Bounds for Banach-Mazur distances between some $C(K)$-spaces
Maciej Korpalski, Grzegorz Plebanek
TL;DR
The paper develops a simple, adaptable method to bound the Banach–Mazur distance between $C(K)$ and $C(L)$, emphasizing the case $L=C[0,\omega]$ and the families $K=[0,\omega]\times k$. By combining a general lower-bound tool with weak-* measure analysis, it establishes universal lower bounds depending on the height of $K$ (e.g., $d_{BM}(C(K),C[0,\omega])\ge m+\sqrt{(m-1)(m+3)}$ when $K^{(m)}\neq\emptyset$) and sharpens bounds in the delicate $k=3$ case. For $k=3$, it proves a computer-assisted lower bound $d_{BM}\ge 3.53125$ and a near-optimal upper bound $d_{BM}\le 3.87512...$ via explicit isomorphisms; the work also provides exact GP24-style bounds for $k\ge2$. Collectively, the results advance the understanding of isomorphism classes among $C(K)$ spaces and illustrate effective use of linear-inequality solvers in Banach-space geometry.
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 $K = [0, ω]$ is a convergent sequence with a limit and $L = [0,ω]\times 3$ consists of three convergent sequences. In this case, we obtain \[3.53125 \leq d_{BM}(C([0,ω]\times 3),C[0,ω]) \leq 3.87513\]