Bounds for Banach-Mazur distances between some $C(K)$-spaces

TL;DR

This paper establishes new lower bounds for the Banach-Mazur distances between certain spaces of continuous functions, especially focusing on spaces over convergent sequences and their products, refining previous bounds and exploring specific cases.

Contribution

It provides new lower bounds and refinements for Banach-Mazur distances between $C(K)$-spaces, especially for spaces over convergent sequences and their products.

Findings

Lower bounds for $d_{BM}(C(K), C(L))$ are established.

Refinements of previous results on Banach-Mazur distances are presented.

Specific bounds for $C([0, ext{"omega"}] imes 3)$ and $C([0, ext{"omega"}])$ are given.

Abstract

We present several results providing lower bounds for the Banach-Mazur distance \[d_{BM}(C(K), C(L))\] between Banach spaces of continuous functions on compact spaces. The main focus is on the case where $C(L)$ represents the classical Banach space $c$ of convergent sequences. In particular, we obtain generalizations and refinements of recent results from \cite{GP24} and \cite{MP25}. Currently, it seems that one of the most interesting questions is when $K = [0, \omega]$ is a convergent sequence with a limit and $L = [0,\omega]\times 3$ consists of three convergent sequences. In this case, we obtain \[3.53125 \leq d_{BM}(C([0,\omega]\times 3),C[0,\omega]) \leq 3.87513\]