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Minimality, homogeneity and topological 0-1 laws for subspaces of a Banach space

Valentin Ferenczi
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Abstract

If a Banach space is saturated with basic sequences whose linear span embeds into the linear span of any subsequence, then it contains a minimal subspace. It follows that any Banach space is either ergodic or contains a minimal subspace. For a Banach space $X$ with an (unconditional) basis, topological 0-1 law type dichotomies are stated for block-subspaces of $X$ as well as for subspaces of $X$ with a successive FDD on its basis. A uniformity principle for properties of block-sequences, results about block-homogeneity, and a possible method to construct a Banach space with an unconditional basis, which has a complemented subspace without an unconditional basis, are deduced.

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Valentin Ferenczi. 2005-02-17. Minimality, homogeneity and topological 0-1 laws for subspaces of a Banach space. https://arxiv.org/abs/math/0502054

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