In this manuscript we find another class of real Banach spaces which admit vector-valued Banach limits different from the classes found in [6, 7]. We also characterize the separating subsets of ℓ∞(X). For this we first need to study when the space of almost convergent sequences is closed in the space of bounded sequences, which turns out to happen only when the underlying space is complete. Finally, a study on the extremal structure of the set of vector-valued Banach limits is conducted when the underlying normed space is a Hilbert space.We also reach the conclusion that the set of vector-valued Banach limits is not a convex component of BCL(ℓ∞(X),X), provided that X is a 1-injective Banach space satisfying that the underlying compact Hausdorff topological space has isolated points.
@article{bwmeta1.element.doi-10_1515_math-2015-0067,
author = {Francisco Javier Garc\'\i a-Pacheco},
title = {Extremal properties of the set of vector-valued Banach limits},
journal = {Open Mathematics},
volume = {13},
year = {2015},
zbl = {1332.46018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_math-2015-0067}
}
Francisco Javier García-Pacheco. Extremal properties of the set of vector-valued Banach limits. Open Mathematics, Tome 13 (2015) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_math-2015-0067/
[1] Banach, S., Théorie des opérations linéaires, Chelsea Publishing company, New York, 1978 | Zbl 0067.08902
[2] Ahmad, Z.U., Mursaleen, M., An application of Banach limits, Proc. Amer. Math. Soc., 1988, 103, 244-246 | Zbl 0652.40009
[3] Semenov, E., Sukochev, F., Extreme points of the set of Banach limits, Positivity, 2013, 17, 163-170 [WoS] | Zbl 1287.46014
[4] Semenov, E., Sukochev, F., Invariant Banach limits and applications, J. Funct. Anal., 2010, 259, 1517-1541 [WoS] | Zbl 1205.46012
[5] Semenov, E., Sukochev, F., Usachev, A., Structural properties of the set of Banach limits, Dokl. Math., 2011, 84, 802-803 [WoS] | Zbl 1245.46007
[6] Armario, R., García-Pacheco, F.J., Pérez-Fernández, F.J., On Vector-Valued Banach Limits, Funct. Anal. Appl., 2013, 47, 315-318 [WoS] | Zbl 1326.40005
[7] Armario, R. García-Pacheco, F.J., Pérez-Fernández, F.J., Fundamental Aspects of Vector-Valued Banach Limits, Izv. Math., 2016, 80, (in press) | Zbl 1326.40005
[8] Rosenthal, H., On injective Banach spaces and the spaces C .S/, Bull. Amer. Math. Soc., 1969, 75, 824-828 | Zbl 0179.45702
[9] Wolfe, J., Injective Banach spaces of continuous functions, Trans. Amer. Math. Soc., 1978, 235, 115-139 | Zbl 0369.46026
[10] Lorentz, G., A contribution to the theory of divergent sequences, Acta Math., 1948, 80, 167-190 | Zbl 0031.29501
[11] Boos, J., Classical and Modern Methods in summability, Oxford University Press, 2000 | Zbl 0954.40001
[12] Mursaleen, M., On some new invariant matrix methods of summability, Quart. Jour. Math. Oxford, 1983, 34, 77-86 | Zbl 0539.40006
[13] Mursaleen, M., On A-invariant mean and A-almost convergence, Analysis Mathematica, 2011, 37, 173-180 [WoS] | Zbl 1240.40023
[14] Mursaleen, M., Applied Summability Methods, Springer Briefs, Heidelberg New York Dordrecht London, 2014 | Zbl 1302.40001
[15] Raimi, R.A., Invariant means and invariant matrix methods of summability, Duke Math. J., 1963, 30, 81-94 | Zbl 0125.03201
[16] Aizpuru, A., Armario, R., García-Pacheco, F.J., Pérez-Fernández, F.J., Banach limits and uniform almost summability, J. Math. Anal. Appl., 2011, 379, 82-90 | Zbl 1223.46004
[17] Aizpuru, A., Armario, R., García-Pacheco, F.J., Pérez-Fernández, F.J., Vector-Valued Almost Convergence and Classical Properties in Normed Spaces, Proc. Indian Acad. Sci. Math., 2014, 124, 93-108 [WoS] | Zbl 1300.46012
[18] García-Pacheco, F.J., Convex components and multi-slices in topological vector spaces, Ann. Funct. Anal., 2015, 6, 73-86 [WoS] | Zbl 06441309
[19] Day, M.M., Normed linear spaces, 3rd edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21, Springer-Verlag, New York-Heidelberg, 1973