Weak Approximation Properties Induced by Unconditionally p-Compact Sets

This paper investigates several equivalent formulations of the unconditionally p-approximation property. Inspired by these formulations, we introduce a new approximation property, called the weak unconditionally p-approximation property, and develop a number of equivalent characterizations for it. We also examine its relationship with the corresponding duality problem and obtain a partial answer. In addition, the weak bounded approximation property is reinterpreted within the framework of unconditionally p-compact sets, leading to new characterizations. Finally, we establish a three-space result for the weak unconditionally p-approximation property by showing that this property is preserved under suitable assumptions on a complemented subspace and the corresponding quotient space.