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On the number of non-isomorphic subspaces of a Banach space. (English) Zbl 1082.46008

The main result proved here states that if \(X\) is a Banach space with unconditional basis \((e_i)_{i\in\mathbb{N}}\), then either there exists a perfect set \(P\) of infinite subsets of \(\mathbb{N}\) such that for any two distinct \(A,B\in P\), \([e_i]_{i\in A}\not=[e_i]_{i\in B}\), or for a residual set of infinite subsets \(A\) of \(\mathbb{N}\), \([e_i]_{i\in A}\) is isomorphic to \(X\), and in that case, \(X\) is isomorphic to its square, to its hyperplanes, uniformly isomorphic to \(X\oplus[e_i]_{i\in D}\) for any \(D\subset\mathbb{N}\), and isomorphic to a denumerable Schauder decomposition into uniformly isomorphic copies of itself. As a corollary, the authors obtain the following Gowers-type dichotomy: any Banach space contains \(2^{\mathbb{N}}\) pairwise non-isomorphic subspaces or is saturated with subspaces isomorphic to their squares.

MSC:

46B03 Isomorphic theory (including renorming) of Banach spaces
03E15 Descriptive set theory
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