C$^{\ast }$-simplicity has no local obstruction

Abstract

In 2016, I solved a problem of de la Harpe from 2006: Is there a nondiscrete C$^{\ast }$-simple group? However the solution was not fully satisfactory, as the C$^{\ast }$-simple groups provided (and their operator algebras) are very close to discrete groups. All previously known examples are of this form. In this article I give yet another construction of nondiscrete C$^{\ast }$-simple groups. The statement in the title then follows. This in particular gives the first examples of nonelementary C$^{\ast }$-simple groups (in Wesolek’s sense).

1 Introduction

A fundamental motivation of operator algebra theory is to give a framework for understanding locally compact groups. Successful achievements in this direction include Glimm’s dichotomy theorem, the Kasparov theory, the Baum–Connes theory, and Popa’s deformation/rigidity theory. Also, via the (reduced) crossed product construction, locally compact groups produce interesting examples of concrete operator algebras. On the one hand, for discrete groups many deep structural results on these operator algebras have been established. On the other hand, their nondiscrete counterparts are not yet on a comparable level.

In this article, we focus on C $^{\ast }$-simplicity [9], the simplicity of the reduced group C$^{\ast }$-algebra, of locally compact groups. For discrete groups, satisfactory characterisations of C$^{\ast }$-simplicity were established in the last decade (see e.g., [1, 6]). However, the results do not (at least directly) extend to nondiscrete groups. C$^{\ast }$-simplicity of nondiscrete groups is still a mysterious property. A main reason for this difficulty is the lack of interesting examples. Indeed, even the existence of such a group – questioned by de la Harpe [3] – was not known until [13]. Although such groups are now known [11, 13], all the currently known examples are very close to discrete groups. (More precisely, they are essentially the projective limit of discrete groups of particularly good form; see [13, Proposition], which is the only previously known result to produce a nondiscrete C$^{\ast }$-simple group. In particular, all these groups are elementary in Wesolek’s sense by [15, Theorem 3.18].)

In this article, we provide a new framework to produce nondiscrete C$^{\ast }$-simple groups. Note that C$^{\ast }$-simple groups must be totally disconnected, by [10, Theorem A]. Thus our attention is naturally restricted to totally disconnected groups. As a result of the new construction, we conclude the statement in the title: every totally disconnected locally compact group realises as an open subgroup of a C$^{\ast }$-simple group. In particular we obtain the first examples of C$^{\ast }$-simple groups which are nonelementary in Wesolek’s sense [15]. We believe that our new construction sheds new light on (nondiscrete) C$^{\ast }$-simplicity, and that our proof gives a new insight into the analysis of the group and reduced crossed product operator algebras of nondiscrete groups.

2 Preliminaries

Here we fix notations, and prove a basic lemma.

Notations

Throughout the article, let G be a totally disconnected locally compact group. We fix a left Haar measure $\mu $ on G. Define $L^{2}(G) := L^{2}(G, \mu )$. Let $\lambda \colon G \curvearrowright L^{2}(G)$ denote the left regular representation. The representation $\lambda $ integrates to the $\ast $-representation of the group algebra $C_{c}(G)$ on $L^{2}(G)$ which is given by the convolution product. The reduced group C$^{\ast }$-algebra $\mathop {{\mathrm C}_{\mathrm r}^{\ast }}(G)$ is the operator norm closure of $C_{c}(G) \subset \mathbb B\left (L^{2}(G)\right )$. For a compact open subgroup K of G, set

$$ \begin{align*} p_{K}:= \mu(K)^{-1} \chi_{K} \in C_{c}(G) \subset \mathop{{\mathrm C}_{\mathrm r}^{\ast}}(G). \end{align*} $$

Observe that $p_{K}$ is the orthogonal projection onto the K-fixed point space $L^{2}(G)^{K}$. A theorem of van Dantzig [14] shows that the set of compact open subgroups $K<G$ forms a local basis at the identity element $e\in G$. Hence the net $(p_{K})_{K<G}$ forms an approximate unit of $\mathop {{\mathrm C}_{\mathrm r}^{\ast }}(G)$. For a closed subgroup H of G, we identify $\mathop {{\mathrm C}_{\mathrm r}^{\ast }}(H)$ with the C$^{\ast }$-subalgebra of the multiplier algebra $\mathcal M\left (\mathop {{\mathrm C}_{\mathrm r}^{\ast }}(G)\right )$ in the obvious way (compare [7]). In particular, when H is open, we have $\mathop {{\mathrm C}_{\mathrm r}^{\ast }}(H) \subset \mathop {{\mathrm C}_{\mathrm r}^{\ast }}(G)$. When $H<G$ is a closed subgroup normalised by a compact subgroup $K<G$, we equip $\mathop {{\mathrm C}_{\mathrm r}^{\ast }}(H)$ with the K-action induced from the conjugation action $K \curvearrowright H$.

The symbol $\otimes $ stands for the minimal tensor product of C$^{\ast }$-algebras, the Hilbert space tensor product and the tensor product of unitary representations. Denote by $\mathop {\rtimes _{\mathrm r}}$ the reduced C$^{\ast }$-crossed product. (The underlying actions should be always clear from the context.) For a C$^{\ast }$-algebra A equipped with a compact group action $K \curvearrowright A$, denote by $A^{K}$ the fixed point algebra of the K-action.

On conditional expectations

The following lemma should be well known to experts. For completeness of the article, we include the proof.

Lemma 2.1. Let $K<G$ be a compact open subgroup. Let $\alpha \colon G \curvearrowright A$ be a C$^{\ast }$-dynamical system. Then there is a faithful conditional expectation

$$ \begin{align*} E_{K}\colon p_{K} (A \mathop{\rtimes {}_{\mathrm r}} G) p_{K} \rightarrow p_{K} A p_{K}=A^{K} p_{K} \end{align*} $$

satisfying $E_{K}(p_{K} a\lambda _{s} p_{K}) = \chi _{K}(s) p_{K} a p_{K}$ for all $a\in A, s\in G$.

The analogous statement holds true in the von Neumann algebra setting. Moreover, in this setting, $E_{K}$ can be chosen to be normal.

Proof. We show only the C$^{\ast }$-algebra case. The proof in the von Neumann algebra case is identical to that in the C$^{\ast }$-algebra case.

Take a covariant representation $(\pi , v)$ of $(A, \alpha )$ on $\mathfrak {H}$ such that $\pi $ is faithful. (For instance, take a faithful regular covariant representation of $(A, \alpha )$.) We identify $A \mathop {\rtimes _{\mathrm r}} G$ with a C$^{\ast }$-subalgebra of $\mathbb B\left (\mathfrak {H} \otimes L^{2}(G)\right )$ via the regular covariant representation associated to $\pi $. Then this gives rise to an inclusion $p_{K} (A \mathop {\rtimes _{\mathrm r}} G)p_{K} \subset \mathbb B\left (\mathfrak {H} \otimes L^{2}(G)^{K}\right )$. We also identify $p_{K} A p_{K}$ with the C$^{\ast }$-algebra $\pi \left (A^{K}\right )\otimes \text { id}_{\mathbb C \chi _{K}}$ on $\mathfrak {H} \otimes \mathbb C\chi _{K}$ in the obvious way. Let q denote the orthogonal projection from $\mathfrak {H} \otimes L^{2}(G)^{K}$ onto $\mathfrak {H} \otimes \mathbb C\chi _{K}$. Define

$$ \begin{align*} \widetilde{E} \colon \mathbb B\left(\mathfrak{H} \otimes L^{2}(G)^{K}\right) \rightarrow \mathbb B\left(\mathfrak{H} \otimes \mathbb C \chi_{K}\right) \end{align*} $$

by

$$ \begin{align*} \widetilde{E}(x):= q x q. \end{align*} $$

Then under the foregoing identifications of C$^{\ast }$-algebras, we obtain

$$ \begin{align*} \widetilde{E}(p_{K} (A \mathop{\rtimes {}_{\mathrm r}} G)p_{K}) = p_{K} A p_{K}. \end{align*} $$

Hence the map $\widetilde {E}$ restricts to a conditional expectation

$$ \begin{align*} E_{K} \colon p_{K} (A \mathop{\rtimes {}_{\mathrm r}} G)p_{K} \rightarrow p_{K} A p_{K}. \end{align*} $$

Direct computations show that the map $E_{K}$ satisfies the required equation. Let $\rho \colon G \curvearrowright L^{2}(G)$ denote the right regular representation of G. Observe that $(v\otimes \rho )(G)p_{K}$ commutes with $p_{K}(A \mathop {\rtimes _{\mathrm r}} G) p_{K}$. As the subset $[(v \otimes \rho )(G)p_{K}]\cdot \left (\mathfrak {H} \otimes \mathbb C \chi _{K}\right )$ spans a dense subspace of $\mathfrak {H} \otimes L^{2}(G)^{K}$, the conditional expectation $E_{K}$ is faithful.

3 New construction of nondiscrete C$^{\ast }$-simple groups

Recall that G is a totally disconnected locally compact group. We will construct an ambient C$^{\ast }$-simple group $\mathcal G$ of G.

To avoid confusion, we first introduce the following notations. Let $\Upsilon _{n}$, $n\in \mathbb {N}$, be pairwise distinct copies of the group

$$ \begin{align*} \bigoplus_{K<G}\bigoplus_{G/K} \mathbb{Z}_{2}, \end{align*} $$

where the first direct sum is taken over the set of all compact open subgroups K of G. We equip each $\Upsilon _{n}$ with the G-action induced from the left translation G-actions on $G/K$. Let $\Xi _{n}$, $n\in \mathbb {N}$, be pairwise distinct copies of the integer group $\mathbb {Z}$. We equip each $\Xi _{n}$ with the trivial G-action.

Set

$$ \begin{align*} \Gamma_{1}:= \Upsilon_{1},\qquad \Lambda_{1}:= \Gamma_{1} \ast \Xi_{1}, \end{align*} $$

equipped with the obvious G-actions. Assume that $\Gamma _{n}$ and $\Lambda _{n}$ have been defined. We then define

$$ \begin{align*} \Gamma_{n+1}:= \Lambda_{n} \times \Upsilon_{n+1},\qquad \Lambda_{n+1}:=\Gamma_{n+1} \ast \Xi_{n+1}, \end{align*} $$

equipped with the obvious G-actions. As a result, we obtain the increasing sequence

$$ \begin{align*} \Gamma_{1}< \Lambda {}_{1} < \Gamma_{2} < \Lambda_{2} < \dotsb \end{align*} $$

of discrete groups. Define $\Lambda $ to be the inductive limit of this sequence. As the inclusions are G-equivariant, we have a natural G-action $\alpha $ on $\Lambda $. Now set

$$ \begin{align*} \mathcal G:= \Lambda \rtimes_{\alpha} G. \end{align*} $$

Clearly $\mathcal G$ contains an open subgroup isomorphic to G. Define $\mathcal G_{n} := \Lambda _{n} \rtimes G< \mathcal G$ for $n\in \mathbb N$.

For an open compact subgroup $K<G~(< \mathcal G$), the following observation on $p_{K}{\mathrm C}^{\ast }_{\mathrm r}(\mathcal G)p_{K}$ is useful. Note first that the $\ast $-subalgebra $p_{K} C_{c}(\mathcal G) p_{K}$ is dense in $p_{K}{\mathrm C}^{\ast }_{\mathrm r}(\mathcal G) p_{K}$. Since K is open in $\mathcal G$, the characteristic functions $\chi _{S}$, $S \in K\backslash \mathcal G/K$, form a basis of $p_{K} C_{c}(\mathcal G) p_{K}$. Therefore one can approximate a given element $x\in p_{K}{\mathrm C}^{\ast }_{\mathrm r}(\mathcal G) p_{K}$ arbitrarily well by an element of the form

$$ \begin{align*} \sum_{s\in F} p_{K} x_{s} \lambda_{s} p_{K}, \end{align*} $$

where $n\in \mathbb N$, $F =\{e, s_{1}, \dotsc , s_{l}\}$ is a finite subset in G having the pairwise disjoint K-double cosets, $x_{s} \in \mathop {{\mathrm C}_{\mathrm r}^{\ast }}(\Lambda _{n})$ for all $s\in F$ and $x_{e} \in \mathop {{\mathrm C}_{\mathrm r}^{\ast }}(\Lambda _{n})^{K}$.

Note that by [15, Theorem 3.18], the class of elementary totally disconnected locally compact groups is closed under taking open subgroups and group extensions. Therefore the group $\mathcal G$ is elementary if and only if the original group G is elementary. Typical examples of nonelementary totally disconnected locally compact groups include $\mathrm {PSL}_{d}\left (\mathbb {Q}_{p}\right )$ and $\mathrm {Aut}(T_{d})$, where p is a prime number, $d\in \{3, 4, \dotsc \}$ and $T_{d}$ is a d-regular tree [15, Proposition 6.3].

The following theorem is the main result of this article:

Theorem 3.1. The locally compact group $\mathcal G$ is C$^{\ast }$-simple.

Proof. Let I be a nonzero (closed two-sided) ideal of $\mathop {{\mathrm C}_{\mathrm r}^{\ast }}(\mathcal G)= \mathop {{\mathrm C}_{\mathrm r}^{\ast }}(\Lambda ) \mathop {\rtimes _{{\mathrm r}, \alpha }} G$. Take a nonzero positive element $x\in I$. Let $K<G$ be a compact open subgroup satisfying $a:=p_{K} x p_{K} \neq 0$. We will show that $p_{K} \in I$.

Let

$$ \begin{align*} E_{K}\colon p_{K} \mathop{{\mathrm C}_{\mathrm r}^{\ast}}(\mathcal G) p_{K} \rightarrow \mathop{{\mathrm C}_{\mathrm r}^{\ast}}(\Lambda)^{K} p_{K} \end{align*} $$

be the faithful conditional expectation provided in Lemma 2.1. Since a is positive and nonzero, so is $E_{K}(a)$. By rescaling a if necessary, we may further assume that

$$ \begin{align*} \lVert E_{K}(a)\rVert=1. \end{align*} $$

Choose an $n\in \mathbb {N}$ and $a_{0}\in p_{K} C_{c}(\mathcal G_{n})p_{K}$ satisfying

$$ \begin{align*} \lVert a - a_{0}\rVert <1/2,\qquad \lVert E_{K}(a_{0})\rVert=1. \end{align*} $$

Write

$$ \begin{align*} a_{0}= \sum_{s\in F} p_{K} x_{s} \lambda_{s} p_{K}, \quad x_{s}\in \mathop{{\mathrm C}_{\mathrm r}^{\ast}}(\Lambda_{n}),\ x_{e} \in \mathop{{\mathrm C}_{\mathrm r}^{\ast}}(\Lambda_{n})^{K}, \end{align*} $$

where $F=\{e, s_{1}, \dotsc , s_{l}\}$ is a finite subset of G having the pairwise distinct K-double cosets. Note that $\lVert x_{e}\rVert = \lVert E_{K}(a_{0})\rVert =1$.

Let $\Upsilon _{n+1, K} < \Upsilon _{n+1}$ be the Kth direct summand of $\Upsilon _{n+1}$. Let $\mathop {{\mathrm C}_{\mathrm r}^{\ast }}\left (\Upsilon _{n+1, K}\right ) \cong C\left (\{0, 1\}^ {G/ K}\right )$ be the obvious G-equivariant $\ast $-isomorphism. Define

$$ \begin{align*} U:= \left\{\left(\epsilon_{gK}\right)_{gK\in G/K} \in \{0, 1\}^ {G/ K}: \epsilon_{K}= 0, \ \epsilon_{gK}=1 \text{ for }gK \subset \bigsqcup_{i=1}^{l} K s_{i} K\right\}. \end{align*} $$

Observe that U is a K-invariant (nonempty) clopen subset of $\{0, 1\}^ {G/ K}$. Moreover, for each i we have $\alpha _{s_{i}}(U) \cap U=\emptyset $. We regard $p:= \chi _{U}$ as an element of $\mathop {{\mathrm C}_{\mathrm r}^{\ast }}(\Lambda )$. Then $p\lambda _{s_{i}} p=0$ for $i=1, \dotsc , l$. The projection p is nonzero and commutes with $p_{K}$ and $x_{s}$, $s\in F$. Thus

$$ \begin{align*} p a_{0} p= p x_{e} p_{K}. \end{align*} $$

Since $x_{e}$ and p sit in the first and second tensor product components of $\mathop {{\mathrm C}_{\mathrm r}^{\ast }}\left (\Gamma _{n+1}\right )=\mathop {{\mathrm C}_{\mathrm r}^{\ast }}(\Lambda _{n}) \otimes \mathop {{\mathrm C}_{\mathrm r}^{\ast }}\left (\Upsilon _{n+1}\right )$, respectively, we have

$$ \begin{align*} \lVert px_{e}\rVert =\lVert p\rVert \lVert x_{e}\rVert=1. \end{align*} $$

Let B be the C$^{\ast }$-subalgebra of $\mathop {{\mathrm C}_{\mathrm r}^{\ast }}(\Lambda )^{K}$ generated by $\mathop {{\mathrm C}_{\mathrm r}^{\ast }}\left (\Gamma _{n+1}\right )^{K}$ and $\mathop {{\mathrm C}_{\mathrm r}^{\ast }}\left (\Xi _{n+1}\right )$. By [5, Theorem 2], B is simple. Note that $p x_{e} \in B$. Therefore, by [16, Lemma 2.3], one has a sequence $b_{1}, \dotsc , b_{r} \in B$ satisfying

$$ \begin{align*} \left\lVert\sum_{i=1}^{r} b_{i} b_{i} {}^{\ast} \right\rVert \leq 2,\qquad \sum_{i=1}^{r} b_{i} px_{e} b_{i}^{\ast} =1_{B}. \end{align*} $$

This implies

$$ \begin{align*} \sum_{i=1}^{r} b_{i} p a_{0} p b_{i}^{\ast} =\sum_{i=1}^{r} b_{i} p x_{e} b_{i}^{\ast} p_{K}=p_{K}. \end{align*} $$

Since $\left \lVert \sum _{i=1}^{r} b_{i} p (a- a_{0}) p b_{i}^{\ast } \right \rVert \leq 2 \lVert a- a_{0}\rVert <1$, we have $\lVert p_{K}+ I\rVert _{\mathop {{\mathrm C}_{\mathrm r}^{\ast }}(\mathcal G)/I} <1$. As $p_{K}$ is a projection, this yields $p_{K} \in I$. Since $K<G$ can be chosen arbitrarily small, we conclude $I=\mathop {{\mathrm C}_{\mathrm r}^{\ast }}(\mathcal G)$.

We keep the settings $G, \mathcal G$ and so on until the end of this article.

4 Uniqueness of KMS weight on $\mathop {{\mathrm C}_{\mathrm r}^{\ast }}(\mathcal G)$

By modifying the proof of Theorem 3.1, we also obtain the uniqueness of KMS weight on $\mathop {{\mathrm C}_{\mathrm r}^{\ast }}(\mathcal G)$ with respect to the modular flow. From now on, we freely use the basic facts on the Plancherel weight observed in [10, Section 2.6].

For a locally compact group H, let $\Delta _{H} \colon H \rightarrow \mathbb R_{>0}$ denote the modular function of H. Define $H_{0} := \ker (\Delta _{H})<H$. Note that for totally disconnected H, it is not hard to see that $H_{0}$ is open in H and that $\Delta _{H}(H) \subset \mathbb Q$. Observe that for our G and $\mathcal G$,

$$ \begin{align*} \Delta_{G}(G)= \Delta_{\mathcal G}(\mathcal G), \qquad \mathcal G_{0}= \Lambda \rtimes G_{0}. \end{align*} $$

Let $\varphi $ denote the Plancherel weight on $\mathop {{\mathrm C}_{\mathrm r}^{\ast }}(\mathcal G)$. Let $\sigma ^{\varphi }$ be the modular flow on $\mathop {{\mathrm C}_{\mathrm r}^{\ast }}(\mathcal G)$:

$$ \begin{align*} \left[\sigma^{\varphi}_{t}(f)\right](s):= \Delta_{\mathcal G}(s)^{{\mathrm i}t} f(s)\quad\text{for }f\in C_{c}(\mathcal G), \ s\in \mathcal G, \ t\in \mathbb R. \end{align*} $$

Throughout the paper, a weight on a C$^{\ast }$-algebra is always assumed to be densely defined, lower semicontinuous and nonzero (i.e., proper) without being stated. (See [8] or [10, Section 2.6] for the definitions.)

For a weight $\psi $ on a C$^{\ast }$-algebra A, as in [8, Definition 1.1], denote by $\mathcal M_{\psi }$ the linear span of $\psi ^{-1}([0, \infty ))$. Note that $\mathcal M_{\psi }$ is a hereditary $\ast $-subalgebra of A. In addition, when $\psi $ is tracial, $\mathcal M_{\psi }$ is a norm dense ideal of A, and hence it contains all projections in A. We call the $\ast $-subalgebra

$$ \begin{align*} \left\{ a\in A: \begin{array}{l}\mathcal M_{\psi} a \cup a \mathcal M_{\psi} \subset \mathcal M_{\psi}, \\ \psi(a x) = \psi(xa) \text{ for all } x\in \mathcal M_{\psi}\end{array}\right\} \subset A \end{align*} $$

the centraliser of $\psi $. We set

$$ \begin{align*} C_{cc}(\mathcal G):= \bigcup_{K<G} p_{K} C_{c}(\mathcal G) p_{K}. \end{align*} $$

Here the union is taken over all compact open subgroups $K<G$. Note that $C_{cc}(\mathcal G)$ is a $\ast$-subalgebra of $C_{c}(\mathcal G)$. Observe that for any $\sigma ^{\varphi }$-KMS weight $\psi $ on $\mathop {{\mathrm C}_{\mathrm r}^{\ast }}(\mathcal G)$, as every $p_{K}$ is a projection fixed by $\sigma ^{\varphi }$, we have $C_{cc}(\mathcal G) \subset \mathcal M_{\psi }$. Indeed, as $p_{K}$ is a projection, one has an analytic element $a\in \mathcal M_{\psi }$ with $p_{K}\leq p_{K} a^{\ast } a p_{K}$. Then, by the KMS condition, we have

$$ \begin{align*} \psi(p_{K}) \leq \psi(p_{K} a^{\ast} a p_{K})= \psi\left( \sigma^{\varphi}_{{\mathrm i}/2}(a) p_{K}\sigma^{\varphi}_{{\mathrm i}/2}(a^{\ast})\right) \leq \psi\left( \sigma^{\varphi}_{{\mathrm i}/2}(a) \sigma^{\varphi}_{{\mathrm i}/2}(a^{\ast})\right)= \psi(a^{\ast} a). \end{align*} $$

Theorem 4.1. Up to scalar multiple, the Plancherel weight $\varphi $ is the only $\sigma ^{\varphi }$-KMS weight on $\mathop {{\mathrm C}_{\mathrm r}^{\ast }}(\mathcal G)$. When $\mathcal G$ is nonunimodular, there is no tracial weight on $\mathop {{\mathrm C}_{\mathrm r}^{\ast }}(\mathcal G)$.

Proof. We consider the following claim:

Claim. Let $\psi $ be a weight on $\mathop {{\mathrm C}_{\mathrm r}^{\ast }}(\mathcal G)$ whose centraliser contains $C_{cc}(\mathcal G_{0})$ and satisfies $C_{cc}(\mathcal G) \subset \mathcal M_{\psi }$. Then for any compact open subgroup $K<G$ and any $s\in \mathcal G \setminus K$, we have

$$ \begin{align*} \psi(\lambda_{s} p_{K})=0. \end{align*} $$

Note that by the foregoing observations, any $\sigma ^{\varphi }$-KMS weights and tracial weights on $\mathop {{\mathrm C}_{\mathrm r}^{\ast }}(\mathcal G)$ satisfy the assumption of the claim.

We first prove the theorem under the assumption that the claim holds true. In the case that $\psi $ is a $\sigma ^{\varphi }$-KMS weight, we will show that $\psi $ is a scalar multiple of $\varphi $. Take any two compact open subgroups $K_{1}, K_{2} < G$. Define $K:=K_{1} \cap K_{2}$ and take $s_{1}, \dotsc , s_{l}, t_{1}, \dotsc , t_{r} \in G$ satisfying $K_{1}= \bigsqcup _{i=1}^{l} s_{i} K, K_{2} = \bigsqcup _{i=1}^{r} t_{i} K$. Then

$$ \begin{align*} p_{K_{1}}= \frac{1}{l}\sum_{i=1}^{l} \lambda_{s_{i}} p_{K},\qquad p_{K_{2}} = \frac{1}{r}\sum_{i=1}^{r} \lambda_{t_{i}} p_{K}. \end{align*} $$

Since $r \mu (K_{1})=l \mu (K_{2})$, the hypothesis implies

$$ \begin{align*} C:=\psi(p_{K_{1}}) \mu(K_{1})= \psi(p_{K_{2}}) \mu(K_{2}). \end{align*} $$

By [10, Lemma 2.23] (see also [8]), we obtain $\psi = C\varphi $. Next consider the case that $\mathcal G$ is nonunimodular and that $\psi $ is a tracial weight. In this case, the equality in the claim implies that the weight $\psi $ vanishes on $C_{cc}(\mathcal G)$. Since the projections $p_{K}$, where $K<G$ are compact open subgroups, form an approximate unit of $\mathop {{\mathrm C}_{\mathrm r}^{\ast }}(\mathcal G)$, it follows from the tracial condition and lower semicontinuity of $\psi $ that $\psi =0$. This proves the statement of the theorem. Hence it suffices to show the claim.

We now prove the claim. Let $\psi , K<G, s\in \mathcal G \setminus K$ be as in the claim. Write $s= g u$, $g\in \Lambda , u\in G$. To show the claimed equation $\psi (\lambda _{s} p_{K})=0$, we first recall from the proof of Theorem 3.1 that when $u \not \in K$, one has a nonzero projection $p \in \mathop {{\mathrm C}_{\mathrm r}^{\ast }}(\Gamma _{1})^{K}$ satisfying $p \lambda _{u} p=0$. In fact, p is taken from the group algebra $\mathbb C[\Gamma _{1}]$. When $u\in K$, define $p:= \lambda _{e} \in \mathbb C[\Gamma _{1}]$. Choose $n\in \mathbb N$ satisfying $g\in \Lambda _{n}$. Consider the subgroup $\Sigma := \Lambda _{n} \ast \Xi _{n+1} \ast \Xi _{n+2} < \Lambda $. Denote by $\tau $ the canonical tracial state on $\mathop {{\mathrm C}_{\mathrm r}^{\ast }}(\Sigma )$. As observed in [12, Lemma 3.8], thanks to [4, Lemma 5], one can proceed with the Powers averaging argument [9] for $\Sigma $ by using only elements in $\Xi _{n+1} \ast \Xi _{n+2}$. This implies that for any $\varepsilon>0$, one has $t_{1}, \dotsc , t_{r} \in \Xi _{n+1} \ast \Xi _{n+2}$ satisfying

$$ \begin{align*} \left\lVert\frac{1}{r}\sum_{i=1}^{r} \lambda_{t_{i}} x \lambda_{t_{i}}^{\ast} - \tau(x)\right\rVert< \varepsilon \quad\text{for }x= p, \lambda_{g}. \end{align*} $$

Then, as $\Xi _{n+1} \ast \Xi _{n+2}$ commutes with G, we have

$$ \begin{align*} \left\lVert\frac{1}{r^{2}} \sum_{i=1}^{r} \sum_{j=1}^{r} \lambda_{t_{i}} p \lambda_{t_{i}}^{\ast} \lambda_{t_{j}} p_{K} \lambda_{s} p_{K} \lambda_{t_{j}}^{\ast}\lambda_{t_{i}} p \lambda_{t_{i}}^{\ast}\right\rVert &=\left\lVert\frac{1}{r}\sum_{i=1}^{r}\left[\lambda_{t_{i}} p \lambda_{t_{i}}^{\ast} p_{K}\left (\frac{1}{r}\sum_{j=1}^{r} \lambda_{t_{j}} \lambda_{g} \lambda_{t_{j}}^{\ast}\right) \lambda_{u} p_{K} \lambda_{t_{i}} p \lambda_{t_{i}}^{\ast}\right]\right\rVert\\ &< \varepsilon + \frac{\tau\left(\lambda_{g}\right)}{r}\left\lVert\sum_{i=1}^{r}\lambda_{t_{i}} p_{K} p \lambda_{u} p p_{K} \lambda_{t_{i}}^{\ast}\right\rVert \\ &= \varepsilon. \end{align*} $$

Here the last equation holds true because the condition $u\in K$ implies $g \neq e$. Since p is a K-invariant projection and $\Sigma , K \subset \mathcal G_{0}$, the previous inequality yields

$$ \begin{align*} \left\lvert\psi\left(\lambda_{s} p_{K} \sum_{i=1}^{r} \sum_{j=1}^{r} \lambda_{t_{j}}^{\ast} \lambda_{t_{i}} p \lambda_{t_{i}}^{\ast} \lambda_{t_{j}} \right)\right\rvert &= \left\lvert\psi\left( \sum_{i=1}^{r} \sum_{j=1}^{r}\lambda_{s} p_{K} \left(\lambda_{t_{j}}^{\ast} \lambda_{t_{i}} p \lambda_{t_{i}}^{\ast}\right)\left(\lambda_{t_{i}} p \lambda_{t_{i}}^{\ast} \lambda_{t_{j}}p_{K}\right)\right)\right\rvert\\ &= \left\lvert\psi\left( \sum_{i=1}^{r} \sum_{j=1}^{r} \lambda_{t_{i}} p \lambda_{t_{i}}^{\ast} \lambda_{t_{j}} p_{K} \lambda_{s} p_{K} \lambda_{t_{j}}^{\ast}\lambda_{t_{i}} p \lambda_{t_{i}}^{\ast}\right)\right\rvert\\ &\leq r^{2} \psi(p_{K}) \varepsilon. \end{align*} $$

This yields

$$ \begin{align*} \lvert\tau(p)\psi(\lambda_{s} p_{K})\rvert &\leq \left\lvert\psi\left(p_{K} \lambda_{s} p_{K} \left(\tau(p)- \frac{1}{r^{2}} \sum_{i=1}^{r} \sum_{j=1}^{r} \lambda_{t_{j}}^{\ast} \lambda_{t_{i}} p \lambda_{t_{i}}^{\ast} \lambda_{t_{j}}\right)\right)\right\rvert + \psi(p_{K})\varepsilon \\ &\leq 2\psi(p_{K}) \varepsilon. \end{align*} $$

Since $\varepsilon>0$ was chosen arbitrarily small (independent on p), we conclude

$$ \begin{align*} \psi(\lambda_ s p_{K})=0. \end{align*} $$

5 On factoriality and types of group von Neumann algebras $L(\mathcal G)$

In this section we observe the factoriality of the group von Neumann algebra $L(\mathcal G)$. We then determine its Murray–von Neumann–Connes type. For the definition of Connes’ S-invariant, we refer the reader to [2, Section III].

Theorem 5.1. The von Neumann algebra $L(\mathcal G)$ is a nonamenable factor of type

$$ \begin{align*} \begin{cases} \textrm{II}_{1} &\text{when }G\text{ is discrete},\\ \textrm{II}_{\infty} &\text{when }G\text{ is nondiscrete and unimodular},\\ \textrm{III} &\text{otherwise}, \end{cases} \end{align*} $$

whose Connes’ S-invariant is the closure of $\Delta _{G}(G)$ in $\mathbb R_{\geq 0}$.

Proof. We first show that $L(\mathcal G)$ and $L(\mathcal G_{0})$ are factors. By [10, Proposition 2.25], the centraliser of the Plancherel weight on $L(\mathcal G)$ is equal to $L(\mathcal G_{0})$. Hence it suffices to show the factoriality of $L(\mathcal G_{0})$. Let $\varphi $ be the Plancherel weight on $L(\mathcal G_{0})$. Note that $\varphi $ is a faithful normal semifinite tracial weight on $L(\mathcal G_{0})$. Hence if $L(\mathcal G_{0})$ is not a factor, then one has a normal semifinite tracial weight $\psi $ which is dominated by $\varphi $ but is not a scalar multiple of $\varphi $. This contradicts the uniqueness of the tracial weight on $\mathop {{\mathrm C}_{\mathrm r}^{\ast }}(\mathcal G_{0})$ (up to scalar multiple), which follows from the proof of Theorem 4.1.

We next show the nonamenability of $L(\mathcal G)$. Take any compact open subgroup $K<G$. Then by Lemma 2.1, the corner $p_{K} L(\mathcal G)p_{K}$ of $L(\mathcal G)$ admits a conditional expectation

$$ \begin{align*} E_{K}\colon p_{K} L(\mathcal G) p_{K} \rightarrow L(\Lambda)^{K} p_{K}. \end{align*} $$

Since $L(\Lambda )^{K} p_{K}$ is nonamenable, so is $L(\mathcal G)$.

Finally we determine the Murray–von Neumann–Connes type of $L(\mathcal G)$. When G is discrete, it is clear from Theorem 4.1 that $L(\mathcal G)$ is of type $\text {II}_{1}$. (Alternatively, in the discrete-group case, the statement follows from the fact that $\mathcal G$ is a nonamenable ICC discrete group.) When G is nondiscrete and unimodular, observe that the Plancherel weight on $L(\mathcal G)$ is tracial and unbounded. Since $L(\mathcal G)$ is nonamenable, it must be of type $\text {II}_{\infty }$. The nonunimodular case and the last statement follow from Connes’ theorem [2] (see [10, Theorem 2.27]).