Approximation properties.

*(English)*Zbl 1067.46025
Johnson, W. B. (ed.) et al., Handbook of the geometry of Banach spaces. Volume 1. Amsterdam: Elsevier (ISBN 0-444-82842-7/hbk). 271-316 (2001).

A Banach space \(X\) is said to have the approximation property (\(AP\) for short) if its identity operator can be uniformly approximated on compact subsets of \(X\) by finite rank operators. Since the works of Banach, Mazur, and Schauder, there is ample literature on the theory surrounding the \(AP\) and its different variants. A detailed study was made by A. Grothendieck in his famous memoir [“Produits tensoriels topologiques et espaces nucléaires.” Mem. Am. Math. Soc. 16 (1955; Zbl 0055.09705; Zbl 0064.35501; Zbl 0123.30301); erratum Ann. Inst. Fourier 6, 117–120 (1955/56; Zbl 0072.12003)]. The approximation problem (does every Banach space have the approximation property?), closely related to the basis problem (does every separable Banach space have a Schauder basis?), has been considered to be one of the central open problems of functional analysis during the about forty years until the celebrated example of P. Enflo [Acta Math. 130, 309–317 (1973; Zbl 0267.46012)] of a (separable) Banach space which fails the \(AP\). Since then, there has been further major progress related to different variants of the \(AP\).

The present survey article gives an excellent concise overview of the area with elegant (sketches of) proofs of basic results and with an inspiring discussion of many important open problems. The article is not all-encompassing. For instance, it does not emphasize the applications of the \(AP\) (there are important applications, e.g., to nuclear operators), the interplay with tensor products is not considered nor the \(AP\) of order \(p\) with \(0<p<1\) or \(p>1\) (\(p=1\) gives the \(AP\)), introduced and studied by Saphar and Reinov (see the survey article of O. Reinov [Math. Nachr. 119, 257–264 (1984; Zbl 0601.46019)] and his recent works).

Section 1 studies results which relate basis theory to the \(AP\) with an emphasis on the structure of complemented subspaces of Banach spaces with (unconditional) bases. Section 2 contains Grothendieck’s classics on the \(AP\) and various counterexamples based on Enflo’s construction. We like to add that recently a related result from Grothendieck’s classics was strengthened up to characterize the \(AP\) in terms of the approximability of weakly compact operators in [Å. Lima, O. Nygaard and E. Oja, Isr. J. Math. 119, 325–348 (2000; Zbl 0983.46024)]. Sections 3–7 discuss the bounded \(AP\) (the approximating net of finite rank operators can be chosen bounded), the commuting bounded \(AP\) (the approximating operators can be chosen to commute), the \(\pi\)-property (the approximating net for the bounded \(AP\) can be given by projection operators), the finite-dimensional decomposition property, the uniform \(AP\) (the property that corresponds to the bounded \(AP\) in the local theory of Banach spaces), and connections between these variants of the \(AP\). Section 8 deals with the compact \(AP\) (here the identity is allowed to be approximated by compact operators, instead of finite rank operators) and its natural versions. To results of Section 8 we would remark that, by [G. Godefroy and P. D. Saphar, Ill. J. Math. 32, 672–695 (1988; Zbl 0631.46015)], if \(X^\ast\) or \(X^{\ast\ast}\) has the Radon–Nikodým property and if \(X^\ast\) has the compact \(AP\) given by weak-star continuous operators, then \(X^\ast\) has the metric compact \(AP\). We also remark an improvement of the Figiel–Johnson example from [O. I. Reĭnov, Math. Notes 33, 427–434 (1983); translation from Mat. Zametki 33, No. 6, 833–846 (1983; Zbl 0543.46009)]: there is a separable Banach space \(X\) with the \(AP\) whose identity cannot be uniformly approximated on compact subsets of \(X\) by any bounded net of weakly compact operators. The final Section 9 studies approximation properties in non-separable spaces. It is shown that some of them are separably determined: e.g., \(X\) has the metric \(AP\) if and only if every separable subspace of \(X\) is contained in a separable subspace of \(X\) with the metric \(AP\). Recently, similar results for approximation properties having some special geometric features (like unconditionality) were proved in [E. Oja, Trans. Am. Math. Soc. 352, 2801–2823 (2000; Zbl 0954.46010)].

For the entire collection see [Zbl 0970.46001].

The present survey article gives an excellent concise overview of the area with elegant (sketches of) proofs of basic results and with an inspiring discussion of many important open problems. The article is not all-encompassing. For instance, it does not emphasize the applications of the \(AP\) (there are important applications, e.g., to nuclear operators), the interplay with tensor products is not considered nor the \(AP\) of order \(p\) with \(0<p<1\) or \(p>1\) (\(p=1\) gives the \(AP\)), introduced and studied by Saphar and Reinov (see the survey article of O. Reinov [Math. Nachr. 119, 257–264 (1984; Zbl 0601.46019)] and his recent works).

Section 1 studies results which relate basis theory to the \(AP\) with an emphasis on the structure of complemented subspaces of Banach spaces with (unconditional) bases. Section 2 contains Grothendieck’s classics on the \(AP\) and various counterexamples based on Enflo’s construction. We like to add that recently a related result from Grothendieck’s classics was strengthened up to characterize the \(AP\) in terms of the approximability of weakly compact operators in [Å. Lima, O. Nygaard and E. Oja, Isr. J. Math. 119, 325–348 (2000; Zbl 0983.46024)]. Sections 3–7 discuss the bounded \(AP\) (the approximating net of finite rank operators can be chosen bounded), the commuting bounded \(AP\) (the approximating operators can be chosen to commute), the \(\pi\)-property (the approximating net for the bounded \(AP\) can be given by projection operators), the finite-dimensional decomposition property, the uniform \(AP\) (the property that corresponds to the bounded \(AP\) in the local theory of Banach spaces), and connections between these variants of the \(AP\). Section 8 deals with the compact \(AP\) (here the identity is allowed to be approximated by compact operators, instead of finite rank operators) and its natural versions. To results of Section 8 we would remark that, by [G. Godefroy and P. D. Saphar, Ill. J. Math. 32, 672–695 (1988; Zbl 0631.46015)], if \(X^\ast\) or \(X^{\ast\ast}\) has the Radon–Nikodým property and if \(X^\ast\) has the compact \(AP\) given by weak-star continuous operators, then \(X^\ast\) has the metric compact \(AP\). We also remark an improvement of the Figiel–Johnson example from [O. I. Reĭnov, Math. Notes 33, 427–434 (1983); translation from Mat. Zametki 33, No. 6, 833–846 (1983; Zbl 0543.46009)]: there is a separable Banach space \(X\) with the \(AP\) whose identity cannot be uniformly approximated on compact subsets of \(X\) by any bounded net of weakly compact operators. The final Section 9 studies approximation properties in non-separable spaces. It is shown that some of them are separably determined: e.g., \(X\) has the metric \(AP\) if and only if every separable subspace of \(X\) is contained in a separable subspace of \(X\) with the metric \(AP\). Recently, similar results for approximation properties having some special geometric features (like unconditionality) were proved in [E. Oja, Trans. Am. Math. Soc. 352, 2801–2823 (2000; Zbl 0954.46010)].

For the entire collection see [Zbl 0970.46001].

Reviewer: Eve Oja (Tartu)

##### MSC:

46B28 | Spaces of operators; tensor products; approximation properties |

46-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to functional analysis |