Fundamental aspects of vector-valued Banach limits

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 $\ell_\infty(X)$.