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

Abstract

We investigate Borel reducibility between equivalence relations $E(X,p)=X^{\mathbb{N}}/{\ell }_p(X)$'s where $X$ is a separable Banach space. We show that this reducibility is related to the so called Hölder$(\alpha )$ 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 $C({\mathbb{R}}^{+})/C_0({\mathbb{R}}^{+})$ is Borel bireducible to ${\mathbb{R}}^{\mathbb{N}}/c_0$.

29 pages