Incomparable, non-isomorphic and minimal Banach spaces

Abstract

Abstract. A Banach space contains either a minimal subspace or a continuum of incomparable subspaces. General structure results for analytic equivalence relations are applied in the context of Banach spaces to show that if E 0 does not reduce to isomorphism of the subspaces of a space, in particular, if the subspaces of the space admit a classification up to isomorphism by real numbers, then any subspace with an unconditional basis is isomorphic to its square and hyperplanes, and the unconditional basis has an isomorphically homogeneous subsequence.

Some equivalence relations which are borel reducible to isomorphism between separable banach spaces

“…Rosendal [30] noticed that hereditarily indecomposable Banach spaces are ergodic, and also proved that an unconditional basis of a non-ergodic Banach space must have a subsequence such that all further subsequences span isomorphic susbpaces. In [9] and [30] it is proved that a non-ergodic Banach space X with an unconditional basis must be isomorphic to its square, its hyperplanes and more generally to X ⊕ Y for any subspace Y generated by a subsequence of the basis.…”

“…The crucial point in our method is that the spaces considered are isomorphic if and only if their canonical bases are equivalent. As Rosendal proved that equivalence of Schauder bases is K σ -complete [30], we cannot hope to go further up in the hierarchy of complexity than K σ with this method. So we now turn to a situation where isomorphism corresponds to permutative equivalence of the canonical bases.…”

“…Rosendal [30] proved that E Kσ is Borel reducible to equivalence between Schauder basis in Banach spaces. We start by rephrasing his proof in a slightly higher generality (Lemma 2.1 and Corollary 2.2).…”

“…In [9] and [30] it is proved that a non-ergodic Banach space X with an unconditional basis must be isomorphic to its square, its hyperplanes and more generally to X ⊕ Y for any subspace Y generated by a subsequence of the basis. This was already obtained by Kalton [20] for spaces with an unconditional basis and only countably many classes.…”

“…Indeed: by Gowers' dichotomy theorem [15] and the fact that H.I. spaces are ergodic [30], we may assume that there exists a subspace X 1 with an unconditional basis. By the previously mentioned theorem of Komorowski and Tomczak-Jaegermann [24], some subspace X 2 of X 1 does not have an unconditional basis, but has a basis (or at least a FDD in the case when X does not have non-trivial cotype), from which it follows that it has the approximation property [25].…”