arXiv subjects
Publications and source records attributed to Karim Boulabiar.
We compare two existence principles for maximal ideals, a classical one for vector lattices with a strong unit and a second, newly introduced one for vector lattices with a strong truncation. Although the latter strictly generalizes the former, we show that the two statements are equivalent over ZF set theory.
This paper characterizes lattice-ordered algebras with the order-to-ring ideal property, i.e., those in which every order ideal is a ring ideal. Our main result shows that an Archimedean lattice-ordered algebra has this property if and only if it is an f-algebra admitting a nil-faithful submultiplicative lattice seminorm, where nil-faithful means that elements with zero seminorm are nilpotent. As a consequence, we prove that an Archimedean semiprime f-algebra has the order-to-ring ideal property if and only if it admits a submultiplicative lattice norm. These results correct and extend earlier incorrect work in the literature and yield a necessary and sufficient condition for all orthomorphisms on an Archimedean vector lattice to be central.
Let $E$ be a directed (i.e., positively generated) ordered vector space endowed with an inner product. In this note, we prove that the following statements are equivalent: i) $E$ is a vector lattice and its norm induced by its inner product is a lattice norm. ii) The metric projection onto the positive cone $E^+$ of $E$ exists and it is both isotone and subadditive. Moreover, in this case, the best approximation to any $x \in E$ from $E^+$ coincides with its positive part $x^+$. This result extends previous work to the non-complete setting.
We introduce the notion of bounded quasi-inversion closed semiprime f-algebras and we prove that, if A is such an algebra, then any intermediate algebra in A is an order ideal of A. This extends a recent result by Dominguez who has dealt with the unital case (the problem on C(X)-type spaces has been solved earlier by Dominguez, Gomez-Perez, and Mulero). Our results are illustrated by examples of algebras of continuous functions and algebras of measurable functions.
We introduce the notion of (maximal) multi-truncations on a vector lattice as a generalization of the notion of truncations, an object of recent origin. We obtain a Johnson-Kist type representation of vector lattices with maximal multi-truncations as vector lattices of almost-finite extended-real continuous functions. The spectrum that allow such a representation is a particular set of prime ideals equipped with the hull-kernel topology. Various representations from the existing literature will appear as special cases of our general result.
Following a recent idea by Ball, we introduce the notion of strongly truncated Riesz space with a suitable spectrum. We prove that, under an extra Archimedean type condition, any strongly truncated Riesz space is isomorphic to a uniformly dense Riesz subspace of a $C_{0}\left( X\right) $-space. This turns out to be a direct generalization of the classical Kakutani Representation Theorem on Archimedean Riesz spaces with strong unit. Another representation theorem on normed Riesz spaces, due to Fremlin, will be obtained as a consequence of our main result.
Let $L$ be a (non necessarily unital) truncated vector lattice of real-valued functions on a nonempty set $X$. A nonzero linear functional $ψ$ on $L$ is called a truncation homomorphism if it preserves truncation, i.e.,% \[ ψ\left( f\wedge\mathbf{1}_{X}\right) =\min\left\{ ψ\left( f\right) ,1\right\} \text{ for all }f\in L. \] We prove that a linear functional $ψ$ on $L$ is a truncation homomorphism if and only if $ψ$ is a lattice homomorphism and% \[ \sup\left\{ ψ\left( f\right) :f\leq\mathbf{1}_{X}\right\} =1. \] This allows us to prove different evaluating characterizations of truncation homomorphisms. In this regard, a special attention is paid to the continuous case and various results from the existing literature are generalized.
We introduce and study the notion of orthosymmetric spaces over an Archimedean vector lattice as a generalization of finite-dimentional Euclidean inner spaces. A special attention has been paid to linear operators on these spaces.
Let $R$ be a lattice ordered ring along with a truncation in the sense of Ball. We give a necessary and sufficient condition on $R$ for its unitization $R\oplus\mathbb{Q}$ to be again a lattice ordered ring. Also, we shall see that $R\oplus\mathbb{Q}$ is a lattice ordered ring for at most one truncation. Particular attention will be paid to the Archimedean case. More precisely, we shall identify the unique truncation on an Archimedean $\ell$-ring $R$ which makes $R\oplus\mathbb{Q}$ into a lattice ordered ring.
Truncated Riesz spaces was first introduced by Fremlin in the context of real-valued functions. An appropriate axiomatization of the concept was given by Ball. Keeping only the first Ball's Axiom (among three) as a definition of truncated Riesz spaces, the first named author and El Adeb proved that if $E$ is truncated Riesz space then $E\oplus\mathbb{R}$ can be equipped with a non-standard structure of Riesz space such that $E$ becomes a Riesz subspace of $E\oplus\mathbb{R}$ and the truncation of $E$ is provided by meet with $1$. In the present paper, we assume that the truncated Riesz space $E$ has a lattice norm $\left\Vert .\right\Vert $ and we give a necessary and sufficient condition for $E\oplus\mathbb{R}$ to have a lattice norm extending $\left\Vert .\right\Vert $. Moreover, we show that under this condition, the set of all lattice norms on $E\oplus\mathbb{R}$ extending $\left\Vert .\right\Vert $ has essentially a largest element $\left\Vert .\right\Vert _{1}$ and a smallest element $\left\Vert .\right\Vert _{0}$. Also, it turns out that any alternative lattice norm on $E\oplus\mathbb{R}$ is either equivalent to $\left\Vert .\right\Vert _{1}$ or equals $\left\Vert .\right\Vert _{0}$. As consequences, we show that $E\oplus\mathbb{R}$ is a Banach lattice if and only if $E$ is a Banach lattice and we get a representation's theorem sustained by the celebrate Kakutani's Representation Theorem.