A short proof on lifting of projection properties in Riesz spaces

Marek Wójtowicz

Commentationes Mathematicae Universitatis Carolinae (1999)

  • Volume: 40, Issue: 2, page 277-278
  • ISSN: 0010-2628

Abstract

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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 .

How to cite

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Wójtowicz, Marek. "A short proof on lifting of projection properties in Riesz spaces." Commentationes Mathematicae Universitatis Carolinae 40.2 (1999): 277-278. <http://eudml.org/doc/248399>.

@article{Wójtowicz1999,
abstract = {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 $C(X)$-spaces. Similar results are obtained for the Riesz spaces $B_\{n\}(T)$, $n=1, 2, \dots $, of all functions of the $n$th Baire class on a metric space $T$.},
author = {Wójtowicz, Marek},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Dedekind completeness; spaces of continuous functions; spaces of Baire functions; Dedekind completeness; spaces of continuous functions; spaces of Baire functions; Archimedean Riesz space; weak order unit},
language = {eng},
number = {2},
pages = {277-278},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A short proof on lifting of projection properties in Riesz spaces},
url = {http://eudml.org/doc/248399},
volume = {40},
year = {1999},
}

TY - JOUR
AU - Wójtowicz, Marek
TI - A short proof on lifting of projection properties in Riesz spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1999
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 40
IS - 2
SP - 277
EP - 278
AB - 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 $C(X)$-spaces. Similar results are obtained for the Riesz spaces $B_{n}(T)$, $n=1, 2, \dots $, of all functions of the $n$th Baire class on a metric space $T$.
LA - eng
KW - Dedekind completeness; spaces of continuous functions; spaces of Baire functions; Dedekind completeness; spaces of continuous functions; spaces of Baire functions; Archimedean Riesz space; weak order unit
UR - http://eudml.org/doc/248399
ER -

References

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  1. Kuratowski K., Introduction to Set Theory and Topology, Polish Scientific Publishers, Warszawa, 1997. Zbl1081.54501MR0346724
  2. Kuratowski K., Mostowski A., Set Theory, Polish Scientific Publishers, Warszawa, 1996. Zbl0337.02034MR0485384
  3. Luxemburg W.A.J., Zaanen A.C., Riesz Spaces I, North-Holland, Amsterdam, 1971. 

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