Raja's covering index of $L_p$ spaces

TL;DR

This paper investigates Raja's covering index for classical and non-commutative $L_p$ spaces, providing exact calculations, bounds, and asymptotic behavior, thus advancing understanding of covering properties in various Banach space settings.

Contribution

The paper offers explicit formulas, bounds, and asymptotic estimates for Raja's covering index in classical and non-commutative $L_p$ spaces, including exact results for Hilbert spaces and new bounds for non-commutative spaces.

Findings

Exact covering index for Hilbert spaces: $ heta_H(n)=n^{-1/2}$

Upper bounds for $L_p$ spaces: $ heta_{L_p}(n) o n^{-1/p}$

Lower bounds for non-commutative $L_p$ spaces: $ heta_{L_p(M, au)}(n) o n^{-1/r}$

Abstract

We study Raja's covering index $\Theta_X(n)$ for classical $L_p$-spaces and their non-commutative counterparts. For infinite-dimensional Hilbert spaces we compute the covering index exactly, proving \[ \Theta_H(n)=n^{-1/2}\qquad(n\in\mathbb N); \] in particular $\Theta_H(2)=1/\sqrt2$, thus answering a question of Raja about the precise two-piece covering index of $\elltwo$. For scalar-valued Lebesgue spaces $L_p(\mu)$, $1\le p<\infty$, we construct an explicit block decomposition of the unit ball yielding the upper bound $\Theta_{L_p(\mu)}(n)\le n^{-1/p}$ for all $n\in\mathbb{N}$; in particular $\Theta_{\ell_p}(n)\le n^{-1/p}$. For $1<p<\infty$, under the corresponding $p$-AUS renormability hypothesis, this combines with Raja's general lower bound to give the sharp asymptotic estimate $\Theta_{L_p(\mu)}(n)\asymp n^{-1/p}$. We also obtain uniform upper bounds $\Theta_{L_p(\mu;E)}(n)\le n^{-1/p}$ for Bochner spaces $L_p(\mu;E)$ over non-atomic $\sigma$-finite measure spaces, with constants independent of the Banach space $E$; this shows that, at the level of power-type upper estimates, the covering index decays at the same rate regardless of the asymptotic geometry of~$E$ and provides a partial negative answer to a problem of Raja. Finally, using non-commutative Clarkson inequalities, we derive power-type lower bounds $\Theta_{L_p(M,\tau)}(n)\gtrsim n^{-1/r}$ for non-commutative $L_p(M,\tau)$ spaces associated with semifinite von Neumann algebras, where $r=\min\{p,2\}$. We do not attempt to optimise the exponent or constants in the non-commutative setting.