We give an intrinsic characterisation of the separable reflexive Banach spaces that embed into separable reflexive spaces with an unconditional basis all of whose normalised block sequences with the same growth rate are equivalent. This uses methods of E. Odell and T. Schlumprecht.
We study infinite asymptotic games in Banach spaces with a finite-dimensional decomposition (F.D.D.) and prove that analytic games are determined by characterising precisely the conditions for the players to have winning strategies. These results are applied to characterise spaces embeddable into sums of finite dimensional spaces, extending results of Odell and Schlumprecht, and to study various notions of homogeneity of bases and Banach spaces. The results are related to questions of rapidity...
It is known that there is a comeagre set of mutually conjugate measure preserving homeomorphisms of Cantor space equipped with the coinflipping probability measure, i.e., Haar measure. We show that the generic measure preserving homeomorphism is moreover conjugate to all of its powers. It follows that the generic measure preserving homeomorphism extends to an action of (ℚ, +) by measure preserving homeomorphisms, and, in fact, to an action of the locally compact ring 𝔄 of finite adèles. ...
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 E₀ 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...
Minimal Hausdorff (Baire) group topologies of certain groups of transformations naturally occurring in analysis are studied. The results obtained are subsequently applied to show that, e.g., the homeomorphism groups of the rational and of the irrational numbers carry no Polish group topology. In answer to a question of A. S. Kechris it is shown that the group of Borel automorphisms of ℝ cannot be a Polish group either.
We prove a game-theoretic dichotomy for sets of block sequences in vector spaces that extends, on the one hand, the block Ramsey theorem of W. T. Gowers proved for analytic sets of block sequences and, on the other hand, M. Davis’ proof of Σ⁰₃ determinacy.
We study the number of non-isomorphic subspaces of a given Banach space. Our main result is the following. Let be a Banach space with an unconditional basis ; then either there exists a perfect set P of infinite subsets of ℕ such that for any two distinct A,B ∈ P, , or for a residual set of infinite subsets A of ℕ, is isomorphic to , and in that case, is isomorphic to its square, to its hyperplanes, uniformly isomorphic to for any D ⊂ ℕ, and isomorphic to a denumerable Schauder decomposition...
We consider the unitary group U of complex, separable, infinite-dimensional Hilbert space as a discrete group. It is proved that, whenever U acts by isometries on a metric space, every orbit is bounded. Equivalently, U is not the union of a countable chain of proper subgroups, and whenever E ⊆ U generates U, it does so by words of a fixed finite length.
We analyse several examples of separable Banach spaces, some of them new, and relate them to several dichotomies obtained in [11],by classifying them according to which side of the dichotomies they fall.
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