The dual form of the approximation property for a Banach space and a subspace (Q2804304)

Let \(X\) be a Banach space and let \(Y\) be a closed subspace of \(X\). The pair \((X,Y)\) is said to have the approximation property (AP) if there exists a net of finite-rank bounded linear operators on \(X\), all of which leave the subspace \(Y\) invariant, such that the net converges uniformly on compact subsets of \(X\) to the identity operator. Clearly, the classical AP of \(X\) is the same as the AP of \((X,X)\) or of \((X,\{ 0\})\). If the pair \((X,Y)\) has the AP, then \(X\), \(Y\), and the quotient space \(X/Y\) all have the AP.NEWLINENEWLINEThe main result provides a criterion of the AP for the pair \((X,Y)\). The authors call it ``the dual form of the AP'' for \((X,Y)\), and it extends Grothendieck's well-known ``condition de biunivocité'' from Banach spaces \(X\) to pairs \((X,Y)\). Let \(N(X)\) denote the space of nuclear operators on \(X\). If \(X\) has the AP, then \(N(X)=X^*\hat\otimes_\pi X\), the projective tensor product, and the trace is well defined on \(N(X)\). The dual form of the AP for the pair \((X,Y)\) is as follows.NEWLINENEWLINEAssume that \(X\) has the AP. Then the pair \((X,Y)\) has the AP if and only if, for all \(T\in N(X)\) such that \(T(X)\subset Y\) and \(T(Y)=\{ 0\}\), one has \(\text{trace}(T)=0\).NEWLINENEWLINEThe authors prefer the formulation in terms of \(N(X)\), but an interested reader may easily reformulate it in terms of \(X^*\hat\otimes_\pi X\) as in the Grothendieck criterion. (Reviewer's remark: In [J. Funct. Anal. 271, No. 3, 566--576 (2016; Zbl 1348.46021)], the authors proved a far-reaching extension of this dual form of the AP, where the subspace \(Y\) is replaced by a nest of closed subspaces of \(X\).)NEWLINENEWLINEThe main result applies to three-space properties. For instance, the authors prove that if \(X\) has the AP and the subspace \(Y\) is an \({\mathcal{L}}_\infty\)-space, then the pair \((X,Y)\) has the AP, and hence \(X/Y\) has the AP. Also, they prove that if \(X\) is an \({\mathcal{L}}_\infty\)-space and \(Y\) has the AP, then the pair \((X,Y)\) has the AP, and hence \(X/Y\) has the AP. The bounded AP version of the latter result was obtained in [\textit{T. Figiel} et al., Isr. J. Math. 183, 199--231 (2011; Zbl 1235.46027)]. The authors also prove the bounded AP version of the former result, improving slightly a result of \textit{J. M. F. Castillo} and \textit{Y. Moreno} [Isr. J. Math. 198, 243--259 (2013; Zbl 1288.46015)].