On The Duality Problem for the $p$-Compact Approximation Property and Its Inheritance to Subspaces

Abstract

In this paper, for $1<p<\infty$ we define the $v_p$ and $v_{p}^{*}$-topologies on the space of bounded linear operators between Banach spaces, and by way of these topologies we introduce the properties $v_{p}^{*}\text D$ and $\text Bv_{p}^{*}\text D$ for the dual space $E^{'}$. Under the assumption of the property $v_{p}^{*}\text D$ on the dual space $E^{'}$, we obtain a solution of the duality problem for the $p$-CAP with $2<p<\infty$. We show that, if $M$ is a closed subspace of a Banach space $E$ such that $M^{\perp}$ is complemented in the dual space $E^{'}$, then $M$ has the $p$-CAP (respectively, BCAP) whenever $E$ has the $p$-CAP (respectively, BCAP) and the dual space $M^{'}$ has the $v_{p}^{*}\text D$ (respectively, $\text Bv_{p}^{*}\text D$).