Subsymmetric weak$^*$ Schauder bases and factorization of the identity

Abstract

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