On a weak Freudenthal spectral theorem

Marek Wójtowicz

Commentationes Mathematicae Universitatis Carolinae (1992)

  • Volume: 33, Issue: 4, page 631-643
  • ISSN: 0010-2628

Abstract

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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 ℬ 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 ℬ e for every e ∈ X + , and every Riesz space with sufficiently many projections has WFP (THEOREM).

How to cite

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Wójtowicz, Marek. "On a weak Freudenthal spectral theorem." Commentationes Mathematicae Universitatis Carolinae 33.4 (1992): 631-643. <http://eudml.org/doc/247388>.

@article{Wójtowicz1992,
abstract = {Let $X$ be an Archimedean Riesz space and $\mathcal \{P\}(X)$ its Boolean algebra of all band projections, and put $\mathcal \{P\}_\{e\}=\lbrace P e:P\in \mathcal \{P\}(X)\rbrace $ and $\mathcal \{B\}_\{e\}=\lbrace x\in X: x\wedge (e-x)=0\rbrace $, $e\in X^+$. $X$ is said to have Weak Freudenthal Property ($\operatorname\{WFP\}$) provided that for every $e\in X^+$ the lattice $lin\, \mathcal \{P\}_\{e\}$ 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. $\operatorname\{WFP\}$ 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 has $\operatorname\{WFP\}$ (THEOREM).},
author = {Wójtowicz, Marek},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Freudenthal spectral theorem; band; band projection; Boolean algebra; disjointness; disjointness; Archimedean Riesz space; Boolean algebra of all band projections; weak Freudenthal property; strong and weak forms of Freudenthal spectral theorem},
language = {eng},
number = {4},
pages = {631-643},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On a weak Freudenthal spectral theorem},
url = {http://eudml.org/doc/247388},
volume = {33},
year = {1992},
}

TY - JOUR
AU - Wójtowicz, Marek
TI - On a weak Freudenthal spectral theorem
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1992
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 33
IS - 4
SP - 631
EP - 643
AB - Let $X$ be an Archimedean Riesz space and $\mathcal {P}(X)$ its Boolean algebra of all band projections, and put $\mathcal {P}_{e}=\lbrace P e:P\in \mathcal {P}(X)\rbrace $ and $\mathcal {B}_{e}=\lbrace x\in X: x\wedge (e-x)=0\rbrace $, $e\in X^+$. $X$ is said to have Weak Freudenthal Property ($\operatorname{WFP}$) provided that for every $e\in X^+$ the lattice $lin\, \mathcal {P}_{e}$ 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. $\operatorname{WFP}$ 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 has $\operatorname{WFP}$ (THEOREM).
LA - eng
KW - Freudenthal spectral theorem; band; band projection; Boolean algebra; disjointness; disjointness; Archimedean Riesz space; Boolean algebra of all band projections; weak Freudenthal property; strong and weak forms of Freudenthal spectral theorem
UR - http://eudml.org/doc/247388
ER -

References

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  1. Aliprantis C.D., Burkinshaw O., Locally Solid Riesz Spaces, New York-London, Academic Press, 1978. Zbl1043.46003MR0493242
  2. Curtis P.C., A note concerning certain product spaces, Arch. Math. 11 (1960), 50-52. (1960) Zbl0093.12602MR0111008
  3. Duhoux M., Meyer M., Extended orthomorphisms on Archimedean Riesz spaces, Annali di Matematica pura ed appl. 33 (1983), 193-236. (1983) Zbl0526.46010MR0725026
  4. Efimov B., Engelking R., Remarks on dyadic spaces II, Coll. Math. 13 (1965), 181-197. (1965) Zbl0137.16104MR0188964
  5. Lavrič B., On Freudenthal's spectral theorem, Indag. Math. 48 (1986), 411-421. (1986) Zbl0619.46005MR0869757
  6. Luxemburg W.A.J., Zaanen A.C., Riesz Spaces I, North-Holland, Amsterdam and London, 1971. 
  7. Semadeni Z., Banach Spaces of Continuous Functions, Polish Scientific Publishers, Warszawa, 1971. Zbl0478.46014MR0296671
  8. Veksler A.I., Projection properties of linear lattices and Freudenthal's theorem (in Russian), Math. Nachr. 74 (1976), 7-25. (1976) MR0430736
  9. Zaanen A.C., Riesz Spaces II, North-Holland, Amsterdam, New York-Oxford, 1983. Zbl0519.46001MR0704021

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