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Genericity and amalgamation of classes of Banach spaces. (English) Zbl 1109.03047
Summary: We study universality problems in Banach space theory. We show that if \(A\) is an analytic class, in the Effros-Borel structure of subspaces of \(C([0,1])\), of non-universal separable Banach spaces, then there exists a non-universal separable Banach space \(Y\), with a Schauder basis, that contains isomorphs of each member of \(A\) with the bounded approximation property. The proof is based on the amalgamation technique of a class \({\mathcal C}\) of separable Banach spaces, introduced in the paper. We show, among others, that there exists a separable Banach space \(R\) not containing \(L^1(0,1)\) such that the indices \(\beta\) and \(r_{ND}\) are unbounded on the set of Baire-1 elements of the ball of the double dual \(R^{**}\) of \(R\). This answers two questions of H. P. Rosenthal.
We also introduce the concept of a strongly bounded class of separable Banach spaces. A class \({\mathcal C}\) of separable Banach spaces is strongly bounded if for every analytic subset \(A\) of \({\mathcal C}\) there exists \(Y\in{\mathcal C}\) that contains all members of \(A\) up to isomorphism. We show that several natural classes of separable Banach spaces are strongly bounded, among them the class of non-universal spaces with a Schauder basis, the class of reflexive spaces with a Schauder basis, the class of spaces with a shrinking Schauder basis and the class of spaces with Schauder basis not containing a minimal Banach space \(X\).

03E15 Descriptive set theory
46B03 Isomorphic theory (including renorming) of Banach spaces
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46B70 Interpolation between normed linear spaces
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[1] Allexandrov, G.; Kutzarova, D.; Plichko, A., A separable space with no Schauder decomposition, Proc. amer. math. soc., 127, 2805-2806, (1999) · Zbl 0921.46007
[2] Alspach, D.; Argyros, S.A., Complexity of weakly null sequences, Dissertationes math., 321, 1-44, (1992) · Zbl 0787.46009
[3] Argyros, S.A., A universal property of reflexive hereditarily indecomposable Banach spaces, Proc. amer. math. soc., 129, 3231-3239, (2001) · Zbl 0987.46009
[4] Argyros, S.A.; Felouzis, V., Interpolating hereditarily indecomposable Banach spaces, J. amer. math. soc., 13, 243-294, (2001) · Zbl 0956.46014
[5] Argyros, S.A.; Kanellopoulos, V., Optimal sequences of continuous functions converging to a Baire-1 function, Math. ann., 324, 689-729, (2002) · Zbl 1034.46008
[6] Argyros, S.A.; Manoussakis, A., An indecomposable and unconditionally saturated Banach space, Studia math., 159, 1-32, (2003) · Zbl 1062.46013
[7] Argyros, S.A.; Todorčević, S., Ramsey methods in analysis, Adv. courses math. CRM Barcelona, (2005), Birkhäuser Basel · Zbl 1092.46002
[8] Argyros, S.A.; Tolias, A., Methods in the theory of hereditarily indecomposable Banach spaces, Mem. amer. math. soc., 806, (2004) · Zbl 1055.46004
[9] Argyros, S.A.; Godefroy, G.; Rosenthal, H.P., Descriptive set theory and Banach spaces, () · Zbl 1121.46008
[10] Bellenot, S.F.; Haydon, R.; Odell, E., Quasi-reflexive and tree spaces constructed in the spirit of R.C. James, (), 19-43
[11] Benyamini, Y.; Lindenstrauss, J., Geometric nonlinear functional analysis, Amer. math. soc. colloq. publ., vol. 48, (2000), Amer. Math. Soc. Providence, RI · Zbl 0946.46002
[12] Bossard, B., Codages des espaces de Banach separables. familles analytiques ou coanalytiques d’espaces de Banach, C. R. acad. sci. Paris ser. I math., 316, 1005-1010, (1993) · Zbl 0788.46007
[13] B. Bossard, Théorie descriptive des ensembles en géométrie des espaces de Banach, Thése, Univ. Paris VI, 1994
[14] Bossard, B., On a problem of H.P. Rosenthal, Houston J. math., 26, 1-16, (2000) · Zbl 0987.46016
[15] Bossard, B., A coding of separable Banach spaces. analytic and coanalytic families of Banach spaces, Fund. math., 172, 117-152, (2002) · Zbl 1029.46009
[16] Bourgain, J., On separable Banach spaces, universal for all separable reflexive spaces, Proc. amer. math. soc., 79, 241-246, (1980) · Zbl 0438.46005
[17] Bourgain, J., On convergent sequences of continuous functions, Bull. soc. math. belg. Sér. B, 32, 235-249, (1980) · Zbl 0474.54008
[18] Bourgain, J., The szlenk index and operators on \(C(K)\) spaces, Bull. soc. math. belg. Sér. B, 31, 87-117, (1979) · Zbl 0438.46014
[19] Casazza, P.G., Approximation properties, () · Zbl 0743.41027
[20] Davis, W.J.; Figiel, T.; Johnson, W.B.; Pelczynski, A., Factoring weakly compact operators, J. funct. anal., 17, 311-327, (1974) · Zbl 0306.46020
[21] Diestel, J., Sequences and series in Banach spaces, Grad. texts in math., vol. 92, (1984), Springer
[22] Farmaki, V., Classifications of Baire-1 functions and \(c_0\)-spreading models, Trans. amer. math. soc., 345, 819-831, (1994) · Zbl 0867.46014
[23] Godefroy, G., Renormings of Banach spaces, () · Zbl 1009.46003
[24] Gowers, W.T.; Maurey, B., The unconditional basic sequence problem, J. amer. math. soc., 6, 851-874, (1993) · Zbl 0827.46008
[25] Grothendieck, A., Critères de compacité dans LES espaces fonctionelles généraux, Amer. J. math., 74, 168-186, (1952) · Zbl 0046.11702
[26] Haydon, R.; Odell, E.; Rosenthal, H.P., On certain classes of Baire-1 functions with applications to Banach space theory, (), 1-35 · Zbl 0762.46006
[27] Johnson, W.B., A complementably universal conjugate Banach space and its relation to the approximation problem, Israel J. math., 13, 301-310, (1972)
[28] W.B. Johnson, B. Maurey, G. Schechtman, Weakly null sequences in \(L_1\), preprint · Zbl 1125.46014
[29] Kechris, A.S., Classical descriptive set theory, Grad. texts in math., vol. 156, (1995), Springer · Zbl 0819.04002
[30] Kechris, A.S.; Louveau, A., Descriptive set theory and the structure of sets of uniqueness, London math. soc. lecture notes ser., vol. 128, (1989), Cambridge Univ. Press · Zbl 0677.42009
[31] Kechris, A.S.; Louveau, A., A classification of Baire class 1 functions, Trans. amer. math. soc., 318, 209-236, (1990) · Zbl 0692.03031
[32] Kechris, A.S.; Woodin, W.H., Ranks on differentiable functions, Mathematika, 32, 252-278, (1986) · Zbl 0618.03024
[33] Kechris, A.S.; Woodin, W.H., A strong boundedness theorem for dilators, Ann. pure appl. logic, 52, 93-97, (1991) · Zbl 0735.03019
[34] Lindenstrauss, J.; Tzafriri, L., Classical Banach spaces I and II, (1996), Springer
[35] Lusky, W., A note on Banach spaces containing \(c_0\) or \(C_\infty\), J. funct. anal., 62, 1-7, (1985) · Zbl 0581.46016
[36] Maurey, B.; Rosenthal, H.P., Normalized weakly null sequences with no unconditional subsequences, Studia math., 61, 77-98, (1977) · Zbl 0357.46025
[37] Moschovakis, Y.N., Descriptive set theory, (1980), North Holland Amsterdam · Zbl 0433.03025
[38] R. Neidinger, Properties of Tauberian operators, Dissertation, University of Texas at Austin, 1984
[39] Neidinger, R., Factoring operators through hereditarily \(\ell_p\) spaces, Lecture notes in math., vol. 1166, (1985) · Zbl 0587.47023
[40] Odell, E.; Rosenthal, H.P., A double-dual characterization of separable Banach spaces not containing \(\ell_1\), Israel J. math., 20, 375-384, (1975) · Zbl 0312.46031
[41] E. Odell, Th. Schlumprecht, personal communication
[42] E. Odell, Th. Schlumprecht, A separable reflexive universal for the uniformly convex Banach spaces, in preparation · Zbl 1108.46007
[43] Pelczynski, A., Universal bases, Studia math., 19, 247-268, (1960) · Zbl 0185.37401
[44] Prus, S., Finite-dimensional decompositions with p-estimates and universal Banach spaces, Bull. Polish acad. sci. math., 31, 281-288, (1983) · Zbl 0547.46007
[45] Prus, S., Finite-dimensional decompositions of Banach spaces with \((p, q)\)-estimates, Dissertationes math., 263, 1-45, (1987)
[46] Ramsey, F.P., On a problem of formal logic, Proc. London math. soc., 30, 264-286, (1930) · JFM 55.0032.04
[47] Rosenthal, H.P., On factors of \(C([0, 1])\) with non-separable dual, Israel J. math., 13, 361-378, (1972)
[48] Rosenthal, H.P., A characterization of Banach spaces not containing \(\ell_1\), Proc. natl. acad. sci. USA, 71, 2411-2413, (1974) · Zbl 0297.46013
[49] Rosenthal, H.P., The Banach spaces \(C(K)\), () · Zbl 0179.45702
[50] Schechtman, G., On Pelczynski’s paper “universal bases”, Israel J. math., 22, 181-184, (1975) · Zbl 0316.46014
[51] Schlumprecht, Th., An arbitrarily distortable Banach space, Israel J. math., 76, 81-95, (1991) · Zbl 0796.46007
[52] Szlenk, W., The non-existence of a separable reflexive Banach space universal for all separable reflexive Banach spaces, Studia math., 30, 53-61, (1968) · Zbl 0169.15303
[53] Tomczak-Jaegermann, N., Banach spaces of type p have arbitrarily distortable subspaces, Gafa, 6, 1074-1082, (1996) · Zbl 0867.46012
[54] Zippin, M., Banach spaces with separable duals, Trans. amer. math. soc., 310, 371-379, (1988) · Zbl 0706.46015
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