Antidiagonal and square averages for amenable groups
Abstract
We prove that, for commuting unitary representations U,V of a countable amenable group G, the averages DF=1/|F|∑g∈ FUg*Vg have a common selfadjoint weak operator limit along all left and right \Folner{} sequences, with strong* convergence along two-sided sequences. This answers Bergelson's question on commuting actions. For any unitary representation π, we also prove that AF=|F|−1∑g∈ Fπ(g2) converges strongly along right \Folner{} sequences to a selfadjoint central operator supported on the finite type~I part of π(G)''; the adjoints AF* converge strongly along left \Folner{} sequences. This settles the convergence question for square averages posed by Bergelson and Rodríguez-Martín. For measure-preserving actions of finitely generated amenable or countable virtually FC groups, we obtain limn1/|Fn|∑g∈ Fnμ(A∩ Tg2A)>0 whenever μ(A)>0. For each ε>0, the set {g:μ(A∩ Tg2A)>μ(A)2−ε} meets every Bohr neighborhood of the identity in a set of positive lower density, uniformly over \Folner{} sequences. These intersections are also left and right IP* for virtually FC groups. We construct a unitary representation π of a locally finite nilpotent group and a weakly mixing vector f for which {g:Re⟨π(g2)f,f⟩>−ε} has upper Banach density zero for small ε>0. We also give a weakly mixing action of a two-generated solvable group for which the IP* conclusion fails. The recurrence results yield product sets BB inside difference sets, with B of positive upper density.
Related
- Groups with rapid decay and trivial amenable radical are C*-simple
- Selflessness for twisted group C*-algebras of amenable groups and their inclusions
- The universal valued Abelian groups of Niemiec are Lévy, strongly exotic, extremely amenable, and 𝔾r(0) is generically monothetic
- Comparison and Almost Finiteness for Actions of Amenable Groups
- Spherical Averages on nilpotent Groups
BibTeX
@misc{luo2026antidiagonal,
title = {{Antidiagonal and square averages for amenable groups}},
author = {Xiaofang Luo},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02901},
eprint = {2611.02901},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02901},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Antidiagonal and square averages for amenable groups
AU - Xiaofang Luo
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02901
UR - https://arxiv.org/abs/2611.02901
AB - We prove that, for commuting unitary representations U,V of a countable amenable group G, the averages D_F=1/|F|∑_g∈ FU_g^*V_g have a common selfadjoint weak operator limit along all left and right \Folner{} sequences, with strong* convergence along two-sided sequences. This answers Bergelson's question on commuting actions. For any unitary representation π, we also prove that A_F=|F|^−1∑_g∈ Fπ(g^2) converges strongly along right \Folner{} sequences to a selfadjoint central operator supported on the finite type~I part of π(G)''; the adjoints A_F^* converge strongly along left \Folner{} sequences. This settles the convergence question for square averages posed by Bergelson and Rodríguez-Martín. For measure-preserving actions of finitely generated amenable or countable virtually FC groups, we obtain lim_n1/|F_n|∑_g∈ F_nμ(A∩ T_g^2A)>0 whenever μ(A)>0. For each ε>0, the set {g:μ(A∩ T_g^2A)>μ(A)^2−ε} meets every Bohr neighborhood of the identity in a set of positive lower density, uniformly over \Folner{} sequences. These intersections are also left and right IP^* for virtually FC groups. We construct a unitary representation π of a locally finite nilpotent group and a weakly mixing vector f for which {g:Re⟨π(g^2)f,f⟩>−ε} has upper Banach density zero for small ε>0. We also give a weakly mixing action of a two-generated solvable group for which the IP^* conclusion fails. The recurrence results yield product sets BB inside difference sets, with B of positive upper density.
ER -