Banach-Stone-like results for combinatorial Banach spaces

TL;DR

This paper establishes conditions under which combinatorial Banach spaces are isometric, linking this to permutations of underlying families, and shows that certain families cannot be permuted onto each other, extending previous results.

Contribution

It provides new Banach-Stone-like theorems for combinatorial Banach spaces, connecting isometries to permutations of hereditary families under topological assumptions.

Findings

Spaces are isometric iff a permutation induces a homeomorphism between families.

Different regular families on ω cannot be permuted onto each other.

Results strengthen previous Banach-Stone theorems for combinatorial spaces.

Abstract

We show that under a certain topological assumption on two compact hereditary families $\F$ and $\G$ on some infinite cardinal $\kappa$, the corresponding combinatorial spaces $X_\F$ and $X_\G$ are isometric if and only if there is a permutation of $\kappa$ inducing a homeomorphism between $\F$ and $\G$. We also prove that two different regular families $\F$ and $\G$ on $\omega$ cannot be permuted one to the other. Both these results strengthen the main result of \cite{BrechFerencziTcaciuc}.