Dual Banach spaces with AP but without BAP

Abstract

We give negative answers to the AP-to-BAP and BAP-to-MAP questions for dual Banach spaces. More precisely, there is a separable Banach space $V$ with the metric approximation property (MAP) such that $V^*$ has the approximation property (AP) but no bounded approximation property (BAP). There is also a separable MAP space $X$, isomorphic to $\ell_1$, whose dual has BAP but no MAP. This resolves a longstanding open question explicitly stated in \cite{Oja2010}. The constructions rely on a finitely generated group without Property A and a quantitative estimate for optimal bounded approximation constants of projective tensor products.