Ergodic Banach Spaces

Details Ergodic Banach Spaces Ergodic Banach Spaces

2003-04-02

arxiv.org/abs/math/0304018v1

Mathematics

Functional Analysis

Scientific paper

We show that any Banach space contains a continuum of non isomorphic subspaces or a minimal subspace. We define an ergodic Banach space $X$ as a space such that $E_0$ Borel reduces to isomorphism on the set of subspaces of $X$, and show that every Banach space is either ergodic or contains a subspace with an unconditional basis $ which is complementably universal for the family of its block-subspaces. We also use our methods to get uniformity results; for example, in combination with a result of B. Maurey, V. Milman and N. Tomczak-Jaegermann, we show that an unconditional basis of a Banach space, of which every block-subspace is complemented, must be asymptotically $c_0$ or $l_p$.

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