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Singular twisted sums generated by complex interpolation


Authors: Jesús M. F. Castillo, Valentin Ferenczi and Manuel González
Journal: Trans. Amer. Math. Soc. 369 (2017), 4671-4708
MSC (2010): Primary 46M18, 46B70, 46E30
DOI: https://doi.org/10.1090/tran/6809
Published electronically: November 28, 2016
MathSciNet review: 3632546
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Abstract: We present new methods to obtain singular twisted sums $ X\oplus _\Omega X$ (i.e., exact sequences $ 0\to X\to X\oplus _\Omega X \to X\to 0$ in which the quotient map is strictly singular) when $ X$ is an interpolation space arising from a complex interpolation scheme and $ \Omega $ is the induced centralizer.

Although our methods are quite general, we are mainly concerned with the choice of $ X$ as either a Hilbert space or Ferenczi's uniformly convex Hereditarily Indecomposable space. In the first case, we construct new singular twisted Hilbert spaces (which includes the only known example so far: the Kalton-Peck space $ Z_2$). In the second case we obtain the first example of an H.I. twisted sum of an H.I. space.

During our study of singularity we introduce the notion of a disjointly singular twisted sum of Köthe function spaces and construct several examples involving reflexive $ p$-convex Köthe function spaces (which includes the function space version of the Kalton-Peck space $ Z_2$).

We then use Rochberg's description of iterated twisted sums to show that there is a sequence $ \mathcal F_n$ of H.I. spaces so that $ \mathcal F_{m+n}$ is a singular twisted sum of $ \mathcal F_m$ and $ \mathcal F_n$, while for $ l>n$ the direct sum $ \mathcal F_n \oplus \mathcal F_{l+m}$ is a nontrivial twisted sum of $ \mathcal F_l$ and $ \mathcal F_{m+n}$.


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  • [1] Pietro Aiena, Fredholm and local spectral theory, with applications to multipliers, Kluwer Academic Publishers, Dordrecht, 2004. MR 2070395
  • [2] S. A. Argyros and I. Deliyanni, Examples of asymptotic $ l_1$ Banach spaces, Trans. Amer. Math. Soc. 349 (1997), no. 3, 973-995. MR 1390965, https://doi.org/10.1090/S0002-9947-97-01774-1
  • [3] Spiros A. Argyros and Andreas Tolias, Methods in the theory of hereditarily indecomposable Banach spaces, Mem. Amer. Math. Soc. 170 (2004), no. 806, vi+114. MR 2053392, https://doi.org/10.1090/memo/0806
  • [4] Antonio Avilés, Félix Cabello Sánchez, Jesús M. F. Castillo, Manuel González, and Yolanda Moreno, On separably injective Banach spaces, Adv. Math. 234 (2013), 192-216. MR 3003929, https://doi.org/10.1016/j.aim.2012.10.013
  • [5] Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275
  • [6] H. Bustos Domecq, Vector-valued invariant means revisited, J. Math. Anal. Appl. 275 (2002), no. 2, 512-520. MR 1943762, https://doi.org/10.1016/S0022-247X(02)00639-X
  • [7] Félix Cabello Sánchez, Nonlinear centralizers with values in $ L_0$, Nonlinear Anal. 88 (2013), 42-50. MR 3057047, https://doi.org/10.1016/j.na.2013.04.006
  • [8] Félix Cabello Sánchez, Nonlinear centralizers in homology, Math. Ann. 358 (2014), no. 3-4, 779-798. MR 3175140, https://doi.org/10.1007/s00208-013-0942-1
  • [9] Félix Cabello Sánchez, There is no strictly singular centralizer on $ L_p$, Proc. Amer. Math. Soc. 142 (2014), no. 3, 949-955. MR 3148529, https://doi.org/10.1090/S0002-9939-2013-11851-4
  • [10] Félix Cabello Sánchez and Jesús M. F. Castillo, Duality and twisted sums of Banach spaces, J. Funct. Anal. 175 (2000), no. 1, 1-16. MR 1774849, https://doi.org/10.1006/jfan.2000.3598
  • [11] Félix Cabello Sánchez and Jesús M. F. Castillo, Uniform boundedness and twisted sums of Banach spaces, Houston J. Math. 30 (2004), no. 2, 523-536 (electronic). MR 2084916
  • [12] Félix Cabello Sánchez, Jesús M. F. Castillo, and Nigel J. Kalton, Complex interpolation and twisted twisted Hilbert spaces, Pacific J. Math. 276 (2015), no. 2, 287-307. MR 3374059, https://doi.org/10.2140/pjm.2015.276.287
  • [13] Félix Cabello Sánchez, Jesús M. F. Castillo, and Jesús Suárez, On strictly singular nonlinear centralizers, Nonlinear Anal. 75 (2012), no. 7, 3313-3321. MR 2891170, https://doi.org/10.1016/j.na.2011.12.022
  • [14] R. del Campo, A. Fernández, A. Manzano, F. Mayoral, and F. Naranjo, Complex interpolation of Orlicz spaces with respect to a vector measure, Math. Nachr. 287 (2014), no. 1, 23-31. MR 3153923, https://doi.org/10.1002/mana.201200289
  • [15] María J. Carro, Joan Cerdà, and Javier Soria, Commutators and interpolation methods, Ark. Mat. 33 (1995), no. 2, 199-216. MR 1373022, https://doi.org/10.1007/BF02559707
  • [16] P. G. Casazza and N. J. Kalton, Unconditional bases and unconditional finite-dimensional decompositions in Banach spaces, Israel J. Math. 95 (1996), 349-373. MR 1418300, https://doi.org/10.1007/BF02761046
  • [17] Jesús M. F. Castillo and Manuel González, Three-space problems in Banach space theory, Lecture Notes in Mathematics, vol. 1667, Springer-Verlag, Berlin, 1997. MR 1482801
  • [18] Jesús M. F. Castillo and Yolanda Moreno Salguero, Strictly singular quasi-linear maps, Nonlinear Anal. 49 (2002), no. 7, Ser. A: Theory Methods, 897-903. MR 1895375, https://doi.org/10.1016/S0362-546X(01)00136-5
  • [19] Jesús M. F. Castillo and Yolanda Moreno, On the Lindenstrauss-Rosenthal theorem, Israel J. Math. 140 (2004), 253-270. MR 2054847, https://doi.org/10.1007/BF02786635
  • [20] Jesús M. F. Castillo and Yolanda Moreno, On the bounded approximation property in Banach spaces, Israel J. Math. 198 (2013), no. 1, 243-259. MR 3096639, https://doi.org/10.1007/s11856-013-0019-6
  • [21] R. R. Coifman, M. Cwikel, R. Rochberg, Y. Sagher, and G. Weiss, A theory of complex interpolation for families of Banach spaces, Adv. in Math. 43 (1982), no. 3, 203-229. MR 648799, https://doi.org/10.1016/0001-8708(82)90034-2
  • [22] M. Cwikel, B. Jawerth, M. Milman, and R. Rochberg, Differential estimates and commutators in interpolation theory, Analysis at Urbana, Vol. II (Urbana, IL, 1986-1987) London Math. Soc. Lecture Note Ser., vol. 138, Cambridge Univ. Press, Cambridge, 1989, pp. 170-220. MR 1009191
  • [23] Michael Cwikel, Nigel Kalton, Mario Milman, and Richard Rochberg, A unified theory of commutator estimates for a class of interpolation methods, Adv. Math. 169 (2002), no. 2, 241-312. MR 1926224, https://doi.org/10.1006/aima.2001.2061
  • [24] Per Enflo, Joram Lindenstrauss, and Gilles Pisier, On the ``three space problem'', Math. Scand. 36 (1975), no. 2, 199-210. MR 0383047
  • [25] V. Ferenczi, A uniformly convex hereditarily indecomposable Banach space, Israel J. Math. 102 (1997), 199-225. MR 1489106, https://doi.org/10.1007/BF02773800
  • [26] V. Ferenczi, Quotient hereditarily indecomposable Banach spaces, Canad. J. Math. 51 (1999), no. 3, 566-584. MR 1701326, https://doi.org/10.4153/CJM-1999-026-4
  • [27] Manuel González and José M. Herrera, Finitely decomposable Banach spaces and the three-space property, Arch. Math. (Basel) 80 (2003), no. 6, 647-654. MR 1997530, https://doi.org/10.1007/s00013-003-0494-9
  • [28] W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), no. 4, 851-874. MR 1201238, https://doi.org/10.2307/2152743
  • [29] Jan Gustavsson and Jaak Peetre, Interpolation of Orlicz spaces, Studia Math. 60 (1977), no. 1, 33-59. MR 0438102
  • [30] N. J. Kalton, The three space problem for locally bounded $ F$-spaces, Compositio Math. 37 (1978), no. 3, 243-276. MR 511744
  • [31] N. J. Kalton, Convexity, type and the three space problem, Studia Math. 69 (1980/81), no. 3, 247-287. MR 647141
  • [32] Nigel J. Kalton, Nonlinear commutators in interpolation theory, Mem. Amer. Math. Soc. 73 (1988), no. 385, iv+85. MR 938889, https://doi.org/10.1090/memo/0385
  • [33] N. J. Kalton, Differentials of complex interpolation processes for Köthe function spaces, Trans. Amer. Math. Soc. 333 (1992), no. 2, 479-529. MR 1081938, https://doi.org/10.2307/2154047
  • [34] N. J. Kalton, An elementary example of a Banach space not isomorphic to its complex conjugate, Canad. Math. Bull. 38 (1995), no. 2, 218-222. MR 1335101, https://doi.org/10.4153/CMB-1995-031-4
  • [35] Nigel Kalton and Stephen Montgomery-Smith, Interpolation of Banach spaces, Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1131-1175. MR 1999193, https://doi.org/10.1016/S1874-5849(03)80033-5
  • [36] N. J. Kalton and N. T. Peck, Twisted sums of sequence spaces and the three space problem, Trans. Amer. Math. Soc. 255 (1979), 1-30. MR 542869, https://doi.org/10.2307/1998164
  • [37] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I: Sequence spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92, Springer-Verlag, Berlin-New York, 1977. MR 0500056
  • [38] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II: Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. MR 540367
  • [39] Gilles Pisier, Holomorphic semigroups and the geometry of Banach spaces, Ann. of Math. (2) 115 (1982), no. 2, 375-392. MR 647811, https://doi.org/10.2307/1971396
  • [40] G. Pisier and H. Xu, Non-commmutative $ L^p$-spaces, Chapter 34 in ``Handbook in the geometry of Banach spaces vol. 2'', W. B. Johnson and J. Lindenstrauss (eds.) Elsevier 2003, pp. 1459-1518.
  • [41] Richard Rochberg and Guido Weiss, Derivatives of analytic families of Banach spaces, Ann. of Math. (2) 118 (1983), no. 2, 315-347. MR 717826, https://doi.org/10.2307/2007031
  • [42] Martin Schechter, Complex interpolation, Compositio Math. 18 (1967), 117-147 (1967). MR 0223880
  • [43] Jesús Suárez de la Fuente, The Kalton centralizer on $ L_p[0,1]$ is not strictly singular, Proc. Amer. Math. Soc. 141 (2013), no. 10, 3447-3451. MR 3080167, https://doi.org/10.1090/S0002-9939-2013-11599-6
  • [44] Frédérique Watbled, Complex interpolation of a Banach space with its dual, Math. Scand. 87 (2000), no. 2, 200-210. MR 1795744

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Additional Information

Jesús M. F. Castillo
Affiliation: Departamento de Matemáticas, Universidad de Extremadura, Avenida de Elvas s/n, 06011 Badajoz, España
Email: castillo@unex.es

Valentin Ferenczi
Affiliation: Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, rua do Matão 1010, 05508-090 São Paulo SP, Brazil – and – Equipe d’Analyse Fonctionnelle, Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie - Paris 6, Case 247, 4 place Jussieu, 75252 Paris Cedex 05, France
Email: ferenczi@ime.usp.br

Manuel González
Affiliation: Departamento de Matemáticas, Universidad de Cantabria, Avenida de los Castros s/n, 39071 Santander, España
Email: manuel.gonzalez@unican.es

DOI: https://doi.org/10.1090/tran/6809
Received by editor(s): January 16, 2015
Received by editor(s) in revised form: July 10, 2015
Published electronically: November 28, 2016
Additional Notes: This research was supported by Project MTM2013-45643, D.G.I. Spain
The research of the second author was supported by Fapesp project 2013/11390-4, including visits of the first and third authors to the University of São Paulo
Article copyright: © Copyright 2016 American Mathematical Society

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