π-Properties, Uniformly Convexity and Uniform Ball Coverings Properties
Abstract
In this paper, we prove a sufficient condition for the uniform ball-covering property of closed operator subspaces invariant under selected truncations. We use uniform convexity of the range space and near-contractive finite-rank projection approximations to obtain quantitative coverings with finite-rank centres whenever the truncated images are separable. A weak-star version gives the same covering parameters for quotients by arbitrary weak-star closed subspaces. In the completely bounded setting, the covering parameters are independent of the matrix level. In particular, the duals of all closed subspaces of S1(ℓ2) admit coverings with centre norm $2$ and radius 1+ε, for 0<ε<1, at every matrix level. Applications include operator-valued Schur multipliers and, for 1<p<∞, prescribed-support Fourier multipliers on countable amenable groups and finitely generated free groups. The Fourier coverings have finite-support centres in the given operator or completely bounded norms. An endpoint example shows failure of the ball-covering property at p=1,∞.
Related
- Beyond Affine Layers: Bounded Ball Coverings to Preclude Gradient Vanishing
- The higher-dimensional Shepp problem: an exact criterion for random ball coverings of tori
- Convexity of the Berezin range of composition operators induced by automorphisms of the Unit Ball in ℂn
- Intractable enumeration problems are like Russian nesting dolls: structural properties of monomer-dimer coverings on two-dimensional quadratic lattices
- Uniform L∞-Boundedness of Global Attractors for Reaction-Diffusion Equations with Neumann boundary condition in Uniformly Perturbed Non-Smooth Domains
BibTeX
@misc{liu2026πproperties,
title = {{$π$-Properties, Uniformly Convexity and Uniform Ball Coverings Properties}},
author = {Rui Liu and Jie Shen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.12538},
eprint = {2607.12538},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.12538},
}
RIS
TY - UNPB
TI - π-Properties, Uniformly Convexity and Uniform Ball Coverings Properties
AU - Rui Liu
AU - Jie Shen
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2607.12538
UR - https://arxiv.org/abs/2607.12538
AB - In this paper, we prove a sufficient condition for the uniform ball-covering property of closed operator subspaces invariant under selected truncations. We use uniform convexity of the range space and near-contractive finite-rank projection approximations to obtain quantitative coverings with finite-rank centres whenever the truncated images are separable. A weak-star version gives the same covering parameters for quotients by arbitrary weak-star closed subspaces. In the completely bounded setting, the covering parameters are independent of the matrix level. In particular, the duals of all closed subspaces of S_1(ℓ_2) admit coverings with centre norm $2$ and radius 1+ε, for 0<ε<1, at every matrix level. Applications include operator-valued Schur multipliers and, for 1<p<∞, prescribed-support Fourier multipliers on countable amenable groups and finitely generated free groups. The Fourier coverings have finite-support centres in the given operator or completely bounded norms. An endpoint example shows failure of the ball-covering property at p=1,∞.
ER -