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Functional Analysis Valencia 2000
July 3-7, 2000
Technical University of Valencia (UPV) and University of Valencia (UV)
Valencia, Spain

Organizers
R.M. Aron (Kent State U., USA), K.D. Bierstedt (U. Paderborn, Germany), J. Bonet (UPV), J. Cerdà (U. Barcelona, Spain), H. Jarchow (U. Zürich, Switzerland), M. Maestre (UV), J. Schmets (U. Liège, Belgium)

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The bidual of a tensor product of Banach spaces.
by
Ricardo García
Departamento de Matemáticas, Universidad de Extremadura, 06171, Badajoz, España.
Coauthors: Félix Cabello Sánchez (Universidad de Extremadura)

Given a Banach space X, we construct a ``natural" operator \alpha:X''[^(\otimes)]...[^(\otimes)] X'' --> (X'[^(\otimes)]...[^(\otimes)] X)'' by following an idea of Arens.

Our main result is the following.

If X'' has the bounded approximation property (BAP in short), then \alpha embeds X''[^(\otimes)]...[^(\otimes)] X'' as a locally complemented subspace of (X'[^(\otimes)]...[^(\otimes)] X)''.

This has some interesting consequences:

If X'' has the BAP, L(nX'') is a complemented subspace of L(nX)'' and P(nX'') is a complemented subspace of P(nX)''.

This improves results by Jaramillo, Prieto and Zalduendo. (They proved that there is a surjection \beta from P(nX)'' onto P(nX'') under the same hypothesis on X''.)

Sample application. Suppose X'' has the BAP. Then the space of holomorphic functions of bounded type Hb(X'') is a complemented subspace of Hb(X)'' (endowed with the strong topology).

Sample application (independently obtained by González and Gutiérrez). The bidual of c0[^(\otimes)] c0 lacks the Dunford-Pettis property.

Sample application. Suppose X' has cotype 2. Then X''[^(\otimes)] X'' is a closed subspace of (X[^(\otimes)] X)''. This applies to X=K(H), B(H) or P (Pisier's cotype 2 space not having uniformly complemented finite dimensional subspaces) whose biduals do not have even the approximation property.

(T)

Date received: April 12, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caey-20.