Banach-Stone-like results for combinatorial Banach spaces
Christina Brech, Claribet Pi\~na

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 and on some infinite cardinal , the corresponding combinatorial spaces and are isometric if and only if there is a permutation of inducing a homeomorphism between and . We also prove that two different regular families and on cannot be permuted one to the other. Both these results strengthen the main result of \cite{BrechFerencziTcaciuc}.
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