SOME INTERESTING BANACH-SPACES

If a Banach space X has an unconditionally basic skipped-blocking finite-dimensional decomposition (UB-SBFDD), then each of the properties RNP, KMP, PCP, and CPCP is equivalent to X not having a subspace isomorphic with c0. If X fails PCP and is a subspace of a space with a UB-SBFDD, then X has a subspace isomorphic with c0. Many examples axe described that have no c0-subspaces and fail RNP. Some of these have PCP. Some fail PCP and have CPCP. Some are not contained in a space with an unconditional basis, but this remains an open question for others. Sufficient conditions are given for a boundedly complete skipped-blocking decomposition to imply CPCP.