A note on the Banach lattice $c_0({\mathrm{\ell }}_2^n)$, its dual and its bidual

The main purpose of this paper is to study some geometric and topological properties on $c_0$ sum of the finite dimensional Banach lattice ${\mathrm{\ell }}_2^n$, its dual and its bidual. Among other results, we show that the Banach lattices $c_0({\mathrm{\ell }}_2^n)$ has the strong Gelfand-Philips property, but does not have the positive Grothendieck property. We also prove that the closed unit ball of $l_{\mathrm{\infty }}({\mathrm{\ell }}_2^n)$ is an almost limited set.