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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: [email protected], [email protected]
R. GARCÍA
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Univ. de Extremadura, Avda. de Elvas s/n, 06071 Badajoz, Spain. e-mail: [email protected], [email protected]
A. DEFANT
Affiliation:
Institüt fur Mathematik, Universität Oldenburg, Postfach 2503, 26111 Oldenburg, Germany. e-mail: [email protected]
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: [email protected]
J. SUÁREZ
Affiliation:
Escuela Politécnica, Universidad de Extremadura, Avenida de la Universidad s/n, 10071 Cáceres, Spain. e-mail: [email protected]
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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

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