The ? p spaces

The ℒ _p spaces which were introduced by A. Pełczyński and the first named author are studied. It is proved, e.g., that (i)X is an ℒ _p space if and only ifX* is and ℒ _q space (p ⁻¹+q ⁻¹=1). (ii) A complemented subspace of an ℒ _p space is either an ℒ _p or an ℒ₂ space. (iii) The ℒ _p spaces have sufficiently many Boolean algebras of projections. These results are applied to show thatX is an ℒ_∞ (resp. ℒ₁) space if and only ifX admits extensions (resp. liftings) of compact operators havingX as a domain or range space. We also prove a theorem on the “local reflexivity” of an arbitrary Banach space. This research was partially supported by NSF Grant# 8964.