Papers by Valentin Ferenczi
Journal of the London Mathematical Society, Jan 6, 2009
It is proved that the relation of isomorphism between separable Banach spaces is a complete analy... more It is proved that the relation of isomorphism between separable Banach spaces is a complete analytic equivalence relation, i.e., that any analytic equivalence relation Borel reduces to it. Thus, separable Banach spaces up to isomorphism provide complete invariants for a great number of mathematical structures up to their corresponding notion of isomorphism. The same is shown to hold for (1) complete separable metric spaces up to uniform homeomorphism, (2) separable Banach spaces up to Lipschitz isomorphism, and (3) up to (complemented) biembeddability, (4) Polish groups up to topological isomorphism, and (5) Schauder bases up to permutative equivalence. Some of the constructions rely on methods recently developed by S. Argyros and P. Dodos. Contents 14 9. Isomorphism of separable Banach spaces 19 References 22
Duke Mathematical Journal, Jul 15, 2013
We study problems of maximal symmetry in Banach spaces. This is done by providing an analysis of ... more We study problems of maximal symmetry in Banach spaces. This is done by providing an analysis of the structure of small subgroups of the general linear group GL(X), where X is a separable reflexive Banach space. In particular, we provide the first known example of a Banach space X without any equivalent maximal norm, or equivalently such that GL(X) contains no maximal bounded subgroup. Moreover, this space X may be chosen to be super-reflexive.
De Gruyter eBooks, Mar 24, 2012
We provide an overview of a number of results concerning the complexity of isomorphism between se... more We provide an overview of a number of results concerning the complexity of isomorphism between separable Banach spaces. We also include some new results on the lattice structure of the set of spreading models of a Banach space.
Advances in Mathematics, Aug 1, 2005
We show that any Banach space contains a continuum of non-isomorphic subspaces or a minimal subsp... more 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. We show that an unconditional basis of a Banach space, of which every block-subspace is complemented, must be asymptotically c 0 or p , and we deduce some new characterisations of the classical spaces c 0 and p .
Bulletin of The London Mathematical Society, Apr 28, 2004
The following property of a normalized basis in a Banach space is considered: any normalized bloc... more The following property of a normalized basis in a Banach space is considered: any normalized block sequence of the basis has a subsequence equivalent to the basis. Under uniformity or other natural assumptions, a basis with this property is equivalent to the unit vector basis of c 0 or p. An analogous problem concerning spreading models is also addressed.
arXiv (Cornell University), Nov 8, 2007
We prove three new dichotomies for Banach spacesà la W.T. Gowers' dichotomies. The three dichotom... more We prove three new dichotomies for Banach spacesà la W.T. Gowers' dichotomies. The three dichotomies characterise respectively the spaces having no minimal subspaces, having no subsequentially minimal basic sequences, and having no subspaces crudely finitely representable in all of their subspaces. We subsequently use these results to make progress on Gowers' program of classifying Banach spaces by finding characteristic spaces present in every space. Also, the results are used to embed any partial order of size ℵ 1 into the subspaces of any space without a minimal subspace ordered by isomorphic embeddability. Contents 38 9. Open problems 40 References 41
arXiv (Cornell University), Nov 18, 2015
We investigate complex structures on twisted Hilbert spaces, with special attention paid to the K... more We investigate complex structures on twisted Hilbert spaces, with special attention paid to the Kalton-Peck Z2 space and to the hyperplane problem. We consider (nontrivial) twisted Hilbert spaces generated by centralizers obtained from an interpolation scale of Köthe function spaces. We show there are always complex structures on the Hilbert space that cannot be extended to the twisted Hilbert space. If, however, the scale is formed by rearrangement invariant Köthe function spaces then there are complex structures on it that can be extended to a complex structure of the twisted Hilbert space. Regarding the hyperplane problem we show that no complex structure on ℓ2 can be extended to a complex structure on an hyperplane of Z2 containing it.
arXiv (Cornell University), May 19, 2019
We study "disjoint" versions of the notions of trivial, locally trivial, strictly singular and su... more We study "disjoint" versions of the notions of trivial, locally trivial, strictly singular and super-strictly singular quasi-linear maps in the context of Köthe function spaces. Among other results, we show: i) (locally) trivial and (locally) disjointly trivial notions coincide on reflexive spaces; ii) On non-atomic superreflexive Köthe spaces, no centralizer is singular, although most are disjointly singular. iii) No super singular quasi-linear maps exist between superreflexive spaces although Kalton-Peck centralizers are super disjointly singular; iv) Disjoint singularity does not imply super disjoint singularity.
Journal of Functional Analysis, May 1, 2017
A topological setting is defined to study the complexities of the relation of equivalence of embe... more A topological setting is defined to study the complexities of the relation of equivalence of embeddings (or "position") of a Banach space into another and of the relation of isomorphism of complex structures on a real Banach space. The following results are obtained: a) if X is not uniformly finitely extensible, then there exists a space Y for which the relation of position of Y inside X reduces the relation E 0 and therefore is not smooth; b) the relation of position of ℓ p inside ℓ p , or inside L p , p = 2, reduces the relation E 1 and therefore is not reducible to an orbit relation induced by the action of a Polish group; c) the relation of position of a space inside another can attain the maximum complexity E max ; d) there exists a subspace of L p , 1 ≤ p < 2, on which isomorphism between complex structures reduces E 1 and therefore is not reducible to an orbit relation induced by the action of a Polish group.
Journal of The Institute of Mathematics of Jussieu, Jul 1, 2015
We investigate questions of maximal symmetry in Banach spaces and the structure of certain bounde... more We investigate questions of maximal symmetry in Banach spaces and the structure of certain bounded non-unitarisable groups on Hilbert space. In particular, we provide structural information about bounded groups with an essentially unique invariant complemented subspace. This is subsequently combined with rigidity results for the unitary representation of Aut(T) on ℓ 2 (T), where T is the countably infinite regular tree, to describe the possible bounded subgroups of GL(H) extending a well-known nonunitarisable representation of F∞. As a related result, we also show that a transitive norm on a separable Banach space must be strictly convex. DEPARTMENT OF MATHEMATICS, STATISTICS, AND COMPUTER SCIENCE (M/C 249), UNIVERSITY
Proceedings of the American Mathematical Society, Nov 17, 2005
The aim of this note is to prove that if K is any infinite metric compact space, then the Lipschi... more The aim of this note is to prove that if K is any infinite metric compact space, then the Lipschitz free spaces of C(K) and c 0 are isomorphic. This gives an example of non-Lipschitz-homeomorphic Banach spaces whose free Lipschitz spaces are isomorphic. We also derive some results about Lipschitz homogeneity for Banach spaces, from the results of G. Godefroy and N. J. Kalton on Lipschitz free Banach spaces.
Studia Mathematica, 2005
We study the number of non-isomorphic subspaces of a given Banach space. Our main result is the f... more We study the number of non-isomorphic subspaces of a given Banach space. Our main result is the following. Let X be a Banach space with an unconditional basis (e i) i∈N ; then either there exists a perfect set P of infinite subsets of N such that for any two distinct A, B ∈ P , [e i ] i∈A [e i ] i∈B , or for a residual set of infinite subsets A of N, [e i ] i∈A is isomorphic to X, and in that case, X is isomorphic to its square, to its hyperplanes, uniformly isomorphic to X ⊕ [e i ] i∈D for any D ⊂ N, and isomorphic to a denumerable Schauder decomposition into uniformly isomorphic copies of itself. Equipe d'Analyse
arXiv (Cornell University), Oct 13, 2011
Displays of Polish groups on separable real spaces are studied. It is proved that any closed subg... more Displays of Polish groups on separable real spaces are studied. It is proved that any closed subgroup of the infinite symmetric group S_\infty containing a non-trivial central involution admits a display on any of the classical spaces c0, C([0,1]), lp and Lp for 1 <=p <\infty. Also, for any Polsih group G, there exists a separable space X on which {-1,1} x G has a display.
arXiv (Cornell University), Jul 16, 2013
We continue the study of Uniformly Finitely Extensible Banach spaces (in short, UFO) initiated in... more We continue the study of Uniformly Finitely Extensible Banach spaces (in short, UFO) initiated in Moreno-Plichko, On automorphic Banach spaces,
HAL (Le Centre pour la Communication Scientifique Directe), 2019
Megrelishvili defines in [17] light groups of isomorphisms of a Banach space as the groups on whi... more Megrelishvili defines in [17] light groups of isomorphisms of a Banach space as the groups on which the Weak and Strong Operator Topologies coincide, and proves that every bounded group of isomorphisms of Banach spaces with the Point of Continuity Property (PCP) is light. We investigate this concept for isomorphism groups G of classical Banach spaces X without the PCP, specially isometry groups, and relate it to the existence of G-invariant LUR or strictly convex renormings of X. Given a (real) Banach space X, we denote by L(X) the set of bounded linear operators on X, and by GL(X) the group of bounded isomorphisms of X. We also denote by Isom(X) the group of surjective linear isometries of X. If G is a subgroup of GL(X), we write G GL(X). Recall that given a Banach space X, the Strong Operator Topology on L(X) is the topology of pointwise convergence, i.e. the initial topology generated by the family of functions f x : L(X) → X, x ∈ X, given by f x (T) = T x, T ∈ L(X), and the Weak Operator Topology on L(X) is generated by the family of functions f x,x * : L(X) → R, x ∈ X, x * ∈ X * , given by f x,x * (T) = x * (T x), T ∈ L(X). Megrelishvili gives the following definition.
Annales de l'Institut Fourier, 2012
São Paulo Journal of Mathematical Sciences
The article is a survey related to a classical unsolved problem in Banach space theory, appearing... more The article is a survey related to a classical unsolved problem in Banach space theory, appearing in Banach's famous book in 1932, and known as the Mazur rotations problem. Although the problem seems very difficult and rather abstract, its study sheds new light on the importance of norm symmetries of a Banach space, demonstrating sometimes unexpected connections with renorming theory and differentiability in functional analysis, with topological group theory and the theory of representations, with the area of amenability, with Fraïssé theory and Ramsey theory, and led to development of concepts of interest independent of Mazur problem. This survey focuses on results that have been published after 2000, stressing two lines of research which were developed in the last ten years. The first one is the study of approximate versions of Mazur rotations problem in its various aspects, most specifically in the case of the Lebesgue spaces L p. The second one concerns recent developments of multidimensional formulations of Mazur rotations problem and associated results. Some new results are also included.
Israel Journal of Mathematics
We study "disjoint" versions of the notions of trivial, locally trivial, strictly singular and su... more We study "disjoint" versions of the notions of trivial, locally trivial, strictly singular and super-strictly singular quasi-linear maps in the context of Köthe function spaces. Among other results, we show: i) (locally) trivial and (locally) disjointly trivial notions coincide on reflexive spaces; ii) On non-atomic superreflexive Köthe spaces, no centralizer is singular, although most are disjointly singular. iii) No super singular quasi-linear maps exist between superreflexive spaces although Kalton-Peck centralizers are super disjointly singular; iv) Disjoint singularity does not imply super disjoint singularity.
Israel Journal of Mathematics, 2017
We investigate complex structures on twisted Hilbert spaces, with special attention paid to the K... more We investigate complex structures on twisted Hilbert spaces, with special attention paid to the Kalton-Peck Z2 space and to the hyperplane problem. We consider (nontrivial) twisted Hilbert spaces generated by centralizers obtained from an interpolation scale of Köthe function spaces. We show there are always complex structures on the Hilbert space that cannot be extended to the twisted Hilbert space. If, however, the scale is formed by rearrangement invariant Köthe function spaces then there are complex structures on it that can be extended to a complex structure of the twisted Hilbert space. Regarding the hyperplane problem we show that no complex structure on ℓ2 can be extended to a complex structure on an hyperplane of Z2 containing it.
Pacific Journal of Mathematics, 2019
Megrelishvili defines in [17] light groups of isomorphisms of a Banach space as the groups on whi... more Megrelishvili defines in [17] light groups of isomorphisms of a Banach space as the groups on which the Weak and Strong Operator Topologies coincide, and proves that every bounded group of isomorphisms of Banach spaces with the Point of Continuity Property (PCP) is light. We investigate this concept for isomorphism groups G of classical Banach spaces X without the PCP, specially isometry groups, and relate it to the existence of G-invariant LUR or strictly convex renormings of X. Given a (real) Banach space X, we denote by L(X) the set of bounded linear operators on X, and by GL(X) the group of bounded isomorphisms of X. We also denote by Isom(X) the group of surjective linear isometries of X. If G is a subgroup of GL(X), we write G GL(X). Recall that given a Banach space X, the Strong Operator Topology on L(X) is the topology of pointwise convergence, i.e. the initial topology generated by the family of functions f x : L(X) → X, x ∈ X, given by f x (T) = T x, T ∈ L(X), and the Weak Operator Topology on L(X) is generated by the family of functions f x,x * : L(X) → R, x ∈ X, x * ∈ X * , given by f x,x * (T) = x * (T x), T ∈ L(X). Megrelishvili gives the following definition.
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Papers by Valentin Ferenczi