| Title:
|
On a weak Freudenthal spectral theorem (English) |
| Author:
|
Wójtowicz, Marek |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
33 |
| Issue:
|
4 |
| Year:
|
1992 |
| Pages:
|
631-643 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $X$ be an Archimedean Riesz space and $\Cal P(X)$ its Boolean algebra of all band projections, and put $\Cal P_{e}=\{P e:P\in \Cal P(X)\}$ and $\Cal B_{e}=\{x\in X: x\wedge (e-x)=0\}$, $e\in X^+$. $X$ is said to have Weak Freudenthal Property (\text{$\operatorname{WFP}$}) provided that for every $e\in X^+$ the lattice $lin\, \Cal P_{e}$ 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. \text{$\operatorname{WFP}$} is equivalent to $X^+$-denseness of $\Cal P_{e}$ in $\Cal B_{e}$ for every $e\in X^+$, and every Riesz space with sufficiently many projections has \text{$\operatorname{WFP}$} (THEOREM). (English) |
| Keyword:
|
Freudenthal spectral theorem |
| Keyword:
|
band |
| Keyword:
|
band projection |
| Keyword:
|
Boolean algebra |
| Keyword:
|
disjointness |
| MSC:
|
06B10 |
| MSC:
|
06E99 |
| MSC:
|
46A40 |
| idZBL:
|
Zbl 0777.46006 |
| idMR:
|
MR1240185 |
| . |
| Date available:
|
2009-01-08T17:59:19Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118535 |
| . |
| Reference:
|
[1] Aliprantis C.D., Burkinshaw O.: Locally Solid Riesz Spaces.New York-London, Academic Press, 1978. Zbl 1043.46003, MR 0493242 |
| Reference:
|
[2] Curtis P.C.: A note concerning certain product spaces.Arch. Math. 11 (1960), 50-52. Zbl 0093.12602, MR 0111008 |
| Reference:
|
[3] Duhoux M., Meyer M.: Extended orthomorphisms on Archimedean Riesz spaces.Annali di Matematica pura ed appl. 33 (1983), 193-236. Zbl 0526.46010, MR 0725026 |
| Reference:
|
[4] Efimov B., Engelking R.: Remarks on dyadic spaces II.Coll. Math. 13 (1965), 181-197. Zbl 0137.16104, MR 0188964 |
| Reference:
|
[5] Lavrič B.: On Freudenthal's spectral theorem.Indag. Math. 48 (1986), 411-421. Zbl 0619.46005, MR 0869757 |
| Reference:
|
[6] Luxemburg W.A.J., Zaanen A.C.: Riesz Spaces I.North-Holland, Amsterdam and London, 1971. |
| Reference:
|
[7] Semadeni Z.: Banach Spaces of Continuous Functions.Polish Scientific Publishers, Warszawa, 1971. Zbl 0478.46014, MR 0296671 |
| Reference:
|
[8] Veksler A.I.: Projection properties of linear lattices and Freudenthal's theorem (in Russian).Math. Nachr. 74 (1976), 7-25. MR 0430736 |
| Reference:
|
[9] Zaanen A.C.: Riesz Spaces II.North-Holland, Amsterdam, New York-Oxford, 1983. Zbl 0519.46001, MR 0704021 |
| . |
