On the type of the von Neumann algebra of an open subgroup of the Neretin group

Abstract

The Neretin group $\mathcal{N}_{d, k}$ is the totally disconnected locally compact group consisting of almost automorphisms of the tree $\mathcal{T}_{d, k}$. This group has a distinguished open subgroup $\mathcal{O}_{d, k}$. We prove that this open subgroup is not of type I. This gives an alternative proof of the recent result of P.-E. Caprace, A. Le Boudec and N. Matte Bon which states that the Neretin group is not of type I, and answers their question whether $\mathcal{O}_{d, k}$ is of type I or not.

1. Introduction

The Neretin group $\mathcal{N}_{d, k}$ was introduced by Yu. A. Neretin in 11 as an analogue of the diffeomorphism group of the circle. This group $\mathcal{N}_{d, k}$ consists of almost automorphisms of the tree $\mathcal{T}_{d, k}$ and is a totally disconnected locally compact Hausdorff group. It has a distinguished open subgroup $\mathcal{O}_{d, k}$; for an accurate definition, see Section 3. Recently, P.-E. Caprace, A. Le Boudec and N. Matte Bon proved that the Neretin group $\mathcal{N}_{d, k}$ is not of type I by constructing two weakly equivalent but inequivalent irreducible representations of $\mathcal{N}_{d, k}$ 4. In their paper, they conjectured that the subgroup $\mathcal{O}_{d, k}$ of the Neretin group $\mathcal{N}_{d, k}$ is not type I either 4, Remark 4.8. Our main theorem answers their question.

Theorem 1.1.

The group von Neumann algebra of $L(\mathcal{O}_{d, k})$ of the open subgroup $\mathcal{O}_{d, k}$ of the Neretin group $\mathcal{N}_{d, k}$ is of type II. In particular, the open subgroup $\mathcal{O}_{d, k}$ of the Neretin group $\mathcal{N}_{d, k}$ is not of type I.

This theorem gives an alternative proof of the fact that the Neretin group $\mathcal{N}_{d, k}$ is not of type I, since the type I property is inherited to open subgroups. In the proof of our main theorem, we construct a nontrivial central sequence in the corner of the group von Neumann algebra $L(\mathcal{O}_{d, k})$.

2. Preliminaries

2.1. von Neumann algebras

We refer the reader to 6 for basics about von Neumann algebras. We review several topologies we use. Let $H$ be a separable Hilbert space. For $\xi \in H$, seminorms $p_{\xi }, p_{\xi }^*$ on $B(H)$ are defined by $p_{\xi }(x) = \| x \xi \|$ and $p_{\xi }^* (x) = \| x^* \xi \|$. The topology defined by these seminorms $\{ p_{\xi } \mid \xi \in H \} \cup \{ p_{\xi }^* \mid \xi \in H \}$ on $B(H)$ is called strong-$*$ operator topology. For $\{ \xi _n \} \in \ell ^2 \otimes H = \{ \{ \xi _n \} \mid \xi _n \in H, \sum _{n=1}^{\infty } \| \xi _n \| ^2 < \infty \}$, seminorms $q_{\{ \xi _n \} }, q_{\{ \xi _n \} }^*$ are defined by $q_{\{ \xi _n \} } (x) = ( \sum _{n=1}^{\infty } \| x \xi _n \| ^2 )^{\frac{1}{2}}$ and $q_{\{ \xi _n \} }^* (x) = ( \sum _{n=1}^{\infty } \| x^* \xi _n \| ^2 )^{\frac{1}{2}}$. The topology defined by these seminorms $\{ q_{\{ \xi _n \} } \mid \{ \xi _n \} \in \ell ^2 \otimes H \} \cup \{ q_{\{ \xi _n \} }^* \mid \{ \xi _n \} \in \ell ^2 \otimes H \}$ on $B(H)$ is called ultrastrong-$*$ topology. Note that these two topologies coincide on bounded subsets of $B(H)$.

We also review definitions of types of von Neumann algebras (see 3, Section 1.3). A von Neumann algebra $M$ is of type I if it is isomorphic to $\prod _{j \in J} \mathcal{A}_j \mathbin{\bar{\otimes }} B(H_j)$ for some set $J$ of cardinal numbers, where $\mathcal{A} _j$ is an abelian von Neumann algebra and $H_j$ is a Hilbert space of dimension $j$. A von Neumann algebra $M$ is of type II$_\mathbf{1}$ if it has no nonzero summand of type I and there exists a separating family of normal tracial states. A von Neumann algebra $M$ is of type II$_\mathbf{\infty }$ if it has no nonzero summand of type I or $\mathrm{II_1}$ but there exists an increasing net of projections $\{ p_i \} _{i \in I} \subset M$ converging strongly to $1_M$ such that $p_i M p_i$ is of type $\mathrm{II}_1$ for every $i \in I$. A von Neumann algebra $M$ is of type II if it is a direct sum of a type II$_1$ and a type II$_{\infty }$ von Neumann algebra. A von Neumann algebra $M$ is of type III if it has no nonzero summand of type I, $\mathrm{II_1}$ or $\mathrm{II_{\infty }}$. Every von Neumann algebra $M$ has a unique decomposition $M \cong M_{\mathrm{I}} \oplus M_{\mathrm{II}} \oplus M_{\mathrm{III}}$ where $M_{\mathrm{I}}, M_{\mathrm{II}}, M_{\mathrm{III}}$ are of type I, type II, type III respectively.

We review types of von Neumann algebras from the perspective of central sequences and obtain a criterion of having no nonzero type I summand.

Definition 2.1.

Let $M$ be a separable von Neumann algebra. A central sequence of $M$ is a sequence $\{ u_n \}$ of unitary elements in $M$ such that $[x, u_n]$ converges to $0$ in the ultrastrong-$*$ topology for all $x \in M$. A central sequence $\{ u_n \}$ of $M$ is trivial if there exists a sequence $\{ z_n \}$ of unitary elements of the center of $M$ such that $u_n - z_n$ converges to $0$ in the ultrastorong-$*$ topology.

Remark 2.2.

A sequence $\{ u_n \}$ of unitary elements in $M$ is a central sequence if and only if there exists $M_0 \subset M$ such that $M_0 '' = M$ and for all $x \in M_0$, $[x, u_n] \to 0$ in the ultrastrong-$*$ topology.

A. Connes showed that any type I factor has no nontrivial central sequence 5, Corollary 3.10 and this fact can be easily extended to type I von Neumann algebras.

Lemma 2.3.

Let $M$ be a separable von Neumann algebra. If $M$ is of type I, then every central sequence of $M$ is trivial.

Proof.

We may assume that $M$ is isomorphic to $\mathcal{A} \mathbin{\bar{\otimes }} B(H)$ for some separable abelian von Neumann algebra $\mathcal{A}$ and some separable Hilbert space $H$. Let $\{ u_n \}$ be a central sequence in $M$. Take some unit vector $\eta _0 \in H$ and let $p \in B(H)$ be the projection onto $\mathbb{C} \eta _0$. Then there exist $a_n \in \mathcal{A}$ such that $(1 \otimes p) u_n (1 \otimes p) = a_n \otimes p \in \mathcal{A} \mathbin{\bar{\otimes }} pB(H)p \cong \mathcal{A} \mathbin{\bar{\otimes }} \mathbb{C} p$. Since $\mathcal{A}$ is abelian, there exists a unitary element $v_n \in \mathcal{A}$ such that $a_n = v_n |a_n|$. We will show $u_n - v_n \otimes 1 \to 0$ in the strong-$*$ topology. First, we will show $u_n - a_n \otimes 1 \to 0$ in the strong-$*$ topology. Fix a faithful representation $\mathcal{A} \subset B(K)$ and take $\xi \in K, \eta \in H$ arbitrarily. Then, for sufficiently large $n$,

$$\begin{align*} u_n (\xi \otimes \eta ) &\approx (1 \otimes (\eta \otimes \eta _0 ^*)) u_n (\xi \otimes \eta _0) \\ &= (1 \otimes (\eta \otimes \eta _0 ^*)) (a_n \otimes p) (\xi \otimes \eta _0) \\ &= (a_n \otimes 1) (\xi \otimes \eta ), \end{align*}$$

where $\eta \otimes \eta _0 ^*$ is a Schatten form; $\eta \otimes \eta _0 ^* (\zeta ) = \langle \zeta , \eta _0 \rangle \eta$. Similarly, one has $u_n^* (\xi \otimes \eta ) \approx (a_n^* \otimes 1) (\xi \otimes \eta )$ for sufficiently large $n$. Finally, we should prove $|a_n| \to 1$ in $\mathcal{A}$ in the ultrastrong-$*$ topology; if this holds, then $a_n \otimes 1 - v_n \otimes 1 = v_n ((|a_n| - 1) \otimes 1) \to 0$ in the ultrastrong-$*$ topology. Since $t \mapsto \sqrt {t \vee 0 }$ is a linear growth function, it suffices to prove $a_n^* a_n \to 1$ in the strong-$*$ topology. For arbitrary $\xi \in K$,

$$\begin{align*} \| a_n^*a_n \xi - \xi \| &= \| (a_n^* a_n \otimes p) \xi \otimes \eta _0 - \xi \otimes \eta _0 \| \\ &= \| (1 \otimes p) u_n^* (1 \otimes p) u_n (1 \otimes p) \xi \otimes \eta _0 - \xi \otimes \eta _0 \| \\ &\to 0. \end{align*}$$

Therefore, a central sequence $\{ u_n \}$ in $M$ is trivial.

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Lemma 2.4.

Let $M$ be a separable von Neumann algebra. Suppose there exist a faithful normal state $\varphi$ and two central sequences $\{ u_n \}, \{ v_n \}$ such that $\varphi ( (u_n v_n u_n^* v_n^*)^k)$ converges to $0$ for every $k \in \mathbb{Z} \setminus \{ 0 \}$. Then $M$ has no nonzero type I summand.

Proof.

For simplicity, we write $u_n v_n u_n^* v_n^*$ as $w_n$. Note that for every $f \in C(\mathbb{T})$, $\varphi (f(w_n)) \to \int _{\mathbb{T}} f(z) \, dz$ where $\mathbb{T} = \{ z \in \mathbb{C} \mid | z | = 1 \}$, since trigonometric polynomials are dense in $C(\mathbb{T})$. Let $p \in M$ be a central projection such that $pM$ is of type I. Since every central sequence in a type I von Neumann algebra is trivial and $\{ p u_n \}$ and $\{ p v_n \}$ are central sequences in $pM$, $p w_n$ converges to $p$ in the ultrastrong-$*$ topology. Then for every $f \in C(\mathbb{T})$, $\varphi (p f(w_n)) \to \varphi (p) f(1)$. Take $\varepsilon > 0$ arbitrarily and $f \in C(\mathbb{T})$ such that $f \geq 0$, $f(1) = 1$ and $\int _\mathbb{T} f(z) \, dz < \varepsilon$. Then $\varphi (f(w_n)) \geq \varphi (p f(w_n))$, so $\varphi (p) \leq \int _{\mathbb{T}} f(z) \, dz < \varepsilon$. Since $\varepsilon$ is arbitrary, $\varphi (p) = 0$, i.e., $p = 0$. Therefore $M$ has no nonzero type I summand.

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2.2. Hecke algebras

We refer the reader to 9 and 10 for definitions and basic properties of Hecke algebras.

Suppose $(G,H)$ is a Hecke pair and $H \backslash G$ is a discrete space. Then the Hecke algebra $\mathcal{H} (G, H)$ acts on $\ell ^2 (H \backslash G)$ from left; define $\lambda \colon \mathcal{H} (G, H) \to B(\ell ^2(H \backslash G))$ by

$$\begin{equation*} \left[ \lambda (f) \xi \right] (Hx) = \sum _{Hy \in H \backslash G} f(Hxy^{-1}) \xi (Hy) \end{equation*}$$

for $f \in \mathcal{H} (G, H)$ and $\xi \in \ell ^2 (H \backslash G)$. We may omit $\lambda$ and write $\mathcal{H} (G, H) \subset B(\ell ^2(H \backslash G))$.

Let $\rho \colon G \to B(\ell ^2 (H \backslash G))$ be the right quasi-regular representation defined by $[ \rho _s \xi ] (x) = \xi (xs)$. One can easily check that $\mathcal{H} (G, H) \subset \rho (G) '$. Moreover, one has $\mathcal{H} (G, H)'' = \rho (G) '$ (see 1, Theorem 1.4). The unit vector $\delta _H \in \ell ^2 (H \backslash G)$ is a separating vector for $\mathcal{H} (G, H)$, since $\delta _H$ is a $\rho (G)$-cyclic vector. Moreover, if $R(x) = R(x^{-1})$ for every $x \in G$, then it is not hard to see that $\delta _H$ is a tracial vector, i.e., the vector state associated with $\delta _H$ is a trace on $\lambda (\mathcal{H} (G, H))$. In particular, the vector state $x \mapsto \langle x \delta _H, \delta _H \rangle$ is a faithful tracial state of $\mathcal{H} (G, H)$ for a unimodular locally compact group $G$ and its compact open subgroup $H$.

For a finite group $G$ and its subgroup $H \leq G$, note that the Hecke algebra $\mathcal{H} (G, H)$ is identical to $p_H \mathbb{C} [G] p_H$ where $p_H = \frac{1}{|H|} \sum _{h \in H} h \in \mathbb{C} [G]$ is a projection (see 9, Corollary 4.4).

Proposition 2.5 is a special case of 10, Proposition 1.3.

Proposition 2.5.

Let $G$ be a finite group acting on a finite group $V$, and let $\Gamma$ be a subgroup of $G$ leaving a subgroup $V_0$ of $V$ invariant. Then we have a canonical embedding $\mathcal{H} (V, V_0)^{\Gamma } \hookrightarrow \mathcal{H} (V \rtimes G, V_0 \rtimes \Gamma )$. Moreover, the canonical traces are consistent with this embedding.

Proof.

We will prove that there exists a canonical, trace preserving embedding $(p_{V_0} \mathbb{C} [V] p_{V_0})^{\Gamma } \hookrightarrow p_{V_0 \rtimes \Gamma } \mathbb{C} [V \rtimes G] p_{V_0 \rtimes \Gamma }$ where $p_H = \frac{1}{|H|} \sum _{h \in H} h$ for a subgroup $H$. Since $\Gamma$ leaves $V_0$ invariant, $p_{V_0}$ commutes with every element of $\Gamma$ in $\mathbb{C} [V_0 \rtimes \Gamma ]$. In particular, $p_{V_0}$ commutes with $p_\Gamma$ and $p_{V_0 \rtimes \Gamma } = p_{V_0} p_{\Gamma } = p_{\Gamma } p_{V_0}$. Note that $p_{\Gamma }$ commutes with every element in $\mathbb{C} [V] ^{\Gamma }$. Therefore, multiplication with $p_{\Gamma }$ is a $*$-homomorphism from $(p_{V_0} \mathbb{C} [V] p_{V_0})^{\Gamma } \cong p_{V_0} \mathbb{C} [V]^{\Gamma } p_{V_0}$ to $p_{V_0 \rtimes \Gamma } \mathbb{C} [V]^{\Gamma } p_{V_0 \rtimes \Gamma } \subset p_{V_0 \rtimes \Gamma } \mathbb{C} [V \rtimes G] p_{V_0 \rtimes \Gamma }$. This map preserves the canonical trace, since it is spatially implemented by the canonical isometry $W \colon \ell ^2 (V_0 \backslash V) \to \ell ^2 ((V_0 \rtimes \Gamma ) \backslash (V \rtimes G))$, and $W^* \delta _{V_0 \rtimes \Gamma } = \delta _{V_0}$. Since the canonical traces are faithful, this $*$-homomorphism is an embedding.

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Corollary 2.6.

In addition to the assumptions of Proposition 2.5, suppose $G$ leaves $V_0$ invariant. Then there is a canonical trace preserving embedding $\mathcal{H} (G, \Gamma ) \hookrightarrow \mathcal{H} (V \rtimes G, V_0 \rtimes \Gamma )$ and $\mathcal{H} (V, V_0) ^{G} \subset \mathcal{H} (G, \Gamma ) '$ in $\mathcal{H} (V \rtimes G, V_0 \rtimes \Gamma )$.

Proof.

The same argument as above shows that the first assertion holds. To show the second assertion, we identify $\mathcal{H} (V, V_0) ^{G}$ and $\mathcal{H} (G, \Gamma )$ with $p_{V_0} \mathbb{C} [V] ^G p_{V_0}$ and $p_{\Gamma } \mathbb{C} [G] p_{\Gamma }$, respectively. The assertion follows from the fact that $p_{V_0} p_{\Gamma } = p_{\Gamma } p_{V_0}$ and $\mathbb{C} [V] ^G \subset \mathbb{C} [G] '$.

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2.3. Locally compact groups

In this paper, topological groups are assumed to be Hausdorff. Let $G$ be a locally compact second countable group and $\mu$ be its left Haar measure. The left regular representation of $G$ is a unitary representation $\lambda \colon G \to \mathcal{U} (L^2(G))$ defined by $(\lambda _g f) (h) = f(g^{-1}h)$ for $f \in L^2(G)$ where $L^2(G)$ is a Haar square integrable function on $G$. The von Neumann algebra $\{ \lambda _g \mid g \in G \} '' \subset B(L^2(G))$ is called the group von Neumann algebra. The representation $\lambda$ extends to a representation of $L^1(G)$: $\lambda (f) g = f * g$ for $f \in L^1(G)$ and $g \in L^2(G)$.

A unitary representation $( \pi , H)$ of $G$ is called of being type I if the associated von Neumann algebra $\pi (G) '' \subset B(H)$ is of type I. A locally compact group $G$ is called of being type I if all its unitary representations are of type I. See 2, Chapter 6, 7 for more details and properties of type I groups.

3. Neretin groups

Let $d, k \geq 2$ be integers and $\mathcal{T}_{d, k}$ be a rooted tree such that the root has $k$ adjacent vertices and the others have $d+1$ adjacent vertices. An almost automorphism of $\mathcal{T}_{d, k}$ is a triple $(A, B, \varphi )$ where $A, B \subset \mathcal{T}_{d, k}$ are finite subtrees containing the root with $| \partial A| = | \partial B |$ and $\varphi \colon \mathcal{T}_{d, k} \setminus A \rightarrow \mathcal{T}_{d, k} \setminus B$ is an isomorphism. The Neretin group $\mathcal{N}_{d, k}$ is the quotient of the set of all almost automorphisms by the relation which identifies two almost automorphisms $(A_1, B_1, \varphi _1), \, (A_2, B_2, \varphi _2)$ if there exists a finite subtree $\tilde{A} \subset \mathcal{T}_{d, k}$ containing the root such that $A_1, \, A_2 \subset \tilde{A}$ and $\varphi _1 | _{\mathcal{T}_{d, k} \setminus \tilde{A}} = \varphi _2 | _{\mathcal{T}_{d, k} \setminus \tilde{A}}$. One can easily check that $\mathcal{N}_{d, k}$ is a group.

Let $d$ be the graph metric on $\mathcal{T}_{d, k}$, $v_0$ be the root of $\mathcal{T}_{d, k}$ and $B_n \coloneq \{ v \in \mathcal{T}_{d, k} \mid d(v_0, v) \leq n \}$ for $n \geq 0$. Every automorphism of $\mathcal{T}_{d, k}$ leaves $B_n$ invariant. For each $n \geq 0$, $\mathcal{O}_{d, k}^{(n)}$ denotes the subgroup consisting of automorphisms on $\mathcal{T}_{d, k} \setminus B_n$ and we write $\mathcal{O}_{d, k} \coloneq \bigcup _{n=0}^{\infty } \mathcal{O}_{d, k}^{(n)}$. Each $\mathcal{O}_{d, k} ^{(n)}$ is a subgroup of $\mathcal{N}_{d, k}$ containing $K \coloneq \operatorname {Aut}{(\mathcal{T}_{d, k})}$. Let $V_n : = \partial B_n = \{ v \in \mathcal{T}_{d, k} \mid d(v, v_0) = n \}$. Note that $\mathcal{O}_{d, k} ^{(n)} \cong \operatorname {Aut}{(\mathcal{T}_{d, d})} \wr \mathfrak{S}_{|V_n|} = \operatorname {Aut}{(\mathcal{T}_{d, d})} ^{|V_n|} \rtimes \mathfrak{S}_{|V_n|}$ and $\mathcal{O}_{d, d} ^{(l)} \wr \mathfrak{S} _{|V_n|} < \mathcal{O}_{d, k} ^{(n+l)}$.

The Neretin group $\mathcal{N}_{d, k}$ admits a totally disconnected locally compact group topology such that the inclusion map $K \hookrightarrow \mathcal{N}_{d, k}$ is continuous and open 7, Theorem 4.4. The Neretin group $\mathcal{N}_{d, k}$ is compactly generated and simple; see 7.

The group $\mathcal{O}_{d, k}$ is an open subgroup of $\mathcal{N}_{d, k}$. It is unimodular and amenable since $\mathcal{O}_{d, k}$ is an increasing union $\bigcup _{n=1}^{\infty } \mathcal{O}_{d, k} ^{(n)}$ of its compact subgroups.

4. Proof of theorem

We normalize the Haar measure $\mu$ on $\mathcal{O}_{d, k}$ so that $\mu (K) = 1$. Let $p = \lambda (\chi _K)$ be the projection onto the subspace of left $K$-invariant functions. This subspace can be identified with $\ell ^2 (K \backslash \mathcal{O}_{d, k})$. The Hecke algebra $\mathcal{H} (\mathcal{O}_{d, k}, K) \subset B(\ell ^2 (K \backslash \mathcal{O}_{d, k}))$ is a dense subalgebra of the corner $pL(\mathcal{O}_{d, k})p \subset B(\ell ^2(K \backslash \mathcal{O}_{d, k}))$ with respect to the weak operator topology. We will show that $p L(\mathcal{O}_{d, k}) p$ is of type II.

Since $K$ acts on $V_n$, there exists a canonical group homomorphism $K \to \operatorname {Aut}{(V_n)} \cong \mathfrak{S}_{|V_n|}$. The range of this homomorphism is denoted by $P_n = \operatorname {Aut}{(B_n)} < \mathfrak{S} _{|V_n|}$. Similarly, let $Q_n$ be the range of the canonical group homomorphism $\operatorname {Aut}{(\mathcal{T}_{d, d})} \to \operatorname {Aut}{( W_n )}$, where $W_n$ is the subset $\{ v \in \mathcal{T}_{d, d} \mid d(v, v_0) = n \}$ of $\mathcal{T}_{d, d}$. One has $\mathcal{H} (\mathcal{O}_{d, k}, K) \cong \cup _{n=1}^{\infty } \mathcal{H} (\mathcal{O}_{d, k}^{(n)}, K)$ and $\mathcal{H} (\mathcal{O}_{d, k}^{(n)}, K) \cong \mathcal{H} (\mathfrak{S}_{|V_n|}, P_n)$. We use this identification freely. For finite groups $G_1, G_2$ and their subgroups $H_i < G_i$, $\mathcal{H} (G_1, H_1) \otimes \mathcal{H} (G_2, H_2) \cong \mathcal{H} (G_1 \times G_2, H_1 \times H_2)$. Proposition 2.5 for $G=\mathfrak{S} _{|V_n|}, \Gamma = P_n, V = \mathfrak{S} _{d^l}^{|V_n|}, V_0 = Q_l^{|V_n|}$ implies

$$\begin{align*} ((\mathcal{H} (\mathcal{O}_{d, d}^{(l)}, \operatorname {Aut}{(\mathcal{T}_{d, d})}))^{\otimes |V_n|})^{P_n} &\cong (\mathcal{H} (\mathfrak{S} _{d^l}^{|V_n|}, Q_l^{|V_n|}))^{P_n} \\ &\hookrightarrow \mathcal{H} (\mathfrak{S} _{d^l}^{|V_n|} \rtimes \mathfrak{S} _{|V_n|}, Q_l^{|V_n|} \rtimes P_n) \\ &= \mathcal{H} (\mathfrak{S} _{d^l}^{|V_n|} \rtimes \mathfrak{S} _{|V_n|}, P_{n+l}) \\ &\subset \mathcal{H} (\mathfrak{S}_{|V_{n+l}|}, P_{n+l}) \end{align*}$$

for $l \in \mathbb{N}$. Moreover, Corollary 2.6 implies $(\mathcal{H} (\mathfrak{S} _{d^l}^{|V_n|}, Q_l^{|V_n|}))^{\mathfrak{S} _{|V_n|}} \subset \mathcal{H} (\mathfrak{S} _{|V_n|}, P_n)'$. Since $\mathcal{H} (\mathcal{O}_{d, d} ^{(l)}, \operatorname {Aut}{(\mathcal{T}_{d, d})}) \cong \mathcal{H} (\mathfrak{S} _{d^l}, Q_l)$ and $(\mathfrak{S} _{d^l}, Q_l)$ is not a Gelfand pair for $l \geq 3$ (see 8, Theorem 1.2), $\mathcal{H} (\mathcal{O}_{d, d} ^{(3)}, \operatorname {Aut}{(\mathcal{T}_{d, d})})$ is noncommutative.

Let $\tau$ be the vector state associated with $\delta _{K} \in \ell ^2 (K \backslash \mathcal{O}_{d, k})$. This is a trace, since $\mathcal{O}_{d, k}$ is a unimodular locally compact group and $K$ is its compact open subgroup. Note that $\tau (x^{\otimes |V_n|}) = (\tau (x))^{|V_n|}$ for $x \in \mathcal{H} (\mathcal{O}_{d, d} ^{(3)}, \operatorname {Aut}{(\mathcal{T}_{d, d})})$ where $\tau$ also denotes the canonical trace on $\mathcal{H} (\mathcal{O}_{d, d} ^{(3)}, \operatorname {Aut}{(\mathcal{T}_{d, d})})$. Since $\mathcal{H} (\mathcal{O}_{d, d} ^{(3)}, \operatorname {Aut}{(\mathcal{T}_{d, d})})$ is a noncommutative finite dimensional algebra, there exist two unitaries $u, v \in \mathcal{H} (\mathcal{O}_{d, d} ^{(3)}, \operatorname {Aut}{(\mathcal{T}_{d, d})})$ such that $| \tau ((u^*v^*uv)^k) | < 1$ and $| \tau ((v^*u^*vu)^k) | < 1$ for all $k \in \mathbb{Z} \setminus \{ 0 \}$. Set $u_n \coloneq u^{\otimes |V_n|} \in \mathcal{H} (\mathcal{O}_{d, k} ^{(n)}, K)'$ and $v_n \coloneq v^{\otimes |V_n|} \in \mathcal{H} (\mathcal{O}_{d, k} ^{(n)}, K)'$. Then for every $x \in \mathcal{H} (\mathcal{O}_{d, k}, K)'' = pL(\mathcal{O}_{d, k})p$, $[x,u_n] \to 0$ and $[x,v_n] \to 0$ in the ultrastrong-$*$ topology. Thus $\{ u_n \}$ and $\{ v_n \}$ are central sequences. In addition, $\tau ((u_n v_n u_n^* v_n ^*)^k ) = \tau ( (u v u^* v ^*)^k )^n \to 0$ as $n \to \infty$ for every $k \in \mathbb{Z} \setminus \{ 0 \}$. So by Lemma 2.4, $p L(\mathcal{O}_{d, k}) p$ has no nonzero type I summand and it is of type II.

Let $K_n \coloneq \{ \varphi \in K \mid \varphi | _{B_n} = \mathrm{id} _{B_n} \}$ and $p_n \coloneq \frac{1}{\mu (K_n)} \lambda (\chi _{K_n}) \in L(\mathcal{O}_{d, k})$. Then $\{ p_n \}$ converges $1_{L(\mathcal{O}_{d, k})}$ in the strong operator topology. Applying the same argument as above to $p_n L(\mathcal{O}_{d, k}) p_n$, one finds that $p_n L(\mathcal{O}_{d, k}) p_n$ is of type II. Therefore $L(\mathcal{O}_{d, k})$ is of type II.