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On Schauder equivalence relations
Author(s) -
Ma Xin
Publication year - 2017
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201600074
Subject(s) - mathematics , equivalence relation , schauder basis , banach space , borel equivalence relation , equivalence (formal languages) , pure mathematics , cardinality (data modeling) , quotient , discrete mathematics , borel measure , probability measure , computer science , data mining
We study Schauder equivalence relations, which are Borel equivalence relations induced by actions of Banach spaces with Schauder bases. Firstly, we show thatR ω / ℓ 1andR ω / c 0are minimal Schauder equivalence relations. Then, we prove that neither of them is Borel reducible to the quotientR ω / T where T is the Tsirelson space. This implies that they cannot form a basis for the Schauder equivalence relations. In addition, we apply an argument of Farah to show that every basis for the Schauder equivalence relations, if such exist, has to be of cardinality 2 ℵ 0 .