Subprojective Banach spaces. (English) Zbl 1338.46013
A Banach space \(X\) is said to be subprojective if any of its infinite-dimensional subspaces contains a further infinite-dimensional subspace complemented in \(X\). The paper under review presents nice results on stability type properties concerning subprojectivity. Here are some of them. Let \(E, X_1, X_2, \ldots\) be Banach spaces with \(E\) possessing a \(1\)-unconditional basis. Then \((\sum_n X_n)_E\) is subprojective if and only if each of the spaces \(E, X_1, X_2, \ldots\) is subprojective. The subprojectivity is not a three-space property. Every subprojective space contains a subspace with an unconditional basis. There is a Banach space with an unconditional basis and without a subprojective subspace. The space of all bounded linear operators on an infinite-dimensional Banach space is not subprojective. Some results concern tensor products and twisted sums of subprojective Banach spaces.
Reviewer: Mikhail M. Popov (Chernivtsi)
MSC:
| 46B03 | Isomorphic theory (including renorming) of Banach spaces |
| 46B08 | Ultraproduct techniques in Banach space theory |
| 46B28 | Spaces of operators; tensor products; approximation properties |
Keywords:
Banach space; complemented subspace; tensor product; space of operators; subprojective Banach spacesReferences:
| [1] | Aiena, P., Fredholm and Local Spectral Theory, with Applications to Multipliers (2004), Kluwer: Kluwer Dordrecht · Zbl 1077.47001 |
| [2] | Alspach, Dale E., Good \(l_2\)-subspaces of \(L_p, p > 2\), Banach J. Math. Anal., 3, 2, 49-54 (2009) · Zbl 1194.46014 |
| [3] | Arazy, J., Basic sequences, embeddings, and the uniqueness of the symmetric structure in unitary matrix spaces, J. Funct. Anal., 40, 302-340 (1981) · Zbl 0457.47035 |
| [4] | Arazy, J.; Lindenstrauss, J., Some linear topological properties of the spaces \(C_p\) of operators on Hilbert space, Compos. Math., 30, 81-111 (1975) · Zbl 0302.47034 |
| [5] | Bessaga, C.; Pełczynski, A., On bases and unconditional convergence of series in Banach spaces, Studia Math., 17, 151-164 (1958) · Zbl 0084.09805 |
| [6] | Bu, Q.; Buskes, G.; Popov, A.; Tcaciuc, A.; Troitsky, V., The 2-concavification of a Banach lattice equals the diagonal of the Fremlin tensor square, Positivity, 17, 283-298 (2013) · Zbl 1281.46021 |
| [7] | Defant, A.; Floret, K., Tensor Norms and Operator Ideals (1993), North-Holland: North-Holland Amsterdam · Zbl 0774.46018 |
| [8] | Diestel, J.; Fourie, J.; Swart, J., The Metric Theory of Tensor Products. Grothendieck’s Résumé Revisited (2008), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1186.46004 |
| [9] | Diestel, J.; Jarchow, H.; Tonge, A., Absolutely Summing Operators (1995), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0855.47016 |
| [10] | Dixmier, J., \(C^\ast \)-Algebras (1977), North-Holland: North-Holland Amsterdam · Zbl 0366.17007 |
| [11] | Fabian, M.; Habala, P.; Hajek, P.; Montesinos, V.; Pelant, J.; Zizler, V., Functional Analysis and Infinite-Dimensional Geometry (2001), Springer: Springer New York · Zbl 0981.46001 |
| [12] | Feder, M., On subspaces of spaces with an unconditional basis and spaces of operators, Illinois J. Math., 24, 196-205 (1980) · Zbl 0411.46009 |
| [13] | Figiel, T.; Johnson, W. B.; Tzafriri, L., On Banach lattices and spaces having local unconditional structure, with applications to Lorentz function spaces, J. Approx. Theory, 13, 395-412 (1975) · Zbl 0307.46007 |
| [14] | Flores, J.; Hernandez, F. L.; Spinu, E.; Tradacete, P.; Troitsky, V. G., Disjointly homogeneous Banach lattices: duality and complementation, J. Funct. Anal., 226, 5858-5885 (2014) · Zbl 1315.46024 |
| [16] | Gonzalez, M.; Martinez-Abejon, A.; Salas-Brown, M., Perturbation classes for semi-Fredholm operators on subprojective and superprojective spaces, Ann. Acad. Sci. Fenn. Math., 36, 481-491 (2011) · Zbl 1232.47015 |
| [17] | Gordon, Y.; Lewis, D.; Retherford, J., Banach ideals of operators with applications, J. Funct. Anal., 14, 85-129 (1973) · Zbl 0272.47024 |
| [18] | Gowers, T.; Maurey, B., Banach spaces with small spaces of operators, Math. Ann., 307, 543-568 (1997) · Zbl 0876.46006 |
| [19] | Jensen, H., Scattered \(C^\ast \)-algebras, Math. Scand., 41, 308-314 (1977) · Zbl 0372.46059 |
| [20] | Jensen, H., Scattered \(C^\ast \)-algebras II, Math. Scand., 43, 308-310 (1978) · Zbl 0423.46050 |
| [21] | Johnson, W.; Maurey, B.; Schechtman, G.; Tzafriri, L., Symmetric structures in Banach spaces, Mem. Amer. Math. Soc., 19, 217 (1979), v+298 pp · Zbl 0421.46023 |
| [22] | Kalton, N. J.; Peck, N. T., Twisted sums of sequence spaces and the three space problem, Trans. Amer. Math. Soc., 255, 1-30 (1979) · Zbl 0424.46004 |
| [23] | Kalton, N. J.; Wojtaszczyk, P., On nonatomic Banach lattices and Hardy spaces, Proc. Amer. Math. Soc., 120, 731-741 (1994) · Zbl 0820.46017 |
| [24] | Kamińska, A.; Maligranda, L.; Persson, L., Convexity, concavity, type and cotype of Lorentz spaces, Indag. Math. (N.S.), 9, 367-382 (1998) · Zbl 0937.46027 |
| [25] | Kusuda, M., A characterization of scattered \(C^\ast \)-algebras and its application to \(C^\ast \)-crossed products, J. Operator Theory, 63, 417-424 (2010) · Zbl 1224.46116 |
| [26] | Lacey, E., The Isometric Theory of Classical Banach Spaces (1974), Springer: Springer Berlin · Zbl 0285.46024 |
| [27] | Lindenstrauss, Y.; Tzafriri, L., Classical Banach Spaces I (1977), Springer: Springer Berlin · Zbl 0362.46013 |
| [28] | Lindenstrauss, Y.; Tzafriri, L., Classical Banach Spaces II (1979), Springer: Springer Berlin · Zbl 0403.46022 |
| [29] | Meyer-Nieberg, P., Banach Lattices (1991), Springer: Springer Berlin · Zbl 0743.46015 |
| [30] | Muller, P. F.X.; Schechtman, G., On complemented subspaces of \(H_1\) and VMO, (Geometric Aspects of Functional Analysis (1987-1988) (1989), Springer: Springer Berlin), 113-125 · Zbl 0687.46014 |
| [31] | Oikhberg, T.; Spinu, E., Domination of operators in the non-commutative setting, Studia Math., 219, 35-67 (2013) · Zbl 1296.47033 |
| [32] | Oja, E., Sur la réflexivité des produits tensoriels et les sous-espaces des produits tensoriels projectifs, Math. Scand., 51, 275-288 (1982) · Zbl 0488.46058 |
| [33] | Ostrovskii, M., Topologies on the set of all subspaces of a Banach space and related questions of Banach space geometry, Quaest. Math., 17, 259-319 (1994) · Zbl 0807.46014 |
| [34] | Pedersen, G., \(C^\ast \)-Algebras and Their Automorphism Groups (1979), Academic Press: Academic Press London, New York · Zbl 0416.46043 |
| [35] | Pełczynski, A.; Semadeni, Z., Spaces of continuous functions. III. Spaces \(C(\Omega)\) for \(Ω\) without perfect subsets, Studia Math., 18, 211-222 (1959) · Zbl 0091.27803 |
| [36] | Pisier, G.; Xu, Q., Non-commutative \(L_p\)-spaces, (Handbook of Banach Spaces, vol. 2 (2003), North-Holland: North-Holland Amsterdam), 1459-1517 · Zbl 1046.46048 |
| [37] | Randrianantoanina, N., Non-commutative martingale VMO-spaces, Studia Math., 191, 39-55 (2009) · Zbl 1175.46007 |
| [38] | Samuel, C., Sur les sous-espaces de \(\ell_p \hat{\otimes} \ell_q\), Math. Scand., 47, 247-250 (1980) · Zbl 0448.46049 |
| [39] | Semadeni, Z., Banach Spaces of Continuous Functions, vol. I (1971), PWN - Polish Scientific Publishers: PWN - Polish Scientific Publishers Warsaw · Zbl 0225.46030 |
| [40] | Tomczak-Jaegermann, N., Banach-Mazur Distances and Finite-Dimensional Operator Ideals (1989), Longman Scientific and Technical: Longman Scientific and Technical Harlow, UK · Zbl 0721.46004 |
| [41] | Whitley, R., Strictly singular operators and their conjugates, Trans. Amer. Math. Soc., 113, 252-261 (1964) · Zbl 0124.06603 |
| [42] | Wojtaszczyk, P., The Banach space \(H_1\), (Functional Analysis: Surveys and Recent Results, III. Functional Analysis: Surveys and Recent Results, III, Paderborn, 1983. Functional Analysis: Surveys and Recent Results, III. Functional Analysis: Surveys and Recent Results, III, Paderborn, 1983, North-Holland Math. Stud., vol. 90 (1984), North-Holland: North-Holland Amsterdam), 1-33 · Zbl 0584.46040 |
| [43] | Wojtaszczyk, P., Banach Spaces for Analysts (1991), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0724.46012 |
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