### The lattice-isometric copies of ${\ell}_{\infty}\left(\text{\Gamma}\right)$ in quotients of Banach lattices.

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A simple way of obtaining separable quotients in the class of weakly countably determined (WCD) Banach spaces is presented. A large class of Banach lattices, possessing as a quotient c0, l1, l2, or a reflexive Banach space with an unconditional Schauder basis, is indicated.

In previous papers, it is proved, among other things, that every infinite dimensional sigma-Dedekind complete Banach lattice has a separable quotient. It has come to my attention that L. Weis proved this result without assuming sigma-Dedekind completeness; the proof is based, however, on the deep theorem of J. Hagler and W.B. Johnson concerning the structure of dual balls of Banach spaces and therefore cannot be applied simply to the case of locally convex solid topologically complete Riesz spaces....

Let $X$ be an Archimedean Riesz space and $\mathcal{P}\left(X\right)$ its Boolean algebra of all band projections, and put ${\mathcal{P}}_{e}=\{Pe:P\in \mathcal{P}\left(X\right)\}$ and ${\mathcal{B}}_{e}=\{x\in X:x\wedge (e-x)=0\}$, $e\in {X}^{+}$. $X$ is said to have Weak Freudenthal Property ($WFP$) provided that for every $e\in {X}^{+}$ the lattice $lin\phantom{\rule{0.166667em}{0ex}}{\mathcal{P}}_{e}$ is order dense in the principal band ${e}^{dd}$. This notion is compared with strong and weak forms of Freudenthal spectral theorem in Archimedean Riesz spaces, studied by Veksler and Lavrič, respectively. $WFP$ is equivalent to ${X}^{+}$-denseness of ${\mathcal{P}}_{e}$ in ${\mathcal{B}}_{e}$ for every $e\in {X}^{+}$, and every Riesz space with sufficiently many projections...

This paper presents an elementary proof and a generalization of a theorem due to Abramovich and Lipecki, concerning the nonexistence of closed linear sublattices of finite codimension in nonatomic locally solid linear lattices with the Lebesgue property.

Let $L$ be an Archimedean Riesz space with a weak order unit $u$. A sufficient condition under which Dedekind [$\sigma $-]completeness of the principal ideal ${A}_{u}$ can be lifted to $L$ is given (Lemma). This yields a concise proof of two theorems of Luxemburg and Zaanen concerning projection properties of $C\left(X\right)$-spaces. Similar results are obtained for the Riesz spaces ${B}_{n}\left(T\right)$, $n=1,2,\cdots $, of all functions of the $n$th Baire class on a metric space $T$.

Let $X$ and $E$ be a Banach space and a real Banach lattice, respectively, and let $\Gamma $ denote an infinite set. We give concise proofs of the following results: (1) The dual space ${X}^{*}$ contains an isometric copy of ${c}_{0}$ iff ${X}^{*}$ contains an isometric copy of ${\ell}_{\infty}$, and (2) ${E}^{*}$ contains a lattice-isometric copy of ${c}_{0}\left(\Gamma \right)$ iff ${E}^{*}$ contains a lattice-isometric copy of ${\ell}_{\infty}\left(\Gamma \right)$.

It is known that a Banach lattice with order continuous norm contains a copy of ${\ell}_{1}$ if and only if it contains a lattice copy of ${\ell}_{1}$. The purpose of this note is to present a more direct proof of this useful fact, which extends a similar theorem due to R.C. James for Banach spaces with unconditional bases, and complements the ${c}_{0}$- and ${\ell}_{\infty}$-cases considered by Lozanovskii, Mekler and Meyer-Nieberg.

Let X, Y be two Banach spaces. We say that Y is a quasi-quotient of X if there is a continuous operator R: X → Y such that its range, R(X), is dense in Y. Let X be a nonseparable Banach space, and let U, W be closed subspaces of X and Y, respectively. We prove that if X has the Controlled Separable Projection Property (CSPP) (this is a weaker notion than the WCG property) and Y is a quasi-quotient of X, then the structure of Y resembles the structure of a separable Banach space: (a) Y/W is norm-separable...

Sea X un espacio de Banach con una base incondicional de Schauder no numerable, y sea Y un subespacio arbitrario no separable de X. Si X no contiene una copia isomorfa de l(J) con J no numerable entonces (1) la densidad de Y y la débil*-densidad de Y* son iguales, y (2) la bola unidad de X* es débil* sucesionalmente compacta. Además, (1) implica que Y contiene subconjuntos grandes formados por elementos disjuntos dos a dos, y una propiedad similar se verifica para las bases incondicionales no numerables...

Let $E$ be a fixed real function $F$-space, i.e., $E$ is an order ideal in ${L}_{0}(S,\Sigma ,\mu )$ endowed with a monotone $F$-norm $\parallel \parallel $ under which $E$ is topologically complete. We prove that $E$ contains an isomorphic (topological) copy of $\omega $, the space of all sequences, if and only if $E$ contains a lattice-topological copy $W$ of $\omega $. If $E$ is additionally discrete, we obtain a much stronger result: $W$ can be a projection band; in particular, $E$ contains a complemented copy of $\omega $. This solves partially the open problem set recently by W....

For two Banach spaces X and Y, we write $di{m}_{\ell}\left(X\right)=di{m}_{\ell}\left(Y\right)$ if X embeds into Y and vice versa; then we say that X and Y have the same linear dimension. In this paper, we consider classes of Banach spaces with symmetric bases. We say that such a class ℱ has the Cantor-Bernstein property if for every X,Y ∈ ℱ the condition $di{m}_{\ell}\left(X\right)=di{m}_{\ell}\left(Y\right)$ implies the respective bases (of X and Y) are equivalent, and hence the spaces X and Y are isomorphic. We prove (Theorems 3.1, 3.3, 3.5) that the class of Orlicz sequence spaces generated by regularly...

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