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Youssef Azouzi

Publications and source records attributed to Youssef Azouzi.

17 recordsLinked to original sources
arXiv2025

Strong completeness of Lp-type vector lattices

Let E be a Dedekind complete Riesz space with weak unit e, equipped with a conditional expectation operator T. We prove that the spaces Lp(T), with their natural vector-valued norms, are strongly complete, extending the p=2 case of Kuo, Kalauch, and Watson. This resolves a question that has remained open for several years. We begin by studying a general type of convergence and its unbounded modification, unifying and generalizing order, norm, and absolute weak convergence while providing simpler proofs. As an application, we consider vector-valued norms and their unbounded variants, generalizing strong convergence in Lp-spaces and convergence in probability. This framework establishes the completeness of Lp(T) and of the universal completion E^{u}, reinforcing the uo-completeness of universally complete vector lattices. Finally we apply our main theorem to obtain a new result in ergodicity for conditional preserving systems.

math.FA↗
arXiv2025

Order denseness in free Banach lattices

We prove a fundamental property: the free vector lattice $FVL[E]$ over a Banach space E is order dense in the free p-convex Banach lattice $FBL^{(p)}[E],~~1 ^leq p \leq \infty,$ if and only if E is finite-dimensional. In a recent work, Oikhberg, Tradacete, Taylor, and Troitsky claimed that order denseness holds for all Banach spaces. We point out a gap in their proof, and consequently, any conclusions relying on this claim require reexamination -- a task we also undertake in this present paper. A key tool in our approach, which also leads to a partial answer to an open question recently posed by these authors.

math.FA↗
arXiv2024

A Kakutani-Rokhlin decomposition for conditionally ergodic process in the measure-free setting of vector lattices

Recently the Kac formula for the conditional expectation of the first recurrence time of a conditionally ergodic conditional expectation preserving system was established in the measure free setting of vector lattices (Riesz spaces). We now give a formulation of the Kakutani-Rokhlin decomposition for conditionally ergodic systems in terms of components of weak order units in a vector lattice. In addition, we prove that every aperiodic conditional expectation preserving system can be approximated by a periodic system.

math.DS↗
arXiv2024

The strong Nakano property in Banach lattices

We study the strong Nakano property in Banach lattices with a special focus on free Banach lattices. We show that for every finite dimensional Banach space $E$, the free Banach lattice $FBL[E]$ has the strong Nakano property with a constant independent of the dimension. It is also shown that if $FBL[E]$ has the strong Nakano property, then $E$ cannot contain subspaces isomorphic to neither $c_0$ nor $L_1$.

math.FA↗
arXiv2023

Some characterizations of ergodicity in Riesz spaces

In the recent surge of papers on ergodic theory within Riesz spaces, this article contributes by introducing enhanced characterizations of ergodicity. Our work extends and strengthens prior results from both the authors and Homann, Kuo, and Watson. Specifically, we show that in a conditional expectation preserving system (E,T,S,e), S can be extended to the natural domain of T and operates as an isometry on L^{p}(T) spaces.

math.FA↗
arXiv2023

Uniform Integrability in vector lattices and applications

In this paper, we explore an abstraction of uniform integrability in vector lattices and demonstrate its application by providing a positive solution to an open question posed by Kuo, Rodda, and Watson. Specifically, we show that for finite p and with T as a conditionally expectation operator, spaces Lp are sequentially complete. Furthermore, we demonstrate that a de La Vall\'e Poussin Theorem does not hold in the general setting of vector lattices.

math.FA↗
arXiv2023

A diagonal principle for nets

In our work, we provide a constructive proof of a generalized version of Cantor's diagonal argument for nets. This result expands the well-known technique beyond sequences, allowing it to be applied to a broader context. This result has significant potential applications in various fields, and we demonstrate a few of these in our work. One such application is the solution to a previously open problem posed by Kandi\'c, Marabeh, and Troitsky. Specifically, we show that the set of unbounded norm compact operators from a Banach space to a Banach lattice is closed.

math.FA↗
arXiv2023

Conditional supremum in Riesz spaces and applications

We extend the concept of conditional supremum to the measure-free setting of Riesz spaces via the conditional expectation operator. We explore its properties and show how this tool is crucial in generalizing various results across multiple disciplines to the framework of Riesz spaces. Among other applications, we utilize this concept in finance to derive characterizations of certain financial conditions.

math.FA↗
arXiv2022

Generating functions in Riesz spaces

We introduce and study the concept of generating function for natural elements in a Dedekind complete Riesz space equipped with a conditional expectatnion operator. This allows to study discrete processes in free-measure setting. In particular we improve a result obtained by Kuo, Vardy and Watson concerning Poisson approximation.

math.FA↗
arXiv2022

The Kac formula and Poincar\'{e} recurrence theorem in Riesz spaces

Riesz space (non-pointwise) generalizations for iterative processes are given for the concepts of recurrence, first recurrence and conditional ergodicity. Riesz space conditional versions of the Poincar\'{e} Recurrence Theorem and the Kac formula are developed. Under mild assumptions, it is shown that every conditional expectation preserving process is conditionally ergodic with respect to the conditional expectation generated by the Ces\`aro mean associated with the iterates of the process. Applied to processes in $L^1(\Omega,{\mathcal A},\mu)$, where $\mu$ is a probability measure, new conditional versions of the above theorems are obtained.

math.PR↗
arXiv2021

The sup-completion of a Dedekind complete vector lattice

Every Dedekind complete Riesz space X has a unique sup-completion X^{s}, which is a Dedekind complete lattice cone. This paper aims to present a systematic study this cone by extending several known results to general setting, proving new results and, in particular, introducing for elements of X^{s} finite and infinite parts. This enuables us to get a satisfactory abstract formulation of some classical results in the setting of Riesz spaces. We prove, in pareticular, a Riesz space version of Borel-Cantelli Lemma and present some applications to it.29*

math.FA↗
arXiv2021

Burkholder Theorem in Riesz spaces

The main purpose of this paper is to give a vector lattice version of a Theorem by Burkholder about convergence of martingales. The proof is based on a vector lattice analogue of Austin's sample function theorem, proved recently by Grobler, Labuschagne and Marraffa and on a new characterization of elements of the sup-completion of a universally complete vector lattice which do not belong to the space.

math.FA↗
arXiv2019

Compact operators between lattice normed spaces

In this paper we continue the study of compact-like operators in lattice normed spaces started recently by Aydin, Emelyanov, Erkurşun Özcand and Marabeh. We show among others, that every p-compact operator between lattice normed spaces is p-bounded. The paper contains answers of almost all questions asked by these authors.

math.FA↗
arXiv2017

Completeness for vector lattices

The notion of unboundedly order converges has been recieved recently a particular attention by several authors. The main result of the present paper shows that the notion is efficient and deserves that care. It states that a vector lattice is universally complete if and only if it is unboundedly order complete. Another notion of completeness will be treated is the notion of sup-completion introduced by Donner.

math.FA↗
arXiv2015

The tensor product of function algebras

In this paper we study the tensor product of two $f$-algebras. We show that the Riesz Subspace generated by a subalgebra in an $f$-algebra is an algebra in order to prove that the Riesz tensor product of two $f$-algebras has a structure of an $f$-algebra.

math.FA↗