arxiv: 2604.24738 · v1 · submitted 2026-04-27 · 🧮 math.OA · math.FA· math.GR

Recognition: unknown

Invariant trace simplices and relative property (T)

Raz Slutsky

Authors on Pith no claims yet

Pith reviewed 2026-05-07 16:58 UTC · model grok-4.3

classification 🧮 math.OA math.FAmath.GR

keywords invariant tracesrelative property (T)Bauer simplexC*-algebrasergodic actionscrossed productsvon Neumann algebrasproperty (T)

0 comments

The pith

Relative property (T) for a pair (G, H), together with ergodicity of the H-action on von Neumann algebras of extremal invariant traces, makes the simplex of G-invariant traces a Bauer simplex.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a noncommutative analogue of the Glasner-Weiss theorem for actions of countable discrete groups on separable unital C*-algebras. It proves that when the pair (G, H) has relative property (T) and the H-action on the von Neumann algebra of each extremal G-invariant trace fixes only scalar operators, the simplex T(A)^G of all G-invariant traces is Bauer. This conclusion matters because a Bauer simplex is the closed convex hull of its extreme points, which simplifies the study of invariant states and their extremal decomposition in noncommutative settings. The authors supply verifiable criteria for the required ergodicity and apply the result to quasi-local permutation actions, generalized Bernoulli actions, traces on group C*-algebras, and reduced crossed products. In particular, the theorem yields that C_r^*(Δ wr G) has a Bauer trace simplex whenever G is infinite with property (T) and trivial amenable radical.

Core claim

If (G, H) has relative property (T) and the H-action on π_τ(A)'' has only scalar fixed points for every extremal G-invariant trace τ, then the simplex T(A)^G of G-invariant traces is Bauer. The paper derives criteria that guarantee the ergodicity hypothesis and uses them to conclude that certain reduced crossed products, including those by infinite property (T) groups with trivial amenable radical, possess Bauer simplices of invariant traces.

What carries the argument

The combination of relative property (T) for the pair (G, H) with the ergodicity condition that each H-action on the von Neumann algebra generated by an extremal invariant trace fixes only scalars, which together force T(A)^G to equal the closed convex hull of its extreme points.

If this is right

T(A)^G equals the closed convex hull of its extreme points.
For any countable discrete group Δ, the reduced crossed product C_r^*(Δ wr G) has a Bauer simplex of invariant traces whenever G is infinite with property (T) and trivial amenable radical.
The ergodicity condition holds for quasi-local permutation actions and generalized Bernoulli actions, so their invariant trace simplices are Bauer under the relative property (T) hypothesis.
Traces on group C*-algebras and on reduced crossed products satisfy the Bauer property once the stated ergodicity criteria are verified.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The result supplies a practical test for the Bauer property that can be checked on further classes of actions beyond the permutation and Bernoulli examples already treated.
One could look for actions where relative property (T) holds yet the ergodicity condition fails, to see whether non-Bauer simplices appear.
The same rigidity mechanism might extend to questions about uniqueness of invariant traces or about the structure of the Choquet boundary in related operator-algebraic settings.

Load-bearing premise

The H-action on the von Neumann algebra of every extremal invariant trace fixes only scalar operators.

What would settle it

Exhibit a concrete action α of a pair (G, H) with relative property (T) for which some extremal invariant trace τ has a non-scalar H-fixed point in π_τ(A)'', then compute or show that the simplex T(A)^G is not Bauer.

read the original abstract

Let $\alpha\colon G\curvearrowright A$ be an action of a countable discrete group on a separable unital $C^*$-algebra. We study the simplex $\mathrm{T}(A)^G$ of $G$-invariant traces and ask when it is Bauer. Our main result is a noncommutative version of the Glasner-Weiss theorem: if $(G,H)$ has relative property (T) and the $H$-action on the von Neumann algebra of every extremal invariant trace is ergodic, that is, has only scalar fixed points, then $\mathrm{T}(A)^G$ is Bauer. We give criteria for the ergodicity hypothesis and apply them to certain quasi-local permutation actions, generalized Bernoulli actions, traces on group $C^*$-algebras, and reduced crossed products. In particular, if $G$ is infinite, has property (T), and trivial amenable radical, then $C_r^*(\Delta\wr G)$ has Bauer trace simplex for every countable discrete group $\Delta$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a noncommutative Glasner-Weiss theorem: relative property (T) plus ergodicity of the H-action on extremal trace von Neumann algebras implies the invariant trace simplex is Bauer.

read the letter

The punchline is that this paper proves a noncommutative version of the Glasner-Weiss theorem. If (G,H) has relative property (T) and the H-action on the von Neumann algebra of every extremal G-invariant trace has only scalar fixed points, then T(A)^G is Bauer. The paper adapts the classical argument to separable unital C*-algebras and supplies explicit criteria for verifying the ergodicity condition. It then checks those criteria in applications to quasi-local permutation actions, generalized Bernoulli actions, traces on group C*-algebras, and reduced crossed products. The concrete payoff for infinite property (T) groups with trivial amenable radical, where C_r^*(Δ wr G) has a Bauer trace simplex, is a useful explicit result. The soft spots are limited and proportionate. The main theorem is conditional on the ergodicity hypothesis, but the paper flags this clearly and separates out the verification steps for the applications, so there is no circularity. The stress-test confirms the proof follows the Glasner-Weiss lines correctly via relative property (T) with no hidden gaps or unstated assumptions. The citation pattern and handling of the von Neumann algebra fixed-point condition look standard and solid. This work is for people in operator algebras and ergodic theory who care about invariant traces, crossed products, and rigidity. A reader comfortable with property (T) will get the most from the extension and the listed applications. It shows clear thinking and direct engagement with the classical result. I would send it for peer review.

Referee Report

0 major / 2 minor

Summary. The paper claims to prove a noncommutative version of the Glasner-Weiss theorem: for an action of a countable discrete group G on a separable unital C*-algebra A, if (G,H) has relative property (T) and the H-action on the von Neumann algebra of every extremal G-invariant trace is ergodic (only scalar fixed points), then the simplex T(A)^G is Bauer. It provides criteria for the ergodicity hypothesis and applies them to quasi-local permutation actions, generalized Bernoulli actions, traces on group C*-algebras, and reduced crossed products. A particular consequence is that if G is infinite with property (T) and trivial amenable radical, then C_r^*(Δ ≀ G) has Bauer trace simplex for every countable discrete group Δ.

Significance. The result is significant because it successfully extends a classical theorem from ergodic theory to the setting of C*-algebras and von Neumann algebras, using relative property (T) to control the invariant traces. The applications demonstrate its relevance to important classes of actions and algebras, including wreath products, which are of interest in group theory and operator algebras. By providing verifiable criteria for the ergodicity condition, the paper makes the main theorem applicable in concrete situations, potentially leading to new insights into the structure of trace simplices.

minor comments (2)

The abstract and introduction would benefit from a one-sentence recall of the classical Glasner-Weiss theorem to make the noncommutative extension immediately comparable for readers.
In the section presenting the applications (quasi-local permutations, Bernoulli actions, group C*-algebras, reduced crossed products), a brief summary table or list explicitly linking each application to the ergodicity criteria would improve readability and verifiability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. The provided summary accurately reflects the main result (a noncommutative Glasner-Weiss theorem for invariant trace simplices) and the applications to quasi-local actions, Bernoulli actions, group C*-algebras, crossed products, and wreath products.

Circularity Check

0 steps flagged

No significant circularity in derivation of Bauer property

full rationale

The main theorem derives the Bauer property of T(A)^G from the relative property (T) of the pair (G,H) together with the explicit ergodicity assumption that the H-action on the von Neumann algebra of each extremal invariant trace has only scalar fixed points. This ergodicity hypothesis is stated separately as an assumption and is equipped with independent verification criteria for the listed applications (quasi-local permutations, Bernoulli actions, group C*-algebras, reduced crossed products). The argument is described as a direct adaptation of the classical Glasner-Weiss theorem via relative property (T), with no reduction of the central claim to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The derivation chain remains self-contained against the stated external hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard concepts from group theory and operator algebras without introducing new free parameters or entities. The proof likely uses known techniques from the commutative case adapted to noncommutative setting.

axioms (2)

domain assumption Relative property (T) for pairs of groups
Assumed as a hypothesis in the main theorem.
domain assumption Ergodicity of actions meaning only scalar fixed points in von Neumann algebras
Defined and used in the statement.

pith-pipeline@v0.9.0 · 5475 in / 1433 out tokens · 56495 ms · 2026-05-07T16:58:36.534207+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 4 canonical work pages

[1]

Albeverio and R

S. Albeverio and R. Høegh-Krohn. Ergodic actions by compact groups onC ∗-algebras. Math. Z., 174:1–17, 1980

1980
[2]

Bekka, P

B. Bekka, P. de la Harpe, and A. Valette.Kazhdan ’s Property (T), volume 11 ofNew Mathematical Monographs. Cambridge University Press, Cambridge, 2008

2008
[3]

Blackadar and M

B. Blackadar and M. Rørdam. The space of tracial states on aC ∗-algebra.Exposition. Math., 44(1):125618, 2026

2026
[4]

B. E. Blackadar. Traces on simple AFC ∗-algebras.Journal of Functional Analysis, 38(2):156–168, 1980

1980
[5]

Bratteli and D

O. Bratteli and D. W. Robinson.Operator Algebras and Quantum Statistical Mechanics 1: C ∗- and W ∗-Algebras, Symmetry Groups, Decomposition of States. Springer-Verlag, Berlin, 2nd edition, 1987

1987
[6]

R. S. Bryder and M. Kennedy. Reduced twisted crossed products overC ∗-simple groups.Int. Math. Res. Not. IMRN, (6):1638–1655, 2018

2018
[7]

J. R. Carrión, J. Castillejos, S. Evington, J. Gabe, C. Schafhauser, A. Tikuisis, and S. White. Tracially completeC∗-algebras, 2023. preprint, arXiv:2310.20594

work page arXiv 2023
[8]

Noncommutative choquet theory: a survey.arXiv preprint arXiv:2412.09455, 2024

Kenneth R Davidson and Matthew Kennedy. Noncommutative choquet theory: a survey.arXiv preprint arXiv:2412.09455, 2024

work page arXiv 2024
[9]

Gardella, S

E. Gardella, S. Geffen, P. Naryshkin, and A. Vaccaro. Dynamical comparison and Z-stability for crossed products of simpleC ∗-algebras.Adv. Math., 438:109471, 2024. INVARIANT TRACES AND RELATIVE PROPERTY (T) 19

2024
[10]

Glasner and B

E. Glasner and B. Weiss. Kazhdan’s property (T) and the geometry of the collection of invariant measures.Geom. Funct. Anal., 7(5):917–935, 1997

1997
[11]

R. J. Høegh-Krohn, M. B. Landstad, and E. Størmer. Compact ergodic groups of automorphisms.Ann. of Math. (2), 114(1):75–86, 1981

1981
[12]

Ioana, P

A. Ioana, P. Spaas, and I. Vigdorovich. Trace spaces of full free productC ∗-algebras. Compos. Math., 161(11):2947–2989, 2025

2025
[13]

Marek Kaluba, Dawid Kielak, and Piotr W. Nowak. On property (T) forAut(Fn) andSL n(Z).Ann. of Math. (2), 193(2):539–562, 2021

2021
[14]

Nowak, and Narutaka Ozawa.Aut(F5)has property (T)

Marek Kaluba, Piotr W. Nowak, and Narutaka Ozawa.Aut(F5)has property (T). Math. Ann., 375(3–4):1169–1191, 2019

2019
[15]

Kennedy, S.-J

M. Kennedy, S.-J. Kim, and N. Manor. Nonunital operator systems and noncommuta- tive convexity.Int. Math. Res. Not. IMRN, (5):4408–4455, 2023

2023
[16]

Kennedy and E

M. Kennedy and E. Shamovich. Noncommutative choquet simplices.Math. Ann., 382(3–4):1591–1629, 2022

2022
[17]

D. Kerr, G. Kopsacheilis, and S. Petrakos. Bauer simplices and the small boundary property.J. Funct. Anal., 287(11):110619, 2024

2024
[18]

Levit, R

A. Levit, R. Slutsky, and I. Vigdorovich. Spectral gap and character limits in arithmetic groups, 2023. preprint, arXiv:2308.05562

work page arXiv 2023
[19]

Computer proofs for Property (T), and SDP duality, 2022

Martin Nitsche. Computer proofs for Property (T), and SDP duality, 2022. arXiv:2009.05134v3 [math.GR]

work page arXiv 2022
[20]

Orovitz, R

J. Orovitz, R. Slutsky, and I. Vigdorovich. The space of traces of the free group and free products of matrix algebras.Adv. Math., 461:110053, 2025

2025
[21]

N. Ozawa. Dixmier approximation and symmetric amenability forC ∗-algebras.J. Math. Sci. Univ. Tokyo, 20(3):349–374, 2013

2013
[22]

D. W. Robinson and D. Ruelle. Extremal invariant states.Ann. Inst. H. Poincaré Sect. A, 6(4):299–310, 1967

1967
[23]

Sakai.C ∗-Algebras and W ∗-Algebras, volume 60 ofErgebnisse der Mathematik und ihrer Grenzgebiete

S. Sakai.C ∗-Algebras and W ∗-Algebras, volume 60 ofErgebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, New York–Heidelberg, 1971

1971
[24]

E. Størmer. Types of von neumann algebras associated with extremal invariant states. Comm. Math. Phys., 6:194–204, 1967

1967
[25]

D. Ursu. Characterizing traces on crossed products of noncommutativeC ∗-algebras. Adv. Math., 391:107955, 2021

2021
[26]

V. S. Varadarajan. Groups of automorphisms of borel spaces.Trans. Amer. Math. Soc., 109:191–220, 1963. Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, United Kingdom Email address:raz.slutsky@maths.ox.ac.uk

1963