Not too many miles from ${\ell }_{\infty }$ to $L_{\infty }$

Abstract

For either real or complex scalars, we improve the known upper bound for the Banach--Mazur distance between $L_{\infty }[0,1]$ and ${\ell }_{\infty }$, proving $d_{\mathrm{BM}}(L_{\infty }[0,1],{\ell }_{\infty })⩽(2+\sqrt{5}{)}^2=9+4\sqrt{5}=17.944\dots$. The additional input is a stabilised kernel estimate: the copy of ${\ell }_{\infty }$ in $L_{\infty }$ is chosen so that its kernel is an ${\ell }_{\infty }$-sum of singular two-point hyperplanes, and Cohen's bound-two construction is then applied coordinatewise. The quantitative part of the proof is an asymmetric Pełczyński-type factorisation, with the elementary optimisation included below.

arXiv comment6 pp, accepted for publication in Journal of Mathematical Sciences (Japan)