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UNEXPECTED SUBSPACES OF TENSOR PRODUCTS

Published online by Cambridge University Press:  25 October 2006

FÉLIX CABELLO SÁNCHEZ
Affiliation:
Departamento de Matemáticas, Universidad de Extremadura, Avenida de Elvas, 06071 Badajoz, [email protected]
DAVID PÉREZ-GARCÍA
Affiliation:
Área de Matemática Aplicada, Departamento de Matemáticas y Física Aplicadas y Ciencias de la Naturaleza, Escuela Superior de Ciencias Experimentales y Tecnología, Universidad Rey Juan Carlos, Edificio Departamental II, 28933 Móstoles (Madrid), [email protected]
IGNACIO VILLANUEVA
Affiliation:
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, [email protected]
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Abstract

We describe complemented copies of $\ell_2$ both in $C(K_1)\hat{\otimes}_{\pi} C(K_2)$ when at least one of the compact spaces $K_i$ is not scattered and in $L_1(\mu_1)\hat{\otimes}_{\epsilon} L_1(\mu_2)$ when at least one of the measures is not atomic. The corresponding local construction gives uniformly complemented copies of the $\ell_2^n$ in $c_0\hat{\otimes}_{\pi} c_0$. We continue the study of $c_0\hat{\otimes}_{\pi} c_0$ showing that it contains a complemented copy of Stegall's space $c_0(\ell_2^n)$ and proving that $(c_0\ppi c_0)''$ is isomorphic to $\ell_\infty(\ell_\infty^n\hat{\otimes}_{\pi} \ell_\infty^n)$, together with other results. In the last section we use Hardy spaces to find an isomorphic copy of $L_p$ in the space of compact operators from $L_q$ to $L_r$, where $1<p,q,r<\infty$ and $1/r=1/p+1/q$.

Keywords

Type
Notes and Papers
Copyright
The London Mathematical Society 2006

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Footnotes

Partially supported by BMF 2001-1240 and MTM 2004-02635.