Subsymmetric weak${}^{\ast }$ Schauder bases and factorization of the identity

Abstract

Let $X^{\ast }$ denote a Banach space with a subsymmetric weak${}^{\ast }$ Schauder basis satisfying condition~\eqref{eq:condition-c}. We show that for any operator $T:X^{\ast }\to X^{\ast }$, either $T(X^{\ast })$ or $(I-T)(X^{\ast })$ contains a subspace that is isomorphic to $X^{\ast }$ and complemented in $X^{\ast }$. Moreover, we prove that ${\ell }^p(X^{\ast })$, $1\le p\le \infty$ is primary.

16 pages, 1 figure