Traces on reduced group C*-algebras

TL;DR

This paper characterizes when the reduced group C*-algebra admits a non-zero trace, linking it precisely to the openness of the group's amenable radical, thus resolving a previously posed question.

Contribution

It provides a complete characterization of the existence of non-zero traces on reduced group C*-algebras in terms of the group's amenable radical being open.

Findings

Reduced group C*-algebra admits a non-zero trace iff the amenable radical is open.

Answers a question posed by Forrest, Spronk, and Wiersma.

Clarifies the relationship between group properties and C*-algebra traces.

Abstract

In this short note we prove that the reduced group C*-algebra of a locally compact group admits a non-zero trace if and only if the amenable radical of the group is open. This completely answers a question raised by Forrest, Spronk and Wiersma.

Equations

$$\phi (u_{\gamma })=\psi (x)\phi (u_{\gamma })=(\phi (\psi )\phi (u_{\gamma }))(x)=\phi (\psi u_{\gamma })(x)=\phi (u_{\gamma }{\psi }^{\gamma })(x)=\phi (u_{\gamma })\psi (\gamma x)=0\text{.}$$

$$\tau (f)={\int }_{G}f(\gamma )\phi (u_{\gamma })\mathrm{d}g=0\text{,}$$

$$\tau (f)=\tau (1_{\mathrm{Rad}(G)}f)+\tau (1_{G\setminus \mathrm{Rad}(G)}f)=\tau (1_{\mathrm{Rad}(G)}f)=\tau \circ \mathrm{E}(f)\text{.}$$

Full text

Traces on reduced group C-algebras*

by Matthew Kennedy and Sven Raum

††footnotetext: last modified on March 13, 2024††footnotetext: MK research was supported by NSERC Grant Number 418585††footnotetext: MSC classification: 22D25; 46L30 ††footnotetext: Keywords: Traces on group $\mathrm{C}^{*}$-algebras, Furstenberg boundary

Abstract. In this short note we prove that the reduced group C*-algebra of a locally compact group admits a non-zero trace if and only if the amenable radical of the group is open. This completely answers a question raised by Forrest, Spronk and Wiersma.

Introduction

An important fact about the reduced C*-algebra of a discrete group is that it admits at least one non-zero trace. More generally, the reduced C*-algebra of a locally compact group may admit no non-zero traces at all. This is one reason why discrete groups are generally considered to be more tractable in the theory of group C*-algebras.

In a recent preprint, Forrest, Spronk and Wiersma [FSW17, Question 1.1] ask for a characterization of the locally compact groups with reduced C*-algebras that admit a non-zero trace. They provide a partial answer to this question by proving that a compactly generated locally compact group $G$ has this property if and only if its amenable radical $\operatorname{Rad}(G)$ is open.

In this note, we completely settle this question by proving that the result of Forrest-Spronk-Wiersma holds without the assumption that the group is compactly generated. Further, we prove that any trace on the reduced C*-algebra concentrates on the amenable radical.

Theorem 1**.**

Let $G$ be a locally compact group. The reduced C-algebra $\mathrm{C}^{*}_{\mathrm{red}}(G)$ admits a non-zero trace if and only if the amenable radical $\operatorname{Rad}(G)$ of $G$ is open. Further, every trace concentrates on $\operatorname{Rad}(G)$, meaning that it factors through the canonical conditional expectation from $\mathrm{C}^{*}_{\mathrm{red}}(G)$ onto $\mathrm{C}^{*}_{\mathrm{red}}(\operatorname{Rad}(G))$. *

We view Theorem 1 as the natural generalization to locally compact groups of [Bre+14, Theorem 4.1], which states that every trace on the reduced C*-algebra of a discrete group concentrates on the amenable radical.

Our approach to the proof is much different than the approach taken in [FSW17]. We are motivated by the perspective introduced in [KK14], which relates the structure of the reduced group C*-algebra of a discrete group to the dynamics of the topological Furstenberg boundary. In the present setting, it is also necessary to handle the technical difficulties that arise for non-discrete groups.

Theorem 1 immediately yields a characterization of locally compact groups that admit finite weakly regular unitary representations. Recall that a representation is weakly regular if it is weakly contained in the left regular representation.

Corollary 2**.**

*A locally compact group admits a finite weakly regular representation if and only if its amenable radical is open. *

Corollary 2 can be seen as an analogue of a classical result of Kadison and Singer [KS52, Corollary 3] which characterizes the connected locally compact groups without any finite representation.

Acknowledgement

We are grateful to Brian Forrest, Nico Spronk and Matthew Wiersma for sharing a preprint of their work [FSW17].

Proof of Theorem 1

We first prove a generalization to locally compact groups of [Bre+14, Theorem 4.1].

Lemma**.**

*Let $G$ be a locally compact group. Every trace $\tau$ on $\mathrm{C}^{*}_{\mathrm{red}}(G)$ satisfies $\tau(f)=0$ for every function $f\in\mathrm{C}_{\mathrm{c}}(G)$ with support disjoint from the amenable radical $\operatorname{Rad}(G)$. *

Proof.

Let $\tau:\mathrm{C}^{*}_{\mathrm{red}}(G)\rightarrow\mathbb{C}$ be a trace. We continue to denote by $\tau$ the unique extension of $\tau$ to a trace on the multiplier algebra $\mathrm{M}(\mathrm{C}^{*}_{\mathrm{red}}(G))$. By normalizing $\tau$, we can assume that it is unital. The fact that $\tau$ it is tracial implies that it is $G$-equivariant. Hence by the $G$-injectivity of $\mathrm{C}(\partial_{\mathrm{F}}G)$, we can $\tau$ to a $G$-equivariant unital completely positive map $\varphi:\mathrm{M}(\mathrm{C}(\partial_{\mathrm{F}}G)\rtimes_{\mathrm{red}}G)\rightarrow\mathrm{C}(\partial_{\mathrm{F}}G)$.

Proceeding as in [Bre+14], we now show that for $\gamma\in G\setminus\operatorname{Rad}(G)$, $\varphi(u_{\gamma})=0$. By [Fur03, Proposition 7], $\gamma$ acts non-trivially on $\partial_{\mathrm{F}}G$, so there is $x\in\partial_{\mathrm{F}}G$ such that $\gamma x\neq x$. Let $\psi\in\mathrm{C}(\partial_{\mathrm{F}}G)$ be any function satisfying $\psi(x)=1$ and $\psi(\gamma x)=0$. Then

[TABLE]

So if $f\in\mathrm{C}_{\mathrm{c}}(G)\subset\mathrm{C}^{*}_{\mathrm{red}}(G)$ has its support disjoint from $\operatorname{Rad}(G)$, then we obtain

[TABLE]

by the strict continuity of $\varphi$.

Proof of Theorem 1.

Assume that the amenable radical of $G$ is not open and $\tau$ is a trace on $\mathrm{C}^{*}_{\mathrm{red}}(G)$. Let $\mathcal{N}$ be a the filter of open neighbourhoods of $e\in G$. Because $\operatorname{Rad}(G)$ is not open, it does not contain any $U\in\mathcal{N}$. So for every $U\in\mathcal{N}$ there is a positive function $f_{U}\in\mathrm{C}_{\mathrm{c}}(G)$ with support in the non-trivial open set $U\cap\operatorname{Rad}(G)^{\mathrm{c}}$ satisfying $\int_{G}f=1$.

The net $(f_{U})_{U\in\mathcal{N}}$ is a Dirac net for $G$ and hence an approximate identity for $\mathrm{C}^{*}_{\mathrm{red}}(G)$. Since $\mathop{\mathrm{supp}}f_{U}\cap\operatorname{Rad}(G)=\emptyset$ for all $U\in\mathcal{N}$, we obtain $\tau(f_{U})=0$ from the lemma. Since $\{x\in\mathrm{C}^{*}_{\mathrm{red}}(G)\mid\tau(x^{*}x)=0\}$ is an ideal containing the approximate identity $(f_{U})_{U\in\mathcal{N}}$, it follows that $\tau\equiv 0$.

Conversely, assume that the amenable radical $\operatorname{Rad}(G)$ of $G$ is open. Since $\operatorname{Rad}(G)$ is amenable, the left regular representation of $G/\operatorname{Rad}(G)$ on $\ell^{2}(G/\operatorname{Rad}(G))$ provides us with a *-representation of $\mathrm{C}^{*}_{\mathrm{red}}(G)$, since it is weakly contained in the left regular representation of $G$. Its image generates the group von Neumann algebra $\mathrm{L}(G/\operatorname{Rad}(G))\subset\mathcal{B}(\ell^{2}(G/\operatorname{Rad}(G))$. This von Neumann algebra is finite, since the openness of $\operatorname{Rad}(G)$ implies the discreteness of $G/\operatorname{Rad}(G)$. We obtain a trace on $\mathrm{C}^{*}_{\mathrm{red}}(G)$ by composing the representation on $\ell^{2}(G/\operatorname{Rad}(G))$ with the trace on $\mathrm{L}(G/\operatorname{Rad}(G))$.

Finally, for the last statement of the theorem, let $\tau$ be any trace on $\mathrm{C}^{*}_{\mathrm{red}}(G)$. Let $\mathrm{E}:\mathrm{C}^{*}_{\mathrm{red}}(G)\rightarrow\mathrm{C}^{*}_{\mathrm{red}}(\operatorname{Rad}(G))$ denote the natural conditional expectation obtained from the restriction $\mathrm{C}_{\mathrm{c}}(G)\rightarrow\mathrm{C}_{\mathrm{c}}(\operatorname{Rad}(G))$. For $f\in\mathrm{C}_{\mathrm{c}}(G)$, the lemma gives

[TABLE]

Thus $\tau|_{\mathrm{C}_{\mathrm{c}}(G)}=\tau\circ\mathrm{E}|_{\mathrm{C}_{\mathrm{c}}(G)}$. Since $\mathrm{C}_{\mathrm{c}}(G)\subset\mathrm{C}^{*}_{\mathrm{red}}(G)$ is dense, and since $\tau$ and $\tau\circ\mathrm{E}$ are continuous, it follows that $\tau=\tau\circ\mathrm{E}$.