arXiv subjects
Publications and source records attributed to P. Tradacete.
We record two results in the theory of vector and Banach lattices related to order adherence. First, we give a negative answer to Gao and Leung's question on whether the $uo$-adherence of sublattices must be order closed. Second, we present a self-contained counterexample showing that a Banach lattice with a weakly Fatou norm need not admit any equivalent lattice norm with the Fatou property.
Building on a recent construction of G. Plebanek and the third named author, it is shown that a complemented subspace of a Banach lattice need not be linearly isomorphic to a Banach lattice. This solves a long-standing open question in Banach lattice theory.
We study the free Banach lattice $FBL^{(p,\infty)}[E]$ with upper $p$-estimates generated by a Banach space $E$. Using a classical result of Pisier on factorization through $L^{p,\infty}(μ)$ together with a finite dimensional reduction, it is shown that the spaces $\ell^{p,\infty}(n)$ witness the universal property of $FBL^{(p,\infty)}[E]$ isomorphically. As a consequence, we obtain a functional representation for $FBL^{(p,\infty)}[E]$. More generally, our proof allows us to identify the norm of any free Banach lattice over $E$ associated with a rearrangement invariant function space. After obtaining the above functional representation, we take the first steps towards analyzing the fine structure of $FBL^{(p,\infty)}[E]$. Notably, we prove that the norm for $FBL^{(p,\infty)}[E]$ cannot be isometrically witnessed by $L^{p,\infty}(μ)$ and settle the question of characterizing when an embedding between Banach spaces extends to a lattice embedding between the corresponding free Banach lattices with upper $p$-estimates. To prove this latter result, we introduce a novel push-out argument, which when combined with the injectivity of $\ell^p$ allows us to give an alternative proof of the subspace problem for free $p$-convex Banach lattices. On the other hand, we prove that $\ell^{p,\infty}$ is not injective in the class of Banach lattices with upper $p$-estimates, elucidating one of many difficulties arising in the study of $FBL^{(p,\infty)}[E]$.
We investigate the structure of the free $p$-convex Banach lattice $FBL^{(p)}[E]$ over a Banach space $E$. After recalling why such a free lattice exists, and giving a convenient functional representation of it, we focus our study on how properties of an operator $T:E\rightarrow F$ between Banach spaces transfer to the associated lattice homomorphism $\overline{T}:FBL^{(p)}[E]\rightarrow FBL^{(p)}[F]$. Particular consideration is devoted to the case when the operator $T$ is an isomorphic embedding, which leads us to examine extension properties of operators into $\ell_p$, and several classical Banach space properties such as being a G.T. space. A detailed investigation of basic sequences and sublattices of free Banach lattices is provided. In addition, we begin to build a dictionary between Banach space properties of $E$ and Banach lattice properties of $FBL^{(p)}[E]$. In particular, we characterize the existence of lattice copies of $\ell_1$ in $FBL^{(p)}[E]$ and show that $FBL[E]$ has an upper $p$-estimate if and only if $id_{E^*}$ is $(q,1)$-summing ($\frac{1}{p}+\frac{1}{q}=1$). We also highlight the significant differences between $FBL^{(p)}$-spaces depending on whether $p$ is finite or infinite. For example, we show that $FBL^{(\infty)}[E]$ is lattice isometric to $FBL^{(\infty)}[F]$ whenever $E$ and $F$ have monotone finite dimensional decompositions, while, on the other hand, when $p<\infty$ and $E^*$ is smooth, $FBL^{(p)}[E]$ determines $E$ isometrically.
We show that for every weakly compact subset $K$ of $C[0,1]$ with finite Cantor-Bendixson rank, there is a reflexive Banach lattice $E$ and an operator $T:E\rightarrow C[0,1]$ such that $K\subseteq T(B_E)$. On the other hand, we exhibit an example of a weakly compact set of $C[0,1]$ homeomorphic to $ω^ω+1$ for which such $T$ and $E$ cannot exist. This answers a question of M. Talagrand in the 80's.
We study random unconditional convergence for a basis in a Banach space. The connections between this notion and classical unconditionality are explored. In particular, we analyze duality relations, reflexivity, uniqueness of these bases and existence of unconditional subsequences.
A subset $A$ of a Banach space is called Banach-Saks when every sequence in $A$ has a Ces{à}ro convergent subsequence. Our interest here focusses on the following problem: is the convex hull of a Banach-Saks set again Banach-Saks? By means of a combinatorial argument, we show that in general the answer is negative. However, sufficient conditions are given in order to obtain a positive result.