Quiz your maths: do the uniformly continuous functions on the line form a ring?

The paper deals with the interplay between boundedness, order and ring structures in function lattices on the line and related metric spaces. It is shown that the lattice of all Lipschitz functions on a normed space $E$ is isomorphic to its sublattice of bounded functions if and only if $E$ has dimension one. The lattice of Lipschitz functions on $E$ carries a "hidden" $f$-ring structure with a unit, and the same happens to the (larger) lattice of all uniformly continuous functions for a wide variety of metric spaces. An example of a metric space whose lattice of uniformly continuous functions supports no unital $f$-ring structure is provided.Comment: 14 pages, to be published in Proceedings of the American Mathematical Societ

Fine structure of the homomorphisms of the lattice of uniformly continuous functions on the line

We provide a representation of the homomorphisms $U\longrightarrow \mathbb R$, where $U$ is the lattice of all uniformly continuous on the line. The resulting picture is sharp enough to describe the fine topological structure of the space of such homomorphisms.Comment: 11 pages, 1 figur

Nonlinear Centralizers in Homology II. The Schatten classes

The paper computes the spaces of extensions for the Schatten classes when they are regarded in its natural module structure over the algebra of bounded operators on the ground Hilbert space.Comment: 30 page

An example regarding Kalton's paper "Isomorphisms between spaces of vector-valued continuous functions"

The paper alluded to in the title contains the following striking result: Let $I$ be the unit interval and $\Delta$ the Cantor set. If $X$ is a quasi Banach space containing no copy of $c_0$ which is isomorphic to a closed subspace of a space with a basis and $C(I, X)$ is linearly homeomorphic to $C(\Delta, X)$, then $X$ is locally convex, i.e., a Banach space. It is shown that Kalton result is sharp by exhibiting non locally convex quasi Banach spaces X with a basis for which $C(I, X)$ and $C(\Delta, X)$ are isomorphic. Our examples are rather specific and actually in all cases X is isomorphic to $C(\Delta, X)$ if $K$ is a metric compactum of finite covering dimension.Comment: 4 page

Stability constants and the homology of quasi-Banach spaces

We affirmatively solve the main problems posed by Laczkovich and Paulin in \emph{Stability constants in linear spaces}, Constructive Approximation 34 (2011) 89--106 (do there exist cases in which the second Whitney constant is finite while the approximation constant is infinite?) and by Cabello and Castillo in \emph{The long homology sequence for quasi-Banach spaces, with applications}, Positivity 8 (2004) 379--394 (do there exist Banach spaces $X,Y$ for which \Ext(X,Y) is Hausdorff and non-zero?). In fact, we show that these two problems are the same.Comment: This paper is to appear in Israel Journal of Mathematic

On the bounded approximation property on subspaces of ℓp\ell_p when 0<p<10<p<1 and related issues

This paper studies the bounded approximation property (BAP) in quasi Banach spaces. In the first part of the paper we show that the kernel of any surjective operator $\ell_p\to X$ has the BAP when $X$ has it and $0<p\leq 1$, which is an analogue of the corresponding result of Lusky for Banach spaces. We then obtain and study nonlocally convex versions of the Kadec-Pe\l czy\'nski-Wojtaszczyk complementably universal spaces for Banach spaces with the BAP

Complex interpolation and twisted twisted Hilbert spaces

We show that Rochberg's generalizared interpolation spaces $\mathscr Z^{(n)}$ arising from analytic families of Banach spaces form exact sequences $0\to \mathscr Z^{(n)} \to \mathscr Z^{(n+k)} \to \mathscr Z^{(k)} \to 0$. We study some structural properties of those sequences; in particular, we show that nontriviality, having strictly singular quotient map, or having strictly cosingular embedding depend only on the basic case $n=k=1$. If we focus on the case of Hilbert spaces obtained from the interpolation scale of $\ell_p$ spaces, then $\mathscr Z^{(2)}$ becomes the well-known Kalton-Peck $Z_2$ space; we then show that $\mathscr Z^{(n)}$ is (or embeds in, or is a quotient of) a twisted Hilbert space only if $n=1,2$, which solves a problem posed by David Yost; and that it does not contain $\ell_2$ complemented unless $n=1$. We construct another nontrivial twisted sum of $Z_2$ with itself that contains $\ell_2$ complemented

On the category of quotient Banach spaces after Wegner

We study Waelbroeck's category of Banach quotients after Wegner, focusing on its basic homological and functional analytic properties

Twisting non-commutative LpL_p spaces

The paper makes the first steps into the study of extensions ("twisted sums") of noncommutative $L^p$-spaces regarded as Banach modules over the underlying von Neumann algebra $\mathcal M$. Our approach combines Kalton's description of extensions by centralizers (these are certain maps which are, in general, neither linear nor bounded) with a general principle, due to Rochberg and Weiss saying that whenever one finds a given Banach space $Y$ as an intermediate space in a (complex) interpolation scale, one automatically gets a self-extension $0\longrightarrow Y\longrightarrow X\longrightarrow Y \longrightarrow 0.$ For semifinite algebras, considering $L^p=L^p(\mathcal M,\tau)$ as an interpolation space between $\mathcal M$ and its predual $\mathcal M_*$ one arrives at a certain self-extension of $L^p$ that is a kind of noncommutative Kalton-Peck space and carries a natural bimodule structure. Some interesting properties of these spaces are presented. For general algebras, including those of type III, the interpolation mechanism produces two (rather than one) extensions of one sided modules, one of left-modules and the other of right-modules. Whether or not one can find (nontrivial) self-extensions of bimodules in all cases is left open

On Mazur rotations problem and its multidimensional versions

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 Fra\"iss\'e 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 Lp. The second one concerns recent developments of multidimensional formulations of Mazur rotations problem and associated results. Some new results are also included.Comment: 57 pages. This survey will be published in the special issue of the S\~ao Paulo Journal of Mathematical Sciences dedicated to the Golden Jubilee of the Institute of Mathematics and Statistics of the University of S\~ao Paul