On Schauder Equivalence Relations

TL;DR

This paper investigates the structure and boundaries of Schauder equivalence relations generated by Banach spaces with basic sequences, revealing their complexity and mutual incompatibility, especially in relation to Tsirelson spaces.

Contribution

It establishes the boundaries of Schauder equivalence relations, analyzes their properties in Tsirelson spaces, and shows the cardinality of their bases.

Findings

Equivalence relations generated by basic sequences have boundaries.

Equivalence relations from Tsirelson space basis share properties with the space.

l_p and c_0 are not reducible to Tsirelson space equivalence relations.

Abstract

In this paper, we study Schauder equivalence relations, which are Borel equivalence relations generated by Banach spaces with basic sequences. We prove that the set of equivalence relations generated by basic sequences has boundaries. Then we show that equivalence relations generated by the basis in Tsirelson spaces has similar properties of Tsirelson spaces in the Banach space theory. In particular, we prove that both l_p and c_0 are not reducible to the equivalence relation generated by Tsirelson space T with the unit vector basis \{t_n\}. We also show that Borel equivalence relation generated by \alpha-Tsirelson spaces are mutually incompatible. Based on this argument, we show that any basis of Schauder equivalence relations must be of cardinal 2^\omega.