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Ergodic Banach spaces. (English) Zbl 1082.46009

The authors use descriptive set theory to isomorphically classify the subspaces of a Banach space. It follows from the solution of the homogeneous Banach space problem that a Banach space which is not isomorphic to \(\ell_2\) must contain at least two non-isomorphic (infinite-dimensional) subspaces. So a natural question, posed to the authors by G. Godefroy, is: how many non-isomorphic subspaces must a given Banach space contain? Except for \(\ell_2\), no example of a space with only finitely or countably many isomorphism classes of subspaces is known. In this paper, the authors study the possibility, for a given Banach space, to classify the analytic equivalence relations of isomorphisms on its subspaces, up to Borel reducibility.
The work relies on a version of Gowers’s game and Gowers’s “trichotomy” theorem: Every Banach space either contains a hereditarily indecomposable subspace, a subspace with an unconditional basis such that no disjointly supported block-subspaces are isomorphic, or a subspace with an unconditional basis which is quasi-minimal [W. T. Gowers, Ann. Math. (2) 156, No. 3, 797–833 (2002; Zbl 1030.46005)]. Using this, the authors show that for every Banach space which does not contain a minimal subspace (that is, a subspace which embeds in any of its own subspaces), there is a perfect set of subspaces such that two of them do not both embed into each other; in particular, this space contains a continuum of non-isomorphic subspaces.
An important notion for their purpose is defined by the authors: a Banach space is ergodic if the minimal non-smooth Borel equivalence relation \(E_0\) Borel reduces to isomorphism on subspaces (the set of closed linear subspaces being equipped with its Effros–Borel structure). In Theorem 12, the authors 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. They also show that every Banach space \(X\) with an unconditional basis, such that every block-subspace of \(X\) is complemented, is asymptotically \(c_0\) or \(\ell_p\), \(1\leq p<+\infty\).
Note that at the end of the paper, it is announced that the first author and E. M. Galego in [“Some equivalence relations which are Borel reducible to isomorphism between separable Banach spaces” (Preprint) (2004; arXiv:math.FA/0406477), have proven that \(c_0\) and \(\ell_p\), \(1\leq p<2\), are ergodic. The case of \(\ell_p\) for \(p>2\) remains open, and the conjecture is that \(\ell_2\) is the only non-ergodic Banach space.
Reviewer: Daniel Li (Lens)

MSC:

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

Citations:

Zbl 1030.46005
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References:

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