| Title:
|
Remarks on the complementability of spaces of Bochner integrable functions in spaces of vector measures (English) |
| Author:
|
Emmanuele, G. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
37 |
| Issue:
|
2 |
| Year:
|
1996 |
| Pages:
|
217-228 |
| . |
| Category:
|
math |
| . |
| Summary:
|
In the paper [5] L. Drewnowski and the author proved that if $X$ is a Banach space containing a copy of $c_0$ then $L_1({\mu },X)$ is {\it not} complemented in $cabv({\mu },X)$ and conjectured that the same result is true if $X$ is any Banach space without the Radon-Nikodym property. Recently, F. Freniche and L. Rodriguez-Piazza ([7]) disproved this conjecture, by showing that if $\mu$ is a finite measure and $X$ is a Banach lattice not containing copies of $c_0$, then $L_1({\mu },X)$ is complemented in $cabv({\mu },X)$. Here, we show that the complementability of $L_1({\mu },X)$ in $cabv({\mu },X)$ together with that one of $X$ in the bidual $X^{\ast\ast}$ is equivalent to the complementability of $L_1({\mu },X)$ in its bidual, so obtaining that for certain families of Banach spaces not containing $c_0$ complementability occurs (Section 2), thanks to the existence of general results stating that a space in one of those families is complemented in the bidual. We shall also prove that certain quotient spaces inherit that property (Section 3). (English) |
| Keyword:
|
spaces of vector measures and vector functions |
| Keyword:
|
complementability |
| Keyword:
|
Banach lattices |
| Keyword:
|
preduals of W$^\ast$-algebras |
| Keyword:
|
quotient spaces |
| MSC:
|
46B20 |
| MSC:
|
46B30 |
| MSC:
|
46E27 |
| MSC:
|
46E40 |
| MSC:
|
46L99 |
| idZBL:
|
Zbl 0855.46006 |
| idMR:
|
MR1398997 |
| . |
| Date available:
|
2009-01-08T18:23:14Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118827 |
| . |
| Reference:
|
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| . |
