Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-24T05:39:18.664Z Has data issue: false hasContentIssue false
  • English
  • Français

Local complementation and the extension of bilinear mappings

Published online by Cambridge University Press:  05 September 2011

J. M. F. CASTILLO ,
R. GARCÍA ,
A. DEFANT ,
D. PÉREZ-GARCÍA  and
J. SUÁREZ
J. M. F. CASTILLO
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Univ. de Extremadura, Avda. de Elvas s/n, 06071 Badajoz, Spain. e-mail: castillo@unex.es, rgarcia@unex.es
R. GARCÍA
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Univ. de Extremadura, Avda. de Elvas s/n, 06071 Badajoz, Spain. e-mail: castillo@unex.es, rgarcia@unex.es
A. DEFANT
Affiliation:
Institüt fur Mathematik, Universität Oldenburg, Postfach 2503, 26111 Oldenburg, Germany. e-mail: defant@mathematik.uni-oldenburg.de
D. PÉREZ-GARCÍA
Affiliation:
Departamento de Análisis Matemático, Facultad de CC Matemáticas, Univ. Complutense de Madrid, Pza. de Ciencias 3, Ciudad Universitaria, 28040 Madrid, Spain. e-mail: dperez@mat.ucm.es
J. SUÁREZ
Affiliation:
Escuela Politécnica, Universidad de Extremadura, Avenida de la Universidad s/n, 10071 Cáceres, Spain. e-mail: jesus@unex.es
Get access Rights & Permissions [Opens in a new window]

Abstract

We study different aspects of the connections between local theory of Banach spaces and the problem of the extension of bilinear forms from subspaces of Banach spaces. Among other results, we prove that if X is not a Hilbert space then one may find a subspace of X for which there is no Aron–Berner extension. We also obtain that the extension of bilinear forms from all the subspaces of a given X forces such X to contain no uniform copies of ℓpn for p ∈ [1, 2). In particular, X must have type 2 − ϵ for every ϵ > 0. Also, we show that the bilinear version of the Lindenstrauss–Pełczyński and Johnson–Zippin theorems fail. We will then consider the notion of locally α-complemented subspace for a reasonable tensor norm α, and study the connections between α-local complementation and the extendability of α*-integral operators.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 152 , Issue 1 , January 2012 , pp. 153 - 166
Copyright
Copyright © Cambridge Philosophical Society 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Albiac, F. and Kalton, N. J.Topics in Banach Space Theory. of Graduate Texts in Mathematics, vol. 233 (Springer, 2006).Google Scholar
[2]Alspach, D. E.Quotients of c 0 are almost isometric to subspaces of c 0. Proc. Amer. Math. Soc. 76 (1979), 285–289.Google Scholar
[3]Aron, R. and Berner, P.A Hahn–Banach extension theorem for analytic mappings. Bull. Soc. Math. France 106 (1978), 3–24.CrossRefGoogle Scholar
[4]Bennett, G., Dor, L. E., Goodman, V. and Johnson, W. B.On uncomplemented subspaces of L p, 1 < p < 2 Israel J. Math. 26 no. 2 (1977), 178–187.Google Scholar
[5]Bourgain, J.A counterexample to a complementation problem, Comp. Math. 43 (1981), 133–144.Google Scholar
[6]Cabello Sánchez, F. and Castillo, J. M. F.Uniform boundedness and twisted sums of Banach spaces Houston J. Math. 30 (2004), 523–536.Google Scholar
[7]Cabello Sánchez, F., García, R. and Villanueva, I.Extensions of multilinear operator on Banach spaces. Extracta Math. 15 (2000), 291–334.Google Scholar
[8]Carando, D.Extendible polynomials on Banach spaces J. Math. Anal. Appl. 233 (1999), 359–372.Google Scholar
[9]Carando, D. and Dimant, V.Extension of polynomials and John's theorem for symmetric tensor products. Proc. Amer. Math. Soc 135 (2007), 1769–1773.CrossRefGoogle Scholar
[10]Castillo, J. M. F.Snarked sums of Banach spaces Extracta Math 12 (1997), 117–128.Google Scholar
[11]Castillo, J. M. F.Banach spaces, à la recherche du temps perdu Extracta Math 15 (2000), 373–390.Google Scholar
[12]Castillo, J. M. F., García, R. and Jaramillo, J. A.Extension of bilinear forms on Banach spaces Proc. Amer. Math. Soc 129 (2001), 647–656.CrossRefGoogle Scholar
[13]Castillo, J. M. F., García, R. and Jaramillo, J. A.Extension of bilinear forms from subspaces of 1-spaces Ann. Acad. Scientiarum Fennicae 27 (2002), 91–96.Google Scholar
[14]Castillo, J. M. F., García, R. and Suárez, J. Extension and lifting of operators and polynomials Mediterranean J. Math. (to appear).Google Scholar
[15]Davis, W. J., Dean, D. W. and Singer, I.Complemented subspaces and Λ systems in Banach spaces Israel J. Math 6 (1968), 303–309.CrossRefGoogle Scholar
[16]Defant, A. and Floret, K.Tensor norms and Operator ideals. North-Holland Math. Stud. 176 (1993).Google Scholar
[17]Defant, A., García, D., Maestre, M. and Pérez–García, D.Extension of multilinear forms and polynomials from subspaces of 1-spaces. Houston J. Math 33 (2007), no. 3, 839–860.Google Scholar
[18]Defant, A., Maestre, M., and Sevilla–Peris, P.Cotype 2 estimates for spaces of polynomials on sequence spaces. Israel J. Math 129 (2002), 291–315.Google Scholar
[19]Diestel, J., Jarchow, H. and Tonge, A.Absolutely Summing Operators (Cambridge University Press, 1995).CrossRefGoogle Scholar
[20]Dineen, S. Complex analysis on infinite dimensional spaces Monogr. Math. (Berlin, 1999).Google Scholar
[21]Fakhoury, H.Sélections linéaires associées au théorème de Hahn–Banach, J. Funct. Anal. 11 (1972), 436–452.Google Scholar
[22]Fernández Unzueta, M. and Prieto, A.Extension of polynomials defined on subspaces, Math. Proc. Cambridge Philos. Soc. 148 (2010) 505–518.CrossRefGoogle Scholar
[23]Floret, K. and Hunfeld, S.Ultrastability of ideals of homogeneous polynomials and multilinear mappings on Banach spaces, Proc. Amer. Math. Soc. 130, No.5, 1425–1435.Google Scholar
[24]Galindo, P., García, D., Maestre, M. and Mujica, J.Extension of multilinear mappings on Banach spaces, Studia Math. 108 (1994), 55–76.Google Scholar
[25]Godefroy, G., Kalton, N. J. and Lancien, G.Subspaces of c 0ℕ) and Lipschitz isomorphisms, Geometric and Functional Analysis 10 (2000) 798–820.Google Scholar
[26]Jarchow, H., Palazuelos, C., Pérez–García, D. and Villanueva, I.Non-linear Hahn–Banach extensions and summability, J. Math. Anal. Appl. 336 (2007), 1161–1177.Google Scholar
[27]Johnson, W. B. and Zippin, M.Extension of operator from weak*-closed subspaces of ℓ1 an C(K) spaces, Studia Math. 117 (1995), 43–55.Google Scholar
[28]Kalton, N. J. and Peck, N. T.Twisted sums of sequence spaces and the three space problem, Trans. Amer. Math. Soc. 255 (1979), 1–30.Google Scholar
[29]Kalton, N. J.Locally complemented subspaces and p-spaces for 0 < p < 1, Math. Nachr. 115 (1984), 71–97.CrossRefGoogle Scholar
[30]Kirwan, P. and Ryan, R.Extendiblity of homogeneous polynomials on Banach spaces, Proc. Amer. Math. Soc. 126 (1998), 1023–1029.Google Scholar
[31]Lindenstrauss, J. and Pełczyński, A.Contributions to the theory of classical Banach spaces, J. Funct. Anal. 8 (1971), 225–249.Google Scholar
[32]Lindenstrauss, J. and Tzafriri, L.On the complemented subspaces problem, Israel J. Math. 9 (1971), 263–269.Google Scholar
[33]Milman, V. D. and Pisier, G.Banach spaces with a weak cotype 2 property, Israel J. Math. 54 (1986) 139–158.Google Scholar
[34]Moreno, Y. and Plichko, A.On automorphic Banach spaces, Israel J. of Math. 169 (2009), 29–47.Google Scholar
[35]Pérez–García, D.A counterexample using 4-linear forms, Bull. Austral. Math. Soc. 70 (2004), 469–473.CrossRefGoogle Scholar
[36]Pérez–García, D., Wolf, M.M., Palazuelos, C., Villanueva, I. and Junge, M.Quantum Bell inequalities can have arbitrarily large violation, Commun. Math. Phys. 279 (2008), 455–486.Google Scholar
[37]Pietsch, A. gIdeals of multilinear functionals, Proc. 2 Int. Conf. Operator Alg., Ideals and Their Applications in Theoretical Physics, 185199, Teubner-Texte, Leipzig, 1983.Google Scholar
[38]Pisier, G.Factorization of linear operators and geometry of Banch spaces, CBMS 60.Google Scholar
[39]Pisier, G.The volume of convex bodies and Banach space geometry, Cambridge Tracts in Math. 94, Cambridge Univ. Press, Cambridge, 1989.Google Scholar
[40]Tomczak–Jaegermann, N.Banach–Mazur Distances and Finite-Dimensional Operator Ideals, Longman Scientific Technical, 1989.Google Scholar
[41]Zalduendo, I.Extending polynomials – a survey, Publicaciones del Departamento de Análisis Matemático de la Univ. Complutense de Madrid, 41, 1998.Google Scholar