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Acknowledgement of priority: Separable quotients of Banach spaces.

Marek Wójtowicz — 1998

Collectanea Mathematica

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....

On a weak Freudenthal spectral theorem

Marek Wójtowicz — 1992

Commentationes Mathematicae Universitatis Carolinae

Let X be an Archimedean Riesz space and 𝒫 ( X ) its Boolean algebra of all band projections, and put 𝒫 e = { P e : P 𝒫 ( X ) } and e = { x X : x ( e - x ) = 0 } , e X + . X is said to have Weak Freudenthal Property ( WFP ) provided that for every e X + the lattice l i n 𝒫 e is order dense in the principal band e d d . 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 𝒫 e in e for every e X + , and every Riesz space with sufficiently many projections...

A short proof on lifting of projection properties in Riesz spaces

Marek Wójtowicz — 1999

Commentationes Mathematicae Universitatis Carolinae

Let L be an Archimedean Riesz space with a weak order unit u . A sufficient condition under which Dedekind [ σ -]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 ( X ) -spaces. Similar results are obtained for the Riesz spaces B n ( T ) , n = 1 , 2 , , of all functions of the n th Baire class on a metric space T .

Isomorphic and isometric copies of ( Γ ) in duals of Banach spaces and Banach lattices

Marek Wójtowicz — 2006

Commentationes Mathematicae Universitatis Carolinae

Let X and E be a Banach space and a real Banach lattice, respectively, and let Γ 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 , and (2) E * contains a lattice-isometric copy of c 0 ( Γ ) iff E * contains a lattice-isometric copy of ( Γ ) .

The lattice copies of 1 in Banach lattices

Marek Wójtowicz — 2001

Commentationes Mathematicae Universitatis Carolinae

It is known that a Banach lattice with order continuous norm contains a copy of 1 if and only if it contains a lattice copy of 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 -cases considered by Lozanovskii, Mekler and Meyer-Nieberg.

The controlled separable projection property for Banach spaces

Jesús FerrerMarek Wójtowicz — 2011

Open Mathematics

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...

The structure of nonseparable Banach spaces with uncountable unconditional bases.

Carlos FinolMarek Wójtowicz — 2005


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...

Copies of the sequence space ω in F -lattices with applications to Musielak−Orlicz spaces

Marek WójtowiczHalina Wiśniewska — 2016

Commentationes Mathematicae

Let E be a fixed real function F -space, i.e., E is an order ideal in L 0 ( S , Σ , μ ) endowed with a monotone F -norm under which E is topologically complete. We prove that E contains an isomorphic (topological) copy of ω , the space of all sequences, if and only if E contains a lattice-topological copy W of ω . 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 ω . This solves partially the open problem set recently by W....

Cantor-Bernstein theorems for Orlicz sequence spaces

Carlos E. FinolMarcos J. GonzálezMarek Wójtowicz — 2014

Banach Center Publications

For two Banach spaces X and Y, we write d i m ( X ) = d i m ( Y ) 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 d i m ( X ) = d i m ( Y ) 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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