Bounded approximation properties in non-Archimedean Banach spaces (Q2905241)

The paper is concerned with the bounded approximation property. One says that a Banach space \((E,\|\cdot\|)\) over a non-trivially valued field \((K,|\cdot|)\) has the \(\lambda\)-bounded approximation property (\(\lambda\)-BAP, \(\lambda \geq 1\)) if for every compact subset \(X\) of \(E\) and every \(\epsilon >0\) there exists a finite-rank linear operator \(T\) on \(E\) with \(\|T\|\leq \lambda\) such that (1) \(\|Tx-x\|\leq \epsilon\) for all \(x\in X.\) If there exists a compact operator \(T\) with \(\|T\|\leq \lambda\) satisfying (1), then one says that \(E\) has the \(\lambda\)-bounded compact approximation theory (\(\lambda\)-BCAP). Let us mention that in the non-Archimedean (n.a.) case a linear operator \(T\) is called compact if the image of the unit ball is a compactoid. In contrast to the Archimedean case, in the non-Archimedean one \(\lambda\)-BAP and \(\lambda\)-BCAP are equivalent.NEWLINENEWLINEThe paper examines the approximation property in the case when \(K\) is n.a. valued field, complete with respect to the valuation, and \(E\) is a n.a. Banach space over \(K\), in comparison with the Archimedean case (meaning the case \(K=\mathbb R\) or \(K=\mathbb C\)). Every norm polar Banach space has the \(\lambda\)-BAP for every \(\lambda>1.\) If \(K\) is spherically complete or if \(E\) has an orthogonal base (\(E=c_0(I),\) for instance), then \(E\) has \(1\)-BAP, too. An n.a. Banach space is called norm polar if \(\| x\|=\sup\{|f(x)| : f\in E^*, \, |f|\leq \|\cdot\|\}\), \(x\in E\), where \(E^*\) is the algebraic dual of \(E\). If \(K\) is spherically complete, then \(E\) is norm-polar (a Hahn-Banach type theorem holds in this case). If \(K\) is not spherically complete and \(I\) infinite, then the space \(\ell^\infty(I)\) has the \(\lambda\)-BAP for every \(\lambda >1,\) but not for \(\lambda =1.\) This provides a counterexample to the approximation property in the n.a. case. The paper contains also other results on the approximation property in n.a. Banach spaces and, in Section 6, a discussion on the differences between the approximation property in the Archimedean and in the non-Archimedean cases.