$c_0$-subspaces and fourth dual types

Abstract:

For a separable Banach space $X$, we define the notion of a ${c_{0 + }}$-type on $X$ and show that the existence of such a type is equivalent to the embeddability of ${c_0}$ in $X$. All these types are weakly null and fourth dual (i.e. of the form $\tau (x) = \left \| {x + g} \right \|$ for $g \in {X^{****}}$). We define the ${l^{{l^ + }}}$-dual types on $X$ (these are generated by sequences in ${X^{**}}$) and prove that they coincide with the fourth dual types on $X$. We also prove that ${c_{0 + }}$-types are fourth dual types.