Geodesic
Fundamental aspects of vector-valued Banach limits
Izvestiya. Mathematics , Tome 80 (2016) no. 2, pp. 316-328

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This paper is divided into four parts. In the first we study the existence of vector-valued Banach limits and show that a real Banach space with a monotone Schauder basis admits vector-valued Banach limits if and only if it is $1$-complemented in its bidual. In the second we prove two vector-valued versions of Lorentz' intrinsic characterization of almost convergence. In the third we show that the unit sphere in the space of all continuous linear operators from $\ell_\infty(X)$ to $X$ which are invariant under the shift operator on $\ell_\infty(X)$ cannot be obtained via compositions of surjective linear isometries with vector-valued Banach limits. In the final part we show that if $X$ enjoys the Krein–Milman property, then the set of vector-valued Banach limits is a face of the unit ball in the space of all continuous linear operators from $\ell_\infty(X)$ to $X$ which are invariant under the shift operator on $\ell_\infty(X)$.
Keywords: Banach limit, almost convergence, group of isometries, extremal structure. 
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     author = {F. J. Garcia-Pacheco and F. J. Perez-Fernandez},
     title = {Fundamental aspects of vector-valued {Banach} limits},
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F. J. Garcia-Pacheco; F. J. Perez-Fernandez. Fundamental aspects of vector-valued Banach limits. Izvestiya. Mathematics , Tome 80 (2016) no. 2, pp. 316-328. http://geodesic.mathdoc.fr/item/IM2_2016_80_2_a2/