Borel reducibility and Holder($\alpha$) embeddability between Banach spaces

TL;DR

This paper explores the relationship between Borel reducibility of certain equivalence relations derived from Banach spaces and H"older embeddability, providing new results on reducibility and answering an open problem.

Contribution

It establishes connections between Borel reducibility and H"older embeddability, and demonstrates reducibility results for specific Banach space equivalence relations, including an affirmative answer to a known problem.

Findings

Borel reducibility relates to H"older embeddability between Banach spaces.

Many reducibility and unreducibility results for $E(L_r,p)$ and $E(c_0,p)$.

Confirmed that $C({R}^+)/C_0({R}^+)$ is Borel bireducible to ${R}^{N}/c_0$.

Abstract

We investigate Borel reducibility between equivalence relations $E(X,p)=X^{\Bbb N}/\ell_p(X)$'s where $X$ is a separable Banach space. We show that this reducibility is related to the so called H\"older$(\alpha)$ embeddability between Banach spaces. By using the notions of type and cotype of Banach spaces, we present many results on reducibility and unreducibility between $E(L_r,p)$'s and $E(c_0,p)$'s for $r,p\in[1,+\infty)$. We also answer a problem presented by Kanovei in the affirmative by showing that $C({\Bbb R}^+)/C_0({\Bbb R}^+)$ is Borel bireducible to ${\Bbb R}^{\Bbb N}/c_0$.