algebras. Lastly we characterize exactness and nuclearity of general Hecke C$^{\ast}$-algebras.


Introduction
A graph product of groups, first introduced in [31], is a group theoretic construction that generalizes both free products and Cartesian products of groups. It associates to a simplicial graph with discrete groups as vertices a new group by taking the free product of the vertex groups, with added relations depending on the graph. The construction preserves many important group theoretical properties (see [2], [3], [13], [17], [18], [19], [31], [33], [43]) and covers for instance right-angled Coxeter groups and right-angled Artin groups.
In [14] Fima and the first-named author explored the operator algebraic counterpart of graph products and developed the theory of reduced and universal graph products of C * -algebras as well as graph products of von Neumann algebras and quantum groups. The construction generalizes operator algebraic free products and -just as for groups -preserves properties like exactness for reduced graph products of C * -algebras and the Haagerup property for von Neumann algebras. Important examples of graph products are right-angled Hecke algebras (see below), special cases of mixed q-Gaussian algebras [8] as well as several group C * -algebras that have been studied in the literature, see in particular [27] and Section 5.4. In a different direction, C * -algebras associated to graph products of groups with distinguished positive cones were studied in [21].
In this paper we prove a Khintchine inequality for general C * -algebraic graph products. These are inequalities which estimate the operator norm of a reduced operator of a given length with the norm of certain Haagerup tensor products of column and row Hilbert spaces. In the case of free groups and words of length 1 these inequalities go back to Haagerup's fundamental paper [32]. In the case of general free products and arbitrary length a Khintchine type inequality has been proven by Ricard and Xu in [48,Section 2]. In the current paper we obtain a Khintchine inequality for general graph products. We do this by introducing an intertwining technique between graph products and free products.
Theorem 0.1. Let Γ be a finite simplicial graph. Consider a graph product (A, ϕ) = * v,Γ (A v , ϕ v ) of unital C * -algebras with GNS-faithful states ϕ v . Denote by χ d the word length projection of length d. Then for every d ∈ N ≥0 there exists some operator space X d and maps with Dom (π d ) = j d (χ d (A)) such that the following statements hold: (i) X d is a direct sum of Haagerup tensor products of column and row Hilbert spaces and is defined in (2.14); (ii) π d • j d is the identity on χ d (A); (iii) π d : Dom (π d ) → A cb ≤ Cd for some (explicit) constant C, depending only on the graph Γ.
In Theorem 0.1 the bound on π d in property (iii) should be regarded as the Khintchine inequality. Inequalities of this kind have a wide range of applications. We refer in particular to the weak amenability results by Ricard and Xu [48], the early connections to Coxeter groups by Bożejko and Speicher [8] and Nou's result on non-injectivity of q-Gaussians [40]. The second half of this paper gives further applications in the case of (right-angled) Hecke C * -algebras.
Hecke algebras are deformations of group algebras of a Coxeter group W depending on a multiparameter q and have been studied since the 1950s. Their development played an important role in representation theory of algebraic groups (see e.g. [34], [5]). Hecke algebras naturally act on the Hilbert space ℓ 2 (W ) and thus complete to C * -algebras (resp. von Neumann algebras) denoted by C * r,q (W ) (resp. N q (W )) which carry a canonical tracial state. In the case of spherical or affine Coxeter groups these operator algebras have been studied early in [38], [37], [35]. Much later, motivated by the study of weighted L 2 -cohomology of Coxeter groups, the study of general (non-affine) Hecke von Neumann algebras was initiated by Dymara in [25] (see also [22]). Other relevant references are [23], [30], [12] and [15].
As observed in [12], Hecke C * -algebras (resp. Hecke-von Neumann algebras) of right-angled Coxeter systems (W, S) can be realized as graph products of finite-dimensional abelian C * -algebras (resp. von Neumann algebras). Applying Theorem 0.1 we deduce a Haagerup inequality for right-angled Hecke algebras that generalizes Haagerup's inequality for free groups, see [32] and [42,Section 9.6]. These are inequalities that estimate the operator norm of an operator of length d with the L 2 -norm up to some polynomial bound.
Theorem 0.2. Let (W, S) be a right-angled Coxeter group with finite generating set S. Then for every multi-parameter q and x ∈ χ d C * r,q (W ) we have x ≤ Cd x 2 for some (explicit) constant depending only on q and the graph Γ.
The main motivation for the second part of this paper is the main result in [30] where it was shown that single parameter Hecke-von Neumann algebras of irreducible right-angled Coxeter systems (W, S) with |S| ≥ 3 are factors, up to a 1-dimensional direct summand.
Theorem 0.3. [30,Garncarek] Let (W, S) be an irreducible right-angled Coxeter system with |S| ≥ 3. Then the single parameter Hecke-von Neumann algebra N q (W ) is a factor if and only if q ∈ ρ, ρ −1 where ρ is the radius of convergence of the spherical growth series w∈W z |w| of W . Moreover, for q outside this interval, N q (W ) is a direct sum of a factor and C.
We take this theorem as a motivation to study the simplicity and the uniqueness of the tracial state of (right-angled) Hecke C * -algebras. By applying Theorem 0.2 in combination with an averaging argument inspired by [45] we are able to give some partial answers. Note that in the single parameter case q = 1 this has been done in [24] (see also [29] and [20]).
Theorem 0.4. Let (W, S) be an irreducible Coxeter system and q some multi-parameter.
(1) If (W, S) is of spherical or affine type, then C * r,q (W ) is not simple and does not have a unique tracial state for any choice of the parameter q; (2) If (W, S) is of non-affine type and the multi-parameter (q ǫs s ) s∈S with lies in the closure of the region of convergence of the multivariate growth series of W , then C * r,q (W ) is not simple and does not have unique tracial state; (3) If (W, S) is right-angled with |S| ≥ 3, then there exists an open neighbourhood U of the multi-parameter 1 ∈ R S >0 such that C * r,q (W ) is simple for all q ∈ U .
In the case of free products of abelian Coxeter groups we further give a full answer to the simplicity question by making use of Dykema's results [27] on the simplicity of free products of finite dimensional abelian C * -algebras. If the Hecke C * -algebra has a unique trace then these simplicity results imply factoriality of the Hecke-von Neumann algebra, see the Remarks 5.8 and 5.11. We thus extend the range of q for which factoriality of N q (W ) holds. In particular we obtain first results in the multi-parameter case which gives a partial answer to Question 2 in [30]. In the case of free products of abelian Coxeter groups our answer fully settles this question.
In the final part of this paper, we prove that Hecke C * -algebras are exact and characterize their nuclearity in terms of the properties of the underlying group. In particular, we show the following.

Preliminaries and notation
1.1. General notation. We use the notation N := {1, 2, . . .} and N ≥0 := {0, 1, 2, . . .}. By M k,l (C) we denote the k times l matrices and we let M k (C) := M k,k (C). We let Z 2 be the group with two elements. We write δ(P) for the function that is 1 if a statement P is true and which is 0 otherwise. We write B(H) for the bounded operators on a Hilbert space H.
For standard results on von Neumann algebras we refer to [49]. For a tracial von Neumann algebra (M, τ ) with faithful τ we denote the norm induced by the tracial state by The following lemma is standard and shall be used several times. Lemma 1.1. Let A and B be unital C * -algebras with respective faithful traces τ A and τ B . Let A 0 ⊆ A and B 0 ⊆ B be dense * -subalgebras of A and B and let π : A 0 → B 0 be a * -isomorphism such that τ B • π = τ A . Then π extends to a * -isomorphism A → B as well as π τ A (A) ′′ → π τ B (B) ′′ where π τ A and π τ B are the GNS-representations.
Proof. This is essentially [39,Theorem 5.1.4]. Without loss of generality we may assume that A and B are represented on their GNS-spaces L 2 (A, τ A ) and L 2 (B, τ B ) with cyclic vectors Ω A and Ω B . Then U : L 2 (A, τ A ) → L 2 (B, τ B ) : aΩ A → π(a)Ω B , a ∈ A 0 is unitary and π(a) = U aU * . Therefore π extends to a map A → B as well as A ′′ → B ′′ .
We use the symbol ⊗ for the reduced tensor product of C * -algebras (through Takesaki's theorem [39,Section 6.4] also known as the spatial or minimal tensor product) and ⊗ for the tensor product of von Neumann algebras. We denote the Haagerup tensor product of operator spaces by ⊗ h (see below).

1.2.
Column and row Hilbert spaces. For the theory of operator spaces we refer to [28] and [42]. The column Hilbert space of dimension k will be denoted by C k and the row Hilbert space of dimension k will be denoted by R k . C k is the operator space spanned by the matrix units e i,0 , i = 1, . . . , k in M k (C). Its operator space structure is the restriction of the operator space structure of M k (C) as a C * -algebra. Concretely, Similarly, R k is the operator space spanned by matrix units e 0,i , i = 1, . . . , k in M k (C) and it inherits the operator space structure of M k (C). We have, For the Haagerup tensor product of operator spaces we refer to [28,Section 9]. We shall mainly need the completely isometric identifications (see [28,Proposition 3.9.4 and 3.9.5]) for k, l ∈ N, 1.3. Graphs. Let Γ be a simplicial graph with vertex set V Γ and edge set EΓ ⊆ V Γ × V Γ. Simplicial means that Γ has no double edges and that (v, v) / ∈ EΓ for any v ∈ V Γ. We will always assume that Γ is finite and undirected. For v ∈ V Γ we denote by Link (v) the set of all w ∈ V Γ such that (v, w) ∈ EΓ. For X ⊆ V Γ we shall write Link(X) := ∩ v∈X Link(v) for the common link. Further, we define Link(∅) := V Γ (all points). We may view Link(X) as a subgraph of Γ by declaring the edge set of Link(X) to be {(v, w) ∈ Link(X) × Link(X) | (v, w) ∈ EΓ}, so that it inherits precisely the edges of Γ.
A clique in the graph Γ is a subgraph Γ 0 ⊆ Γ in which every two vertices share an edge. We write Cliq or Cliq(Γ) for the set of cliques in Γ and Cliq(l) or Cliq(Γ, l) for the set of cliques with l vertices. We will always assume that the empty graph is in Cliq. Comm (Γ 0 ) is the set of pairs In this paper a word refers to an expression v = v 1 v 2 · · · v n with v i ∈ V Γ, i.e. a concatenation of elements in V Γ which we call the letters. Words will be denoted by bold face. We say that two words are shuffle equivalent (also known as II-equivalent) if they are in the same equivalence class of the equivalence relation generated by We say that two words are equivalent, denoted by the symbol ≃, if they are equivalent through shuffle equivalence and the additional relation: • If v and w are reduced words and v ≃ w then necessarily v and w are shuffle equivalent. For a word w we define its length |w| as the number of letters in the shortest representative of w up to equivalence. We say that a word v starts with 1.4. Graph products of groups. As before let Γ be a simplicial graph. The following construction goes back to Green's thesis [31]. For each v ∈ V Γ let G v be a discrete group. Then the graph product group * v,Γ G v is the discrete group obtained from the free product of G v , v ∈ V Γ by adding additional relations [s, t] = 1 for all s ∈ G v , t ∈ G w with (v, w) ∈ EΓ. Special (extremal) cases of graph products are free products (graph with no edges) and Cartesian products (complete graphs). Right-angled Coxeter and right-angled Artin groups are natural examples of graph products. A right-angled Coxeter group can be seen as a graph product where all groups G v , v ∈ V Γ are equal to Z 2 . A right-angled Artin group is a graph product with G v = Z, v ∈ V Γ. 1.5. Graph products of operator algebras. In [14] the concept of graph products has been translated to the operator algebraic setting. The construction is a generalization of free products by adding commutation relations depending on the underlying graph. We present a slightly different viewpoint of [14] by identifying Hilbert spaces up to shuffle equivalence. This makes the notation much shorter and yields the same construction.
Let Γ be a finite simplicial graph. For v ∈ V Γ let A v be a unital C * -algebra. Let ϕ v be a GNS-faithful state on A v , meaning that the GNS-representation of ϕ v is faithful.
. We have the convention H ∅ := CΩ where Ω is a unit vector called the vacuum vector. If v = v 1 · · · v n and w = w 1 · · · w n are reduced equivalent (hence shuffle equivalent) words then H v ≃ H w naturally by applying flip maps to the vectors dictated by the shuffle equivalence. More precisely, by [14,Lemma 1.3] if v and w are equivalent reduced words, there exists a unique permutation σ of the numbers 1, . . . , n such that v σ(i) = w i and such that if i < j and v i = v j then also σ(i) < σ(j). Then there exists a unitary map Q v,w : H v → H w which sends ξ 1 ⊗ · · · ⊗ ξ n to ξ σ(1) ⊗ · · · ⊗ ξ σ(n) . From now on, we will omit Q v,w in the notation and identify the spaces H v and H w through Q v,w . This significantly simplifies our notation compared to [14] and it is directly verifyable that our constructions below agree with [14].
Let I be a set of representatives of all reduced words modulo shuffle equivalence. Set H = ⊕ v∈I H v . We represent each A v , v ∈ V Γ on H as follows. Take x ∈ A v . Take ξ 1 ⊗ · · · ⊗ ξ d ∈ H w with w = w 1 . . . w d a reduced word. If w does not start with v then we set If w starts with v we may assume w.l.o.g. (by shuffling the letters if necessary and identifying corresponding Hilbert spaces as described above) that w 1 = v and we set This defines a faithful * -representation of A v on H. We set the (reduced) graph product C * -algebra, If for every v ∈ V Γ, A v is a von Neumann algebra and ϕ v is moreover faithful then we define the graph product von Neumann algebra as ( * v,Γ (A v , ϕ v )) ′′ . We will usually write L 2 (A, ϕ) := H, and call this the (graph product) Fock space. The notation is justified as in [14] it is shown that H is the Hilbert space of the standard form of the von Neumann algebraic graph product. An operator a 1 · · · a n with a i ∈ A • v i and v = v 1 · · · v n a reduced word is called a reduced operator of type v. We refer to n as the length of the operator. We define P v as the orthogonal projection of H onto ⊕ v∈Iv H v where I v are representatives of all words that start with v up to shuffle equivalence. For d ∈ N ≥0 , let where a 1 · · · a r is a reduced operator. So χ d is the word length projection of length d.
1.6. Coxeter groups. A Coxeter group W is a group that is freely generated by a set S subject to relations (st) ms,t = 1 for exponents m s,t ∈ {1, 2, . . . , ∞} where m s,s = 1, m s,t ≥ 2 for all s = t and m s,t = m t,s . The condition m s,t = ∞ means that no relation of the form (st) m = 1, m ∈ N is imposed. The pair (W, S) is called a Coxeter system. The data of (W, S) can be encoded in its Coxeter diagram 1 with vertex set S and edge set {(s, t) | m s,t ≥ 3} where every edge between two vertices s, t ∈ S is labeled by the corresponding exponent m s,t . A special subgroup of W is one that is generated by a subset S 0 of S and which will be denoted by W S 0 . Special subgroups are also Coxeter groups with the same exponents as W [22, Theorem 4.1.6]. We briefly write W s = W {s} for s ∈ S and note that by the relation s 2 = e we have W s = Z 2 , the group of two elements. We call the Coxeter system (W, S) irreducible if its Coxeter diagram is connected. This is equivalent to W not having a non-trivial decomposition into a direct product of special subgroups. Irreducible Coxeter groups fall into three classes. Definition 1.2. Let (W, S) be an irreducible Coxeter system. 1 We emphasize that we refer to this as the diagram of the Coxeter group, which is different from the graph defined in Section 1.7.
• It is of spherical type if it is locally finite, i.e. every finitely generated subgroup of W is finite. • It is of affine type if it is finitely generated, infinite and virtually abelian.
• It is of non-affine type if it is neither spherical nor affine.
An irreducible Coxeter group is non-amenable if and only if it is of non-affine type, see [ 1.7. Right-angled Coxeter groups as graph products. A Coxeter group (or Coxeter system) is called right-angled if m s,t ∈ {1, 2, ∞} for all s, t ∈ S. So we have that for s ∈ S, s 2 = e and two letters s, t ∈ S, s = t either commute (case m s,t = 2) or they are free (case m s,t = ∞). Suppose that (W, S) is a right-angled Coxeter system. Set Γ as the graph with vertices V Γ = S and edge set EΓ = {(s, t) | m s,t = 2}. Then we find an isomorphism of groups sending s 1 · · · s n ∈ W, s i ∈ S to s 1 · · · s n ∈ * s,Γ W s (i.e. our notation coincides). Indeed, this follows from the fact that the defining (universal) properties of a right-angled Coxeter group and a graph product over the special subgroups W s ≃ Z 2 are the same.
1.8. Words in Coxeter groups. Let w ∈ W with w = s 1 · · · s n where s 1 , . . . , s n ∈ S. We call this expression reduced if it has minimal length, i.e. n ≤ m for every other representation w = t 1 · · · t m , t 1 , . . . , t m ∈ S. The set of letters that can occur in a reduced expression is independent of the choice of the reduced expression, [22, Proposition 4.1.1]. By |w| := n we define a wordlength on W . If v −1 w = |w| − |v| (resp. wv −1 = |w| − |v|) for v ∈ W we say that w starts (resp. ends) with v. Note that in the right-angled case these definitions agree with the definitions for words and lengths on graphs under the isomorphism (1.2). Coxeter groups satisfy three important conditions on words in S of which we will make use implicitly. We use the usual convention that s means that s is removed from an expression. . Let (W, S) be a Coxeter system, w = s 1 · · · s n an expression for an element w ∈ W and s, t ∈ S. Then the following (equivalent) conditions hold: • Deletion condition: If s 1 · · · s n is not a reduced expression for w, then there exist i < j such that s 1 · · · s i · · · s j · · · s n is also an expression for w. • Exchange condition: If w = s 1 · · · s n is reduced, then either |sw| = n + 1 or there exists 1 ≤ i ≤ n with sw = s 1 · · · s i · · · s n . • Folding condition: If |sw| = |w| + 1 and |wt| = |w| + 1, then either |swt| = |w| + 2 or |swt| = |w|.
In the right-angled case, if we have cancellation of the form s 1 · · · s n = s 1 · · · s i · · · s j · · · s n for s 1 , . . . , s n ∈ S, then s i = s j and s i commutes with s i+1 , . . . , s j−1 .
Let (W, S) be a Coxeter system and let R (W,S) >0 (resp. C (W,S) and {−1, 1} (W,S) ) be the set of tuples q := (q s ) s∈S in R S >0 (resp. in C S and {−1, 1} S ) of positive real numbers (resp. complex numbers and numbers −1 or 1) with the property that q s = q t whenever s and t are conjugate in W .
. For every reduced expression w = s 1 · · · s n of w ∈ W define w | w ∈ W } and such that for every s ∈ S and w ∈ W we have, Denote by ℓ 2 (W ) the Hilbert space of square-summable functions on W with canonical orthonormal basis (δ w ) w∈W . For every s ∈ S the element T (q) s acts boundedly on ℓ 2 (W ) by Moreover, this action extends to a faithful * -representation C q [W ] → B(ℓ 2 (W )). We will usually identify C q [W ] with its image in B(ℓ 2 (W )). The unital * -algebra C q [W ] is called the (Iwahori) Hecke algebra of (W, S) with parameter q. We set the reduced Hecke C * -algebra C * r,q (W ) as the norm closure of C q [W ] in B ℓ 2 (W ) using the representation (1.5). Then set the Hecke-von Neumann algebra are respectively the group algebra, reduced group C * -algebra and group von Neumann algebra of W . Hence Hecke algebras are q-deformations of group (C * -and von Neumann) algebras and the deformation (in principle) depends on q; we comment on this further in Section 4. For every q the vector state x ∈ B(ℓ 2 (W )), restricts to a tracial state τ q on C * r,q (W ) and N q (W ) with τ q (T (q) w ) = 0 for all w ∈ W \ {e}. Finally, define P s , s ∈ S to be the projection of ℓ 2 (W ) onto Note that if the Coxeter group is right-angled then under the graph product identification (1.2) this notation coincides with the notation of Section 1.5.

1.10.
Hecke C * -algebras as graph products. Let (W, S) be a right-angled Coxeter system with graph Γ as defined in Section 1.7. In [12, there exists an isomorphism, Moreover, the isomorphism is given by the canonical map T where v = s 1 · · · s d is reduced. For q = 1 this coincides with the corresponding result on Coxeter groups.
w to get the relations (1.3) and (1.4). This convention is consistent with [12], [15] and [30]. 3 The corollary is only stated for a single parameter qs = q but it also holds in the multi-parameter case with the same proof.

A graph product Khintchine inequality
The aim of this section is to prove a Khintchine inequality for general graph products. Such a Khintchine inequality estimates the operator norm of a reduced operator of length d with the norm of certain Haagerup tensor products of column and row Hilbert spaces. This estimate of norms holds up to a bound that is polynomial in d. We make this more precise in the current section.
Remark 2.1. In the context of free products a Khintchine inequality was already obtained by Ricard and Xu in [48,Section 2]. If in the current paper we would only treat the case of free Coxeter systems and their Hecke deformations (i.e. m(s, t) = ∞, s = t) the results from [48] would be sufficient. Here however, we want a more general theorem. One of the problems that arises while proving such a theorem is that the analogue of [48, Lemma 2.3] in its form fails in a general graph product setting. We remedy this problem by using maps which intertwine graph products with free products.
Let us now prepare for the proof of the main theorem of this section. We fix notation for both a graph product and a free product. As before, let Γ be a finite simplicial graph and let I be a set of representatives of equivalence classes of reduced words with letters in be its graph product with vacuum vector Ω. We also set the free product (i.e. the graph product over Γ with all edges removed) as a C * -subalgebra of A and A f . We let P v be the graph product projection onto words that start with v (as before) and we let P f v be the free product projection onto words which start with v. Let Γ 0 ∈ Cliq(Γ, l) and let w = w 1 · · · w l ∈ I be the word consisting of all letters in V Γ 0 . We write where the latter is the d-fold algebraic tensor product.
Our first aim is to show that reduced operators in A of length d can be decomposed as sums of creation operators, annihilation operators and diagonal operators in a sense to be made precise below. Crucial is that first the annihilation operators act, then the diagonal operators and then the creation operators. That is, we shall be looking for an analogue of the decomposition [ . Then, if possible, we permute the letters of v through shuffle equivalence in the form: where * and ⋄ are the remaining letters and each of the 3 respective terms in between brackets are shuffle equivalent themselves. This means that in between the first brackets there is a word of length k that ends on the clique V Γ 1 , in between the second brackets there is the clique of length l given by V Γ 0 , and at the end there is a word of length d − k − l that starts with the clique V Γ 2 .
Moreover, we want that * · · · * does not end on letters commuting with V Γ 0 and V Γ 1 and ⋄ · · · ⋄ does not start with letters commuting with V Γ 0 and V Γ 2 . This means that the cliques Γ 1 and Γ 2 are maximal for the property that a decomposition like (2.2) exists.
Of course not for every d, l, k, Γ 0 , Γ 1 , Γ 2 and v this permutation exists since v cannot always be written in the from (2.2).
as the permutation of indices 1, . . . , d that satisfies: ( they are the representatives of their equivalence class) and if v i = v j , i < j then σ(i) < σ(j) so that σ comes from a shuffle equivalence. Then σ is unique if it exists.
The permutation σ of Definition 2.3 does not necessarily exist. All expressions below in which a non-existing σ occurs need to be interpreted as 0 and we shall recall this at the relevant places. However, there is no choice for the letter * (e is not allowed as it is in Link(Γ 0 ) and f cannot be moved past ab).
Now we find the following decomposition.
Suppose that we have a non-zero expression of the form Then, for some permutation α of the indices 1, . . . , d coming from a shuffle equivalence and for some 0 ).
Proof. We prove this in a series of claims. To avoid cumbersome notation we shall not write the permutation of the shuffle equivalences in the proof.
Claim 1. The expression (2.3) is up to shuffle equivalence equal to: Moreover, the tail of annihilation operators is maximal in the sense that if for some i ≤ m we have Q Proof of Claim 1. Suppose that we are given an expression as in (2.5). Suppose that for some To do so we may suppose the index i was chosen maximally. Suppose that v i and v i+1 · · · v m do not commute and let v k be the first letter in v i+1 · · · v m that does not commute with v i . Our choice of i yields that Q v k = P v k this would contradict maximality of i. But then (2.5) contains a factor P v i P v k = 0 which means that (2.5) would be zero which in turn is a contradiction.
(2.6) Moreover, the tail of annihilation and diagonal operators is maximal in the sense that if for some Proof of Claim 2. Suppose that we are given a non-zero expression as in (2.6). Suppose that for some i < r we have Q v i = P v i . Then we need to show that v i commutes with v i+1 · · · v r . To do so we may suppose the index i < r was chosen maximally. Suppose that v i and v i+1 · · · v r do not commute and let v k be the first letter in v i+1 · · · v r that does not commute with v i . We claim that our choice of i yields that Q But then (2.6) contains the factor P v i P v k = 0 with v i and v k non-commuting, which means that (2.6) would be zero. As this is a contradiction the claim follows. (2.7) Moreover v r+1 · · · v m forms a clique.
Proof of Claim 3. This is obvious now from Claim 2 and the fact that Proof of the lemma. We may now directly conclude the lemma from Claim 3. The permutation α is then the composition of the shuffle equivalences coming from claims 1, 2 and 3.
Proposition 2.6. Let a 1 · · · a d be a reduced operator of type v = v 1 · · · v d in the graph product (A, ϕ). We have the following equality of operators in B(L 2 (A, ϕ))

8)
where σ (changing over the summation) is as in Definition 2.3. If such σ does not exist then the summand is understood as 0.
Proof. We first note that we may decompose, What we showed in Lemma 2.5 and (2.9) is that the product a 1 · · · a d decomposes as a sum of operators of the form (2.4) (where α depends on the summand). The proof is finished by arguing that the summation in (2.8) runs exactly over all these summands. It is clear that each (non-zero) summand in (2.8) is an expression of the form (2.4) with r = k and l = m − r. Conversely, take an expression of the form (2.4), then the letters v α(r+1) · · · v α(m) form a clique Γ 0 by Lemma 2.5. Set l = #V Γ 0 and k = r. Now in v α(1) , . . . , v α(r) there may be letters at the end that commute with Γ 0 . Let Γ 1 be the clique of letters that appear at the end of v α(1) , . . . , v α(r) that commute with Γ 0 that is maximal in the following sense: there are no letters s appearing at the end of v α(1) · · · v α(r) that commute with Γ 0 and Γ 1 . Such a clique is unique since if both Γ 1 and Γ ′ 1 would be such cliques, then so is Γ 1 ∪ Γ ′ 1 and hence Γ 1 = Γ ′ 1 by maximality. Similarly we may let Γ 2 be a clique of letters that appear at the start of v α(m+1) · · · v α(d) that commute with Γ 0 and that is maximal. Then for this choice of Γ 0 , Γ 1 , Γ 2 we have that σ satisfying (1) -(6) exists and it is moreover the only choice for which it exists. This shows that each non-zero expression (2.4) occurs exactly once in the summation (2.8).
In order to prove our Khintchine inequality we introduce the necessary notation. We define as in [48, Section 2] the following subspaces of B(L 2 (A f , ϕ f )), completely isometrically, where the subscript C (resp. R) denotes the column (resp. row) Hilbert space structure. In particular, if each A • v is one dimensional (as is the case for right-angled Hecke algebras) we have that L 1 (resp. K 1 ) is completely isometrically isomorphic to the column Hilbert space C #V Γ (resp. row Hilbert space R #V Γ ). We set k-fold Haagerup tensor products, For a 1 , . . . , a l with a i ∈ A v i and v = v 1 · · · v l a reduced word (for the graph product) in I consisting precisely of all letters of Γ 0 we define an element of B(L 2 (A f , ϕ f )) by setting for r > l, r is a reduced word in the free product). If r < l then the image in (2.11) is 0.
Lemma 2.7. The operator defined in (2.11) is bounded.
Proof. Let r ∈ N ≥0 . Fix a word w = w 1 · · · w r with w i = w i+1 for all 1 ≤ i < r. Then, is an invariant subspace for the action (2.11). Moreover, note that (2.11) is for r ≥ l just the tensor product operator P w 1 a 1 P w 1 ⊗ · · · ⊗ P w l a l P w l ⊗ 1 ⊗r−l , where P w i a i P w i acts on L 2 (A • w i , ϕ w i ). Clearly this operator is bounded.

Set the diagonal space
to be the linear span of all operators of the form (2.11). A Γ 0 inherits the operator space structure of A f . Set for d ∈ N ≥0 , (2.14) In case A • v , v ∈ V Γ are all 1-dimensional, the space X d can also be understood in terms of bounded operators on a Hilbert space. Indeed, the remarks after (2.10) and [6,Proposition 3.5] give the first two completely isometric isomorphisms of: The third completely isometric isomorphism of (2.15) holds by definition of the operator space structure on A Γ 0 as part of the C * -algebra A f . Next consider the following embedding where the image of a 1 ⊗ · · · ⊗ a d is given as follows: Consider a summand of X d indexed by . Then the restriction of the image of j d to this summand is given by with σ given by Definition 2.3; if such σ is non-existent then the image of j d (a 1 ⊗ · · · ⊗ a d ) in the summand of X d corresponding to (l, k, be the direct sums of product maps. For a 5-tuple (l, k, Γ 0 , Γ 1 , Γ 2 ) as above, let be the product map π f d restricted to the corresponding summand of X d . This map is completely bounded as follows from the definition of the Haagerup tensor product. Consequently, π f d : X d → B(L 2 (A f , ϕ f )) is completely bounded by the number of summands of X d , i.e. π f d cb ≤ (#Cliq(Γ)) 3 d.
• We define the partial isometry, (2.19) as follows. Consider a reduced operator b 1 · · · b n ∈ A of type w. We need to define a permutation σ R = σ w R,l,k,Γ 0 ,Γ 1 coming from a shuffle equivalence satisfying (1) -(4) of Definition 2.3. Moreover we assume that this σ R is chosen such that each of the expressions w σ R (1) · · · w σ R (k) , w σ R (k+1) · · · w σ R (k+l) and w σ R (k+l+1) · · · w σ R (n) are in I. If σ R exists then it is unique and we set The maps Q l,k,Γ 0 ,Γ 1 ,Γ 2 and R l,k,Γ 0 ,Γ 1 preserve orthogonality and inner products and are therefore partial isometries.
where σ is defined as in (1) -(6) of Definition 2.3 and the right hand side should be understood as 0 otherwise.
Proof. Note that both sides of (2.20) equal 0 if a σ as in the statement of the proposition does not exist, c.f. the definition of j d . So from now on we assume that σ exists and that the right hand side of (2.20) is non-zero for some elementary tensor product a 1 ⊗ · · · ⊗ a d ∈ Σ d of type v 1 · · · v d . We argue first that without loss of generality we may assume that the permutation σ on the right hand side of (2.20) is trivial. Indeed, if σ is non-trivial then we may work with the element x ′ = a σ(1) ⊗ · · · ⊗ a σ(d) ∈ Σ d instead of x. Then note that the left hand side of (2.20) for x and x ′ are the same. Similarly the right hand side of (2.20) is the same for x and x ′ .
So assume that σ is trivial. Now take a reduced operator b 1 · · · b r , r ≥ 0 of type w = w 1 · · · w r . We first prove the proposition for the case that the permutation σ Q := σ w Q,l,d−l−k,Γ 0 ,Γ 2 ,Γ 1 exists. In that case set, where (⋄ · · · ⋄) has no letters in V Γ 1 at the start. We have by definition of π f d,l,k,Γ 0 ,Γ 1 ,Γ 2 , Then, for the left hand side of (2.20), Now, for the right hand side of (2.20) we consider an expression, The assumption that σ is trivial yields that v k+l+1 · · · v d starts with V Γ 2 , that the letters v k , . . ., v k+l exhaust V Γ 0 and that the letters at the end of v 1 · · · v k that commute with Γ 0 are precisely given by V Γ 1 . If (2.22) is non-zero then let us argue that there exists a word w ′ 1 · · · w ′ r as defined above. Indeed, if (2.22) is non-zero, then we may shuffle b 1 · · · b r into an operator · · · w ′ r does not have a letter of V Γ 1 up front (because if that happens then applying P ⊥ v i , i ≤ k will give zero). So we conclude that (2.22) can only be non-zero if there exists w ′ 1 · · · w ′ r as defined above, in which case If one of the terms with a • is zero, then also this term was zero in (2.21) and the proposition is proved. If none of these terms are zero, then the image of (2.23) under R l,k,Γ 0 ,Γ 1 equals (2.21) and (2.22) is in ker(R l,k,Γ 0 ,Γ 1 ) ⊥ . This concludes the proposition in case σ Q exists.
If σ Q does not exist, then On the other hand we already noted that (2.23) can only be nonzero if a permutation σ Q exists. So if σ Q is non-existent then also (2.23) is zero, yielding the proposition.
Set the product map We define the map for d ∈ N ≥0 , (2.24) so that by definition π d • j d = ρ d . Now the crucial part is to show that the map π d is well-defined and completely bounded with linear bound in d. This is where we use the announced intertwining argument between graph products and free products. Now we are ready for the main theorem of this section. Recall that the word length projection χ d was defined in (1.1). Theorem 2.9 (Graph product Khintchine inequality). Let Γ be a finite simplicial graph and consider a graph product (A, ϕ) = * v,Γ (A v , ϕ v ) of unital C * -algebras A v with GNS-faithful states ϕ v . Then for every d ∈ N ≥0 there exist maps with Dom(π d ) = j d (χ d (A)) and where X d is defined in (2.14) and (2.15) such that: Proof. Part (i) follows from Proposition 2.6. From Proposition 2.8 we see that on the domain j d (χ d (A)) the map π d is given by the direct sum of the maps In particular, π d is well-defined. As each of these summands is completely contractive and there are at most (#Cliq(Γ)) 3 d summands, we see that π d is completely bounded with the desired complete bound.

Khintchine and Haagerup inequalities for right-angled Hecke C * -algebras
In this section we make the Khintchine inequality from Section 2 explicit in the case of Hecke algebras. As a consequence we derive a Haagerup inequality for right-angled Coxeter groups and their Hecke deformations. Such a Haagerup inequality shows that the L ∞ -norm of an operator of length d can be estimated with the L 2 -norm up to a polynomial bound Q(d). It is a generalisation of Haagerup's inequality for free groups F n , see [32] or also [42,Section 9.6]. It entails that there exists a constant C such that for every x ∈ C[F n ] supported on group elements of length d we have x ≤ Cd x 2 . In particular, here Q(d) = d so that we have a linear estimate in the length d. Haagerup and Khintchine inequalities have found a wide range of applications in operator theory. We will give further applications to C * -simplicity problems in Section 5.3.
Let (W, S) be a finitely generated right-angled Coxeter system and let q = (q s ) s∈S ∈ R (W,S) >0 be a multi-parameter set. Let Γ be the graph associated to (W, S) defined in Section 1.7. From (1.7) we see that we have a canonical isomorphism (C * r,q (W ), τ q ) ≃ * s,Γ (C * r,qs (W s ), τ qs ). Furthermore, we may specialize Section 2 to * s,Γ (C * r,qs (W s ), τ qs ). In particular this defines the space of diagonal operators A Γ 0 as in (2.13) and the operator space X d of (2.14). We first observe that A Γ 0 simplifies. Recall that P s , s ∈ S and P f Γ 0 , Γ 0 ∈ Cliq(Γ) were defined in (1.6) and (2.1).
Lemma 3.1. For a right-angled Coxeter group (W, S) we have A Γ 0 = CP f Γ 0 for any Γ 0 ∈ Cliq(Γ, l). Moreover, for v = s 1 · · · s l , s i ∈ S a reduced word in I with {s 1 , . . . , s l } = V Γ 0 we have, Proof. For s ∈ S we have that P s T (q) s P s = p s (q)P s and C * r,q (W s ) is a two dimensional C * -algebra spanned by the identity and T (q) s (see the comments after [12,Corollary 3.3]). Now let v = s 1 · · · s l be as in the lemma. Let A s = C * r,qs (W s ). For operators a i ∈ A s i , 0 ≤ i ≤ l, we then have that P s i a i P s i is a scalar multiple of P s i so that Diag(a 1 , . . . , a l ) is a scalar multiple of P f Γ 0 , see (2.12). If a i = T (q) s the scalar multiple is s∈V Γ 0 p s (q). Lemma 3.1 shows that we may identify A Γ 0 ≃ C completely isometrically. Then in (2.15) . So for a right-angled Coxeter system we get by Lemma 3.1, (2.14) and (2.15), We equip M k,d−k−l (C) with the inner product where Tr d−k−l is the non-normalized trace that takes the value 1 on rank 1 projections. We further equip X d with the direct sum of these inner products. For x ∈ X d we let Tr . Then, as for any finite dimensional type I von Neumann algebra, we have x ≤ x 2,Tr . (3.2) By Theorem 2.9 we obtain maps r,q (W ))) such that π d • j d is the identity on C * r,q (W ) and π d : Dom(π d ) → X d cb ≤ #Cliq(Γ) 3 d.
We now have the following orthogonality lemma.
Let v, w ∈ W be reduced of length d. If the permutation σ v given in Definition 2.3 exists we have If such σ v does not exist then p l,k,Γ 0 ,Γ 1 ,Γ 2 j d (T (q) v ) = 0. Proof. The final claim of the statement follows from the definition of j d . So it remains to prove (3.3) and we assume that σ v as in Definition 2.3 exists. Since the right hand side of (3.3) has the term δ(v ≃ w) we may assume that also σ w exists. Moreover, by shuffling the letters of v and w if necessary, which does not change the operators T (q) v and T (q) w , we may assume that σ v and σ w are the identity permutation.
For s ∈ S set e s = P f s T (q) s P f,⊥ s and e ′ s = P f,⊥ s T (q) s P f s . These form an orthonormal basis of the respective column Hilbert space L 1 and row Hilbert space K 1 . Now write a reduced expression v = s 1 · · · s d , s i ∈ S. By assumption that σ v was trivial we see that s k+1 , . . . , s k+l commute and form a clique Γ 0 in Γ. From Lemma 3.1 we find that Diag P s k+1 T (q) s k+1 P s k+1 , . . . , P s k+l T (q) s k+l P s k+l = ( It now follows from the definition of j d that From this we can directly conclude (3.3).
Theorem 3.3 (Khintchine inequality for right-angled Hecke C * -algebras). Let (W, S) be a rightangled Coxeter system with finite generating set S and graph Γ.
be a multi-parameter. Then for every d ∈ N ≥0 there exist maps Proof. Statements (i) and (ii) are immediate from (1.7) and Theorem 2.9. We thus prove (iii).
This completes the proof. be a multi-parameter. Then for each d ∈ N ≥0 and x ∈ χ d (C * r,q (W )) we have Proof. By respectively Theorem 3.3 (i), (3.2) and Theorem 3.3 (ii), (iii), we get for every x ∈ χ d (C * r,q (W )), This completes the proof.

On isomorphisms of Hecke algebras and their C * -and von Neumann algebras
In this section we discuss isomorphism properties of Hecke algebras and in particular their dependence on q. These properties are well understood for finite Coxeter systems, which we summarize as follows.
Remark 4.1. By Tits's deformation theorem, the Hecke algebras C q [W ] of a given finite Coxeter system (W, S) are pairwise isomorphic to each other for different q ∈ R (W,S) >0 , see [11,Proposition 10.11.2]. The argument is based on the classification of finite-dimensional semi-simple algebras and the isomorphism is not explicit.
To the knowledge of the authors there is no general statement for arbitrary Coxeter systems (W, S) about the dependence on q ∈ R (W,S) >0 of the isomorphism class of their Hecke deformations. However, in the case of right-angled (not necessarily finite) Coxeter systems one can still prove that all Hecke deformations are isomorphic; even with an explicit isomorphism. See e.g. [38, (2.1.13)], [41,Corollary 9.7] for this result which we present in an alternative way and which is suited for the next sections. . Then the map π q,1 : given by for s ∈ S defines an isomorphism of * -algebras.
Proof. Set α s (q) := (1 − q s ) / (1 + q s ) and β s (q) := 2 √ q s / (1 + q s ). Being its own inverse, for s that uniquely extends to a group homomorphism φ on the free group F (S) in S. Since (W, S) is right-angled one easily checks that φ (st) ms,t = (φ (s) φ (t)) ms,t for all s, t ∈ S. This implies that φ induces a group homomorphism φ ′ : The universal property of the group algebra C 1 [W ] then implies the existence of the unital * -algebra homomorphism π q,1 . It is clearly surjective. The injectivity follows from the universal property of the Hecke algebra C 1 [W ].  >0 . Indeed, in Example 6.3 we show that this cannot be the case. In the case |S| ≥ 3 this can also be proved through the simplicity of C * r,q (W ) in the same way as Remark 4.5, see Section 5. Remark 4.5. Garncarek's factoriality result of Theorem 0.3 illustrates that the isomorphism of Proposition 4.2 does not necessarily extend to an isomorphism of the corresponding Hecke-von Neumann algebras. Indeed, for q / ∈ ρ, ρ −1 the Hecke-von Neumann algebra N q (W ) is not a factor whereas N 1 (W ) is a factor. Hence there can be no isomorphism at all between N q (W ) and N 1 (W ). This was already observed in [22,Section 19, Note 19.2 on p. 358]. The situation is even more delicate, see the next Remark 4.6.
Remark 4.6 (Free factor problem). Consider the right-angled Coxeter group W = (Z 2 ) * l , l ≥ 3. By Theorem 0.3 we see that for the single parameter q ∈ [ 1 l−1 , 1] we have that N q (W ) is a II 1factor. Moreover, [26] together with a calculation in [30,Section 6] shows that for q ∈ [ 1 l−1 , 1] we have that N q (W ) ≃ L(F 2lq(1+q) −2 ) where L(F t ), t ∈ R >1 is the interpolated free group factor, cf. [26], [47]. By [26], [47], the interpolated free group factors are either all isomorphic or they are all non-isomorphic. The problem which of the two in this dichotomy is true is known as the famous free factor problem. Hence, solving the isomorphism question of N q (W ) for different q ∈ [ 1 l−1 , 1] is equivalent to the free factor problem. In Section 5 we shall show that for q ∈ [ 1 l−1 , 1] we have that C * r,q (W ) has unique trace; therefore if any two C * r,q (W ) with q ∈ [ 1 l−1 , 1] are isomorphic we get by Lemma 1.1 that two of the von Neumann algebras N q (W ), q ∈ [ 1 l−1 , 1] would be isomorphic. Since solving the free factor problem using these C * -algebraic methods seems unrealistic 4 we believe that all C * r,q (W ) with q ∈ [ 1 l−1 , 1] are non-isomorphic. We finish this section with the following lemma which is well-known on the algebraic level, see for instance [41,Section 9]. For convenience of the reader we include the proof for the associated C * -and von Neumann algebras here.
>0 . Proof. Note that ǫ s p s (q ′ ) = p s (q). Then from the defining properties of a Hecke algebra (1.3) and (1.4) we have that T w ), so that π q ′ ,q is trace preserving. By Lemma 1.1 π q ′ ,q extends to a * -isomorphism C * r,q (W ) → C * r,q ′ (W ).

Simplicity of Hecke C * -algebras
A C * -algebra is called simple if it does not contain any non-trivial closed two-sided ideal. In this section we investigate the (non-)simplicity of Hecke C * -algebras.
Remark 5.1. Note that in order to study the simplicity of Hecke C * -algebras it suffices to consider irreducible Coxeter systems. Indeed, if (W, S) is a Coxeter system that is not irreducible, it admits a non-trivial decomposition of the form (W, S) = (W T × W T ′ , T ∪ T ′ ). For every q ∈ R (W,S) >0 this induces a decomposition of the corresponding Hecke algebra into an algebraic tensor product By Lemma 1.1 this extends to the C * -and von Neumann algebraic level . Since the (spatial) tensor product of two C * -algebras is simple if and only if both C * -algebras are simple, it suffices to look at irreducible Coxeter systems.

Recall
Garncarek's characterisation of factoriality of the single-parameter Hecke-von Neumann algebras of right-angled Coxeter groups, see Theorem 0.3. Another result related to the simplicity of Hecke C * -algebras appears in [24] (see also [29] and [20]).
Theorem 5.2. An irreducible Coxeter group which is neither of spherical nor affine type is C *simple and has unique tracial state.
In the remaining sections we give partial answers to the question for a characterisation of those Hecke C * -algebras that are simple and have unique tracial state. In the case of a free product of abelian Hecke C * -algebras we obtain a complete answer. 5.1. Non-simplicity of Hecke C * -algebras. Let (W, S) be a Coxeter system with |S| < ∞ and z := (z s ) s∈S ∈ C (W,S) . For every reduced expression w = s 1 ...s n of w ∈ W define z w := z s 1 ...z sn . The growth series of W is the power series in z defined by We denote its region of convergence by R := z ∈ C (W,S) | W (z) converges . For more information on the growth series see [22,Chapter 17]. We further set Lemma 5.3. Let (W, S) be a Coxeter system with |S| < ∞ and q ∈ R ′ . Then there exists a character on C * r,q (W ). In particular C * r,q (W ) is not simple and does not have unique tracial state.
Proof. First assume that q ∈ R ′ with q s ≤ 1 for all s ∈ S. As shown in the proof of [30,Theorem 5.3], there exists a central projection E q ∈ N q (W ) with E q T (q) w = q 1/2 w E q for every w ∈ W (compare also with [22,Lemma 19.2.5] but note that our notational conventions differ slightly). Hence the map χ q ( · ) := τ q ( · E q ) / E q 2 is a character on C * r,q (W ) with χ q (T (q) Its kernel is a non-trivial maximal ideal in C * r,q (W ). For q ∈ R ′ \ R ′ with q s ≤ 1 for all s ∈ S choose a sequence (q n ) n∈N ⊆ R with q n → q. The map χ: T For general q ∈ R ′ the statement follows from the above in combination with Proposition 4.7 It follows from Lemma 5.3 that Hecke C * -algebras coming from irreducible Coxeter systems of spherical or affine type are never simple and never have unique tracial state, see Corollary 5.4. We are aware of the fact that affine type Coxeter systems always give rise to type I Hecke C *algebras (see [38, §4]), which implies non-simplicity. However, the proof of this statement makes use of the (non-trivial) characterisation of affine Coxeter groups in terms of affine Weyl groups and a resulting description of the corresponding Hecke algebras. The proof we present here is elementary. there exists a character on the corresponding Hecke C * -algebra C * r,q (W ). In particular, C * r,q (W ) is not simple and does not have unique tracial state.
Proof. First assume that |S| < ∞ and let q ∈ R (W,S) >0 with q s ≤ 1 for every s ∈ S. Being of spherical or affine type the Coxeter group W is amenable. Hence by [ , the statement follows with Proposition 4.7.
Next assume that |S| = ∞ and so (W, S) is of spherical type. Again, the map χ: T there exists a finite subset T ⊆ S such that the support {w ∈ W | x (w) = 0} of x is contained in the finite subgroup W T of W generated by T . Now W T is also a Coxeter group with the same exponents as W, see [22,Theorem 4.1.6 (i), Theorem 3.4.2 (i)]. As in [22,Lemma 19.2.2] one sees that C * r,q (W T ) embeds into C * r,q (W ) canonically. Under this identification we have x ∈ C * r,q (W T ). But the map T (q) w is a character on the finite-dimensional C * -algebra C * r,q (W T ), hence χ (x) ≤ x . As this holds for every x ∈ C q [W ], χ extends to a character on C * r,q (W ).

Averaging operators.
To show simplicity properties of Hecke C * -algebras coming from right-angled, irreducible Coxeter systems with at least three generators (see subsection 5.3), we will make use of a method inspired by Power's averaging argument in [45]. This requires the introduction of suitable averaging operators on the corresponding Hecke algebra which average over a finite subset of the Coxeter group. The statements of this subsection apply in greater generality. Therefore we will formulate our approach for arbitrary discrete groups. Let G be a discrete group. Denote the left regular representation of G by λ, let τ be the canonical tracial state on C * r (G) and fix some finite set F ⊆ G. Every such F defines an averaging operator Φ on C * r (G) given by The map Φ is trace-preserving, unital and completely positive. In particular it induces a bounded operatorΦ on ℓ 2 (G) ⊖ Cδ e given byΦ (xδ e ) := Φ (x) δ e for all x ∈ C [G] with τ (x) = 0.
Proposition 5.5. Let G be a discrete group with the property that there exist three elements g 1 , g 2 , g 3 ∈ G and a subset D ⊆ G \ {e} such that D ∪ g 1 Dg −1 1 = G \ {e} and the sets D, g 2 Dg −1 2 and g 3 Dg −1 3 are pairwise disjoint. Take F := {e, g 1 , g 2 , g 3 }. Then the operatorΦ on ℓ 2 (G) ⊖ Cδ e has norm strictly smaller than one.
The proof of Proposition 5.5 is based on Ching's following variation of Pukánszky's 14εargument in [46,Lemma 10].
Lemma 5.6. [16, Lemma 4] Let G be a group as in Proposition 5.5. Then for every x ∈ C * r (G). Further, we shall need that for arbitrary vectors ξ 0 , . . . , ξ n in a Hilbert space H we have the following equality, which for n = 1 is known as the parallelogram law: Note that (5.2) can be verified directly by writing out all norms as inner products.
Proof of Proposition 5.5. The norm ofΦ : ℓ 2 (G) ⊖ Cδ e → ℓ 2 (G) ⊖ Cδ e is clearly majorized by 1. Now suppose that Φ = 1. Take a sequence x k ∈ C[G] with τ (x k ) = 0 and x k δ e = 1 such that x k λ g i δ e with g 1 , g 2 , g 3 as in the proposition and g 0 the identity. By Therefore each of the individual summands on the left hand side converge to 0 as k → ∞. Since ξ k 0 = x k δ e and y 2 = yδ e , y ∈ C[G] we see from Lemma 5.6 that This contradicts that x k δ e = 1. We conclude that Φ < 1.

5.3.
Simplicity in the right-angled case for q close to 1. Let us now bring together the statements from Theorem 3.4 and Section 5.2. of 1 = (1 s ) s∈S such that for all q ∈ U the C * -algebra C * r,q (W ) is simple and has unique tracial state.
Proof. As (W, S) is irreducible with |S| ≥ 3 we find elements s, t 0 , . . . , t n ∈ S with n ≥ 1, S = {s, t 0 , . . . , t n }, m s,t 0 = ∞, t 1 = s and m t i ,t i+1 = ∞ for i = 0, . . . , n − 1. Then w 1 := t 0 · · · t n · · · t 0 , w 2 := s, w 3 := t 1 and D := {w ∈ W | |t 0 w| < |w|} satisfy the conditions from Proposition 5.5. For every reduced expression w = s 1 · · · s m in W consider the operator By the same arguments as in the proof of Proposition 4.2 this operator is unitary and does not depend on the reduced expression for w. By abuse of notation we will denote it by π q,1 T (1) w . Choose a positive integer d with |w| ≤ d for all w in F = {e, w 1 , w 2 , w 3 } and define a deformed averaging operator Φ q on C * r,q (W ) by Again, these maps are trace-preserving, unital, completely positive and they induce contractive linear operatorsΦ q on ℓ 2 (W ) ⊖ Cδ e viaΦ q (xδ e ) := Φ q (x) δ e for x ∈ C q [W ]. One easily checks that Φ q → Φ for q → 1. In particular, Proposition 5.5 implies that there exists an open neighborhood U ⊆ R (W,S) >0 of 1 = (1 s ) s∈S such thatΦ q has norm strictly smaller than 1 ∈ R for all q ∈ U .
Denote by χ d the word length projection on C * r,q (W ) from equation (1.1) and put χ ≤d := d r=0 χ r . Let l ≥ 1 and x ∈ χ ≤d (C q [W ]). Note that for every w ∈ F we have χ r (π q,1 (T (1) w )) = 0 for r ≥ d. As Φ l q averages over π q,1 (T w ) | w ∈ F and χ r (x) = 0 for r ≥ d, we get that χ r (Φ l q (x)) = 0 for r ≥ (2l + 1)d. In particular by the triangle inequality and with C := (#Cliq(Γ)) 3 ( s∈S p s (q)) we have by Theorem 3.4. The Cauchy-Schwarz inequality then implies For q ∈ U this converges to 0 as l → ∞. The simplicity and the unique trace property of C * r,q (W ), q ∈ U now follow by a standard argument (see for instance [45]): Let I be a non-zero ideal in C * r,q (W ). Choose 0 = x ∈ I positive. For every ε > 0 we find so 0 = τ q (x) ∈ I. Hence I must be trivial and this shows that C * r,q (W ) is simple if q ∈ U . Further, as Φ l q (x) → τ q (x) by (5.3) and τ ′ Φ l q (x) = τ ′ (x) for every tracial state τ ′ , C * r,q (W ) has τ q as its unique tracial state.
Remark 5.8. (a) Theorem 5.7 immediately implies that for q ∈ U the Hecke-von Neumann algebra N q (W ) is a II 1 -factor by uniqueness of the trace and Lemma 1.1. This extends results by Garncarek [30] to the multi-parameter case; note that the proof from [30] does not trivially extend to the multi-parameter case. (b) Theorem 5.7 does not give any information about the simplicity of Hecke C * -algebras of infinitely generated Coxeter groups unless q equals 1. in Theorem 5.7. Though our aim is different, the approach is inspired by [30,Section 6] and translates the remarks made there into the C *algebraic setting by using results from [27].
Let us assume that (W, S) is a Coxeter system where W is of the form W = Z k 1 2 * · · · * Z k l 2 with l, k 1 ≥ 2 and k 2 , . . . , k l ∈ N. For each 1 ≤ m ≤ l denote by s . The Hecke C * -algebra C * r,q (W ) decomposes as a reduced free product , τ q over the canonical traces. In the notation of [27] we get using [30, Lemma 6.2] that C * r,q (W ) , τ q ≃ * l m=1 C Z km 2 , (·) µ m , (5.4) where µ m (w) = q w km i=1 1 + q s (m) i −1 for w ∈ Z km 2 .
Proposition 5.9. Let (W, S) be a Coxeter system of the form W = Z k 1 2 * · · · * Z k l 2 where l, k 1 , . . . , k l ∈ N and q ∈ R This implies that Q has only positive coefficients. Hence for z ∈ Ω we have Q(z) ≥ 0. Moreover, for z ∈ Ω we have W (z) > 0 and so by (5.7) we have 0 ≤ Q (z) < 1. Further, for any z ∈ C (W,S) with 0 ≤ Q (z) < 1 we can expand and rearrange the terms in the series ∞ m=0 (Q (z)) m to get the (converging) Taylor series of (1 − Q (z)) −1 . The same is true for the product (1 − Q (z)) −1 s∈S (1 + z s ). Combining this with the previous paragraph we get that the Taylor series of the function in (5.5) (which is the growth series W (z)) converges on Ω.
Theorem 5.10. Let (W, S) be a Coxeter system of the form W = Z k 1 2 * · · · * Z k l 2 where l, k 1 ≥ 2 and let q ∈ R (W,S) >0 . Then the following statements are equivalent.
Proof. (1) ⇒ (2): Assume that q ∈ R ′ . Then C * r,q (W ) is not simple and does not have unique tracial state by Lemma 5.3.
(2) ⇒ (1): Assume that C * r,q (W ) is not simple and does not have unique tracial state. Assume further that 0 < q s ≤ 1 for every s ∈ S. Using [27,Proposition 4.10] in combination with (5.4) we get that the set (w 1 , ..., w l ) ∈ Comparing this with Proposition 5.9, we get that q ∈ R ′ . The general case of q ∈ R (W,S) >0 follows by an application of Proposition 4.7.
Remark 5.11. Proposition 5.9 also implies that the reasoning in [30,Section 6] applies to the multi-parameter Hecke-von Neumann algebra N q (W ) with W as in Theorem 5.10. In other words, N q (W ) is a factor if and only if q / ∈ R ′ .
Remark 5.12. In the proof of Theorem 5.10 we only used the simplicity part of [27,Corollary 4.10]. For W as above the full statement of the corollary also provides a detailed description of the ideal structure of C * r,q (W ) for q ∈ R ′ . Further we conclude by [27,Corollary 4.10] that for l ≥ 3 the Hecke C * -algebra C * r,q (W ) has stable rank 1 for every q. In view of Lemma 5.3, Proposition 4.7, Theorem 5.7 and Theorem 5.10 it is a natural question whether or not the characterisation above holds for general Coxeter systems. . Is it true that the Hecke C * -algebra C * r,q (W ) is simple and has unique tracial state if and only if q / ∈ R ′ where R ′ is defined as in (5.1)?
Remark 5.14. Recently a new approach to C * -simplicity results was obtained through Furstenberg/Hamana boundaries, see [36], [9], [4]. In particular, in [4] the Furstenberg boundary of a general unitary representation of a discrete group was defined and investigated in relation to trace-uniqueness properties. Proposition 4.2 shows that Hecke C * -algebras of a right-angled Coxeter group are C * -algebras generated by such a unitary representation. It would be interesting to exploit this connection. However, in light of the results from [4], it is not clear how manageable the Furstenberg-Hamana boundary is. 6. Exactness and nuclearity of Hecke C * -algebras The aim of this last section is to have a look at additional properties of Hecke C * -algebras such as exactness and nuclearity and give (counter-)examples regarding the extension of the isomorphism π q,1 in Proposition 4.2 to the C * -algebraic level. . Then C * r,q (W ) is an exact C * -algebra.
Proof. By [1], the group W acts amenably on a compact space. Therefore, by [10, Theorem 5.1.7] W is exact. With P s as defined in (1.6), we have T (q) s = T (1) s + p s (q)P s for s ∈ S. The Hecke C * -algebra C * r,q (W ) is generated by T (q) s s∈S and we have P s ∈ ℓ ∞ (W ) ⊆ B ℓ 2 (W ) . Hence C * r,q (W ) is contained in the uniform Roe algebra of W which is nuclear by [10, Theorem 5.1.6]. That implies the exactness of C * r,q (W ).
The example illustrates that π q,1 in general does not extend to an isomorphism of (reduced) C * -algebras. In the non-affine case it gets even worse. Here π q,1 and π −1 q,1 never extend to the reduced C * -algebra level for arbitrary choice of q.
Example 6.4. Let (W, S) be an irreducible right-angled Coxeter system of non-affine type. By Theorem 5.7 the reduced group C * -algebra C * r (W ) has unique tracial state. The map π q,1 is not trace preserving with respect to the canonical trace. Hence it does not extend to the reduced C * -algebraic level. To see that π −1 q,1 does in general not extend, one can use the same argument as in Example 6.3 or the trace-uniqueness in Theorem 5.7.