Lipschitz approximable Banach spaces. (English) Zbl 1474.46043
It is shown that a separable Banach space with the bounded compact approximation property is Lipschitz-approximable. Using known examples one deduces that there is a Lipschitz-approximable reflexive separable Banach space which fails the approximation property. Further, some related results on approximation properties in Lipschitz-free Banach spaces are given.
Reviewer: Ondřej Kalenda (Praha)
MSC:
| 46B28 | Spaces of operators; tensor products; approximation properties |
| 46B25 | Classical Banach spaces in the general theory |
References:
| [1] | Borel-Mathurin L., Approximation properties and non-linear geometry of Banach spaces, Houston J. Math. 38 (2012), no. 4, 1135-1148 · Zbl 1281.46019 |
| [2] | Casazza P., Approximation Properties, Handbook of the Geometry of Banach Spaces, Vol. 1, North-Holland, Amsterdam, 2001, pages 271-316 · Zbl 1067.46025 · doi:10.1016/S1874-5849(01)80009-7 |
| [3] | Cho C.-M.; Johnson W. B., A characterization of subspaces \(X\) of \(l_p\) for which \(K(X)\) is an \(M\)-ideal in \(L(X)\), Proc. Amer. Math. Soc. 93 (1985), no. 3, 466-470 · Zbl 0537.47010 |
| [4] | Godefroy G., A survey on Lipschitz-free Banach spaces, Comment. Math. 55 (2015), no. 2, 89-118 · Zbl 1358.46015 |
| [5] | Godefroy G., Extensions of Lipschitz functions and Grothendieck’s bounded approximation property, North-West. Eur. J. Math. 1 (2015), 1-6 · Zbl 1386.46021 |
| [6] | Godefroy G.; Kalton N. J., Lipschitz-free Banach spaces, Studia Math. 159 (2003), no. 1, 121-141 · Zbl 1059.46058 · doi:10.4064/sm159-1-6 |
| [7] | Godefroy G.; Lancien G.; Zizler V., The non-linear geometry of Banach spaces after Nigel Kalton, Rocky Mountain J. Math. 44 (2014), no. 5, 1529-1584 · Zbl 1317.46016 · doi:10.1216/RMJ-2014-44-5-1529 |
| [8] | Godefroy G.; Ozawa N., Free Banach spaces and the approximation properties, Proc. Amer. Math. Soc. 142 (2014), no. 5, 1681-1687 · Zbl 1291.46013 · doi:10.1090/S0002-9939-2014-11933-2 |
| [9] | Jiménez-Vargas A.; Sepulcre J. M.; Villegas-Vallecillos M., Lipschitz compact operators, J. Math. Anal. Appl. 415 (2014), no. 2, 889-901 · Zbl 1308.47023 · doi:10.1016/j.jmaa.2014.02.012 |
| [10] | Kalton N. J., Spaces of Lipschitz and Hölder functions and their applications, Collect. Math. 55 (2004), no. 2, 171-217 · Zbl 1069.46004 |
| [11] | Kalton N. J., The uniform structure of Banach spaces, Math. Ann. 354 (2012), no. 4, 1247-1288 · Zbl 1268.46018 · doi:10.1007/s00208-011-0743-3 |
| [12] | Oja E., On bounded approximation properties of Banach spaces, Banach algebras 2009, Banach Center Publ., 91, Polish Acad. Sci. Inst. Math., Warsaw, 2010, pages 219-231 · Zbl 1211.46015 |
| [13] | Pernecka E.; Smith R. J., The metric approximation property and Lipschitz-free spaces over subsets of \(\mathbb{R}^n\), J. Approx. Theory 199 (2015), 29-44 · Zbl 1333.46017 · doi:10.1016/j.jat.2015.06.003 |
| [14] | Thele R. L., Some results on the radial projection in Banach spaces, Proc. Amer. Math. Soc. 42 (1974), 483-486 · Zbl 0276.46015 · doi:10.1090/S0002-9939-1974-0328550-1 |
| [15] | Willis G., The compact approximation property does not imply the approximation property, Studia Math. 103 (1992), no. 1, 99-108 · Zbl 0814.46017 · doi:10.4064/sm-103-1-99-108 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
