arXiv subjects

V. Ferenczi

Publications and source records attributed to V. Ferenczi.

7 recordsLinked to original sources

arXiv2026

Kalton-Peck space is not isomorphic to its hyperplanes

Let $Z_2$ denote the real Kalton-Peck space. No hyperplane of $Z_2$ admits a complex structure, and $Z_2$ is even in the sense of Ferenczi-Galego. Therefore, $Z_2$ is not linearly isomorphic to its hyperplanes, solving a long-standing conjecture in Banach space theory. The argument uses the Kalton-Swanson symplectic form on $Z_2$, the Castillo-Gonz\'alez-Pino Fredholm theorem in $Z_2$, and a special case of a mod-$2$ theorem of Li-Zhu on skew-adjoint Fredholm relations in symplectic spaces, for which a self-contained proof is included.

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arXiv2026

On complex structures and uniqueness of algebra norms in Banach spaces

For $X$ an infinite dimensional Banach space, we contribute to the study of the Banach algebra $L(X)/S(X)$, where $S(X)$ is the ideal of strictly singular operators. We extend results of Ferenczi-Galego (2007) by proving that $\|I-J\|_S \geq 2$, whenever $I$ is a complex structure on a real space $X$ and $J$ extends a complex structure on a hyperplane of $X$, and where $\|.\|_S$ denotes a certain algebra norm on $L(X)/S(X)$ dominated by the usual quotient norm $\|.\|$. We solve two questions of Kalton-Swanson (1982) by proving that if $X=Z_2$ the Kalton-Peck space, then $L(Z_2)/S(Z_2)$ a) is not complete for $\|.\|_S$ and b) that it is not *-isomorphic to a $C^*$-algebra for $\|.\|$. In particular $L(Z_2)/S(Z_2)$ admits two inequivalent *-algebra norms.

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arXiv2021

Local Banach-space dichotomies and ergodic spaces

We prove a local version of Gowers' Ramsey-type theorem [Ann. Math. 156 (2002)], as well as local versions both of the Banach space first dichotomy (the "unconditional/HI" dichotomy) of Gowers [Ann. Math. 156 (2002)] and of the third dichotomy (the "minimal/tight" dichotomy) due to Ferenczi-Rosendal [J. Funct. Anal. 257 (2009)]. This means that we obtain versions of these dichotomies restricted to certain families of subspaces called D-families, of which several concrete examples are given. As a main example, non-Hilbertian spaces form D-families; therefore versions of the above properties for non-Hilbertian spaces appear in new Banach space dichotomies. As a consequence we obtain new information on the number of subspaces of non-Hilbertian Banach spaces, making some progress towards the "ergodic" conjecture of Ferenczi-Rosendal and towards a question of Johnson.

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arXiv2020

Amalgamation and Ramsey properties of $L_p$ spaces

We study the dynamics of the group of isometries of $L_p$-spaces. In particular, we study the canonical actions of these groups on the space of $δ$-isometric embeddings of finite dimensional subspaces of $L_p(0,1)$ into itself, and we show that for $p \neq 4,6,8,\ldots$ they are $\varepsilon$-transitive provided that $δ$ is small enough. We achieve this by extending the classical equimeasurability principle of Plotkin and Rudin. We define the central notion of a Fraïssé Banach space which underlies these results and of which the known separable examples are the spaces $L_p(0,1)$, $p \neq 4,6,8,\ldots$ and the Gurarij space. We also give a proof of the Ramsey property of the classes $\{\ell_p^n\}_n$, $p\neq 2,\infty$, viewing it as a multidimensional Borsuk-Ulam statement. We relate this to an arithmetic version of the Dual Ramsey Theorem of Graham and Rothschild as well as to the notion of a spreading vector of Matoušek and Rödl. Finally, we give a version of the Kechris-Pestov-Todorcevic correspondence that links the dynamics of the group of isometries of an approximately ultrahomogeneous space $X$ with a Ramsey property of the collection of finite dimensional subspaces of $X$.

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arXiv2019

Isometries of combinatorial Banach spaces

We prove that every isometry between two combinatorial spaces is determined by a permutation of the canonical unit basis combined with a change of signs. As a consequence, we show that in the case of Schreier spaces, all the isometries are given by a change of signs of the elements of the basis. Our results hold for both the real and the complex cases.

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arXiv2019

Differential processes generated by two interpolators

We study couples of interpolators, the differentials they generate and their associated commutator theorems. An essential part of our analysis is the study of the intrinsic symmetries of the process. Since we work without any compatibility or categorical assumption, our results are flexible enough to generalize most known results for commutators or translation operators, in particular those of Cwikel, Kalton, Milman, Rochberg \cite{ckmr} for differential methods and those of Carro, Cerdà and Soria \cite{caceso} for compatible interpolators. We also generalize stability and singularity results in \cite{cfg,ccfg,correa} from the complex method to general differential methods and obtain new incomparability results.

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arXiv2006

On strongly asymptotic $\ell_p$ spaces and minimality

We study Banach spaces X with a strongly asymptotic l_p basis (any disjointly supported finite set of vectors far enough out with respect to the basis behaves like l_p) which are minimal (X embeds into every infinite dimensional subspace). In particular such spaces embed into l_p.

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