The asymptotic behaviour of the metric approximation property on subspaces of \(c_ 0\) (Q1813023)

Roughly speaking, we prove that for a (separable) Banach space \(X\) the following two conditions are equivalent: (i) \(X\) is almost isometrically contained in the class of subspaces of \(c_ 0\) sharing the metric approximation property. (ii) \(X\) is almost isometrically contained in the class of Banach spaces \(Y\) with the following property: There is a sequence \((F_ n)\) of finite rank operators converging strongly to the identity such that for some sequence \((\delta_ n)\) of non-negative numbers tending to zero \[ \| F_ ny_ 1+(\text{Id}-F_ n)y_ 2\|\leq1+\delta_ n\| y_ 1- y_ 2\| \] for all \(y_{1,2}\) contained by \(B_ Y\), the unit ball of \(Y\). This characterization has grown out of an attempt to relate the class of spaces \(X\) for which \(K(Y,X)\) is an M-ideal in \(L(Y,X)\) for every \(Y\) to the class of subspaces of \(c_ 0\) sharing the metric approximation property. This latter class can be characterized by the condition [see \textit{R. Paya} and \textit{W. Werner}, An approximation property related to \(M\)-ideals of compact operators, Proc. Am. Math. Soc. 111, No. 4, 993-1001 (1991)] \[ \sup_{\| x_{1,2}\|\leq1}\| F_ nx_ 1+(\text{Id}-F_ n)x_ 2\|\leq1+\varepsilon_ n \] where \((F_ n)\) again converges strongly to the identity, \(x_{1,2}\) are in \(B_ X\), \(\varepsilon\geq0\) and \(\varepsilon_ n\to0\).