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Borel reducibility and Hölder\((\alpha )\) embeddability between Banach spaces. (English) Zbl 1250.03082

Summary: 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. 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 \(\mathbb R^{\mathbb N}/c_{0}\).

MSC:

03E15 Descriptive set theory
46B20 Geometry and structure of normed linear spaces
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