Ideal topologies and corresponding approximation properties (Q2790821)

The authors propose a unifying approach to numerous approximation properties in Banach spaces using the concept of the (\({\mathcal{I,J}},\tau\))-approximation property, where \({\mathcal{I}}\) and \({\mathcal{J}}\) are operator ideals, and \(\tau\) is an ideal topology. Quoting the authors: ``This concept recovers many classical/recent approximation properties as particular instances and several important known results are particular cases of more general results that are valid in this general framework.'' The paper contains a useful introduction to and a long list of references on various approximation properties.NEWLINENEWLINEAn ideal topology \(\tau\) is a correspondence that, for all Banach spaces \(E\) and \(F\), assigns a linear topology, still denoted by \(\tau\), on the space \({\mathcal{L}}(E;F)\) of all bounded linear operators such that for every operator ideal \({\mathcal{I}}\), \(\overline{{\mathcal{I}}}^\tau\), defined by \(\overline{\mathcal{I}}^\tau(E;F):=\overline{{\mathcal{I}}(E;F)}^\tau\), is an operator ideal. A Banach space \(E\) has the \(({\mathcal{I,J}},\tau)\)-approximation property, \(({\mathcal{I,J}},\tau)\)-AP for short, if \({\mathcal{I}}(F;E)\subset\overline{{\mathcal{J}}(F;E)}^\tau\) for every Banach space \(F\).NEWLINENEWLINEThe classical approximation property coincides with the \(({\mathcal{K,F}},\|\cdot\|)\)-AP, where \({\mathcal{K}}\) and \({\mathcal{F}}\) are the operator ideals of compact and finite-rank operators, respectively, and also with the (\({\mathcal{L,F}},\tau_c\))-AP, where \(\tau_c\) denotes the topology of uniform convergence on compact sets.NEWLINENEWLINELet \({\mathcal{I}}\) be an operator ideal and let NEWLINE\[NEWLINE C_{{\mathcal{I}}}(E)=\{A\subset E : \exists F \exists u\in {\mathcal{I}}(F;E)\;\text{such\;that}\;A\subset u(B_F)\}, NEWLINE\]NEWLINE where \(B_F\) denotes the closed unit ball of \(F\). The sets belonging to \(C_{{\mathcal{I}}}(E)\) are called \({\mathcal{I}}\)-bounded sets. The topology of uniform convergence on \({\mathcal{I}}\)-bounded sets is denoted by \(\tau_{C_{{\mathcal{I}}}}\).NEWLINENEWLINEOne of the main results is as follows.NEWLINENEWLINETheorem 3.9. Let \(\mathcal{I},\mathcal {J}_1,{\mathcal{J}}_2\) be operator ideals such that \({\mathcal{J}}_1\) has the Grothendieck property, \({\mathcal{I}}\supset {\mathcal{J}}^{\text{sur}}_1={\mathcal{J}}^{\text{sur}}_1\circ{\mathcal{J}}^{\text{sur}}_2\) and such that operators belonging to \({\mathcal{I}}\) map \({\mathcal{J}}_2\)-bounded sets to \({\mathcal{J}}_1\)-bounded sets. The following statements are equivalent for a Banach space \(E\):NEWLINENEWLINE(a)\quad \(\text{id}_E\in\overline{{\mathcal{F}}(E;E)}^{\tau_{C_{{\mathcal{J}}_1}}}\),NEWLINENEWLINE(b)\quad \(E\) has the \(({\mathcal{J}}^{\text{sur}}_1,{\mathcal{F}},\|\cdot\|)\)-AP,NEWLINENEWLINE(c)\quad \(E\) has the \(({\mathcal{J}}^{\text{sur}}_1,{\mathcal{F}},\tau_{C_{{\mathcal{J}}_2}})\)-AP,NEWLINENEWLINE(d)\quad \(E\) has the \(({\mathcal{I}},{\mathcal{F}},\tau_{C_{{\mathcal{J}}_2}})\)-AP.NEWLINENEWLINEThe proof of Theorem 3.9 relies on the Lima-Nygaard-Oja isometric version of the Davis-Figiel-Johnson-Pełczyński factorization theorem.NEWLINENEWLINEThe main part (Section 4) of the paper deals with the multilinear case. The authors introduce the notion of projective ideal topology that extends the ideal topology. They generalize to the \(({\mathcal{I,J}},\tau)\)-AP context some recent results on approximation properties in (symmetric) projective tensor products of Banach spaces.NEWLINENEWLINEAmong other unified approaches to approximation properties, the authors point out the convex approximation properties, launched in [\textit{A. Lissitsin} and \textit{E. Oja}, J. Math. Anal. Appl. 379, No. 2, 616--626 (2011; Zbl 1223.46016)]. The reviewer would suggest to look at [\textit{A. Lissitsin}, Arch. Math. 105, No. 2, 163--171 (2015; Zbl 1333.46016)] for an even more general approach that uses systems of seminorms on spaces of operators.