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Completely continuous multilinear operators
on $C(K)$ spaces

Author: Ignacio Villanueva
Journal: Proc. Amer. Math. Soc. 128 (2000), 793-801
MSC (1991): Primary 46E15, 46B25
Published electronically: September 9, 1999
MathSciNet review: 1670435
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Abstract: Given a $k$-linear operator $T$ from a product of $C(K)$ spaces into a Banach space $X$, our main result proves the equivalence between $T$ being completely continuous, $T$ having an $X$-valued separately $\omega^*-\omega^*$ continuous extension to the product of the biduals and $T$ having a regular associated polymeasure. It is well known that, in the linear case, these are also equivalent to $T$ being weakly compact, and that, for $k>1$, $T$ being weakly compact implies the conditions above but the converse fails.

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Additional Information

Ignacio Villanueva
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain

Keywords: $C(K)$ spaces, completely continuous, multilinear operators, Aron-Berner extension
Received by editor(s): March 8, 1998
Received by editor(s) in revised form: April 24, 1998
Published electronically: September 9, 1999
Additional Notes: This work was partially supported by DGES grant PB97-0240.
Communicated by: Dale Alspach
Article copyright: © Copyright 1999 American Mathematical Society

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