Traces on reduced group C*-algebras
Matthew Kennedy, Sven Raum

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.
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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 -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 has this property if and only if its amenable radical 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 be a locally compact group. The reduced C-algebra admits a non-zero trace if and only if the amenable radical of is open. Further, every trace concentrates on , meaning that it factors through the canonical conditional expectation from onto . *
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 be a locally compact group. Every trace on satisfies for every function with support disjoint from the amenable radical . *
Proof.
Let be a trace. We continue to denote by the unique extension of to a trace on the multiplier algebra . By normalizing , we can assume that it is unital. The fact that it is tracial implies that it is -equivariant. Hence by the -injectivity of , we can to a -equivariant unital completely positive map .
Proceeding as in [Bre+14], we now show that for , . By [Fur03, Proposition 7], acts non-trivially on , so there is such that . Let be any function satisfying and . Then
[TABLE]
So if has its support disjoint from , then we obtain
[TABLE]
by the strict continuity of .
Proof of Theorem 1.
Assume that the amenable radical of is not open and is a trace on . Let be a the filter of open neighbourhoods of . Because is not open, it does not contain any . So for every there is a positive function with support in the non-trivial open set satisfying .
The net is a Dirac net for and hence an approximate identity for . Since for all , we obtain from the lemma. Since is an ideal containing the approximate identity , it follows that .
Conversely, assume that the amenable radical of is open. Since is amenable, the left regular representation of on provides us with a *-representation of , since it is weakly contained in the left regular representation of . Its image generates the group von Neumann algebra . This von Neumann algebra is finite, since the openness of implies the discreteness of . We obtain a trace on by composing the representation on with the trace on .
Finally, for the last statement of the theorem, let be any trace on . Let denote the natural conditional expectation obtained from the restriction . For , the lemma gives
[TABLE]
Thus . Since is dense, and since and are continuous, it follows that .
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[Bre+14] Emmanuel Breuillard, Mehrdad Kalantar, Matthew Kennedy and Narutaka Ozawa “ C ∗ superscript C \mathrm{C}^{*} -simplicity and the unique trace property for discrete groups.”, 2014 eprint: ar Xiv:1410.2518
- 2[FSW 17] Brian E. Forrest, Nico Spronk and Matthew Wiersma “Existence of tracial states on reduced group C ∗ superscript C \mathrm{C}^{*} -algebras.”, Preprint, 2017 eprint: ar Xiv:1706.05354
- 3[Fur 03] Alex Furman “On minimal strongly proximal actions of locally compact groups.” In Isr. J. of Math. 136 , 2003, pp. 173–187 DOI: 10.1007/BF 02807197 · doi ↗
- 4[KS 52] Richard V. Kadison and Isadore M. Singer “Some remarks on representations of connected groups.” In Proc. Natl. Acad. Sci. USA 38 , 1952, pp. 419–423 DOI: 10.1073/pnas.38.5.419 · doi ↗
- 5[KK 14] Mehrdad Kalantar and Matthew Kennedy “Boundaries of reduced C ∗ superscript C \mathrm{C}^{*} -algebras of discrete groups.”, Accepted for publication in J. Reine Angew. Math., 2014 eprint: ar Xiv:1410.2518
