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Ultrastability of ideals of homogeneous polynomials and multilinear mappings on Banach spaces

Authors: Klaus Floret and Stephan Hunfeld
Journal: Proc. Amer. Math. Soc. 130 (2002), 1425-1435
MSC (2000): Primary 46B08; Secondary 46B28, 46G25
Published electronically: December 27, 2001
MathSciNet review: 1879966
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Abstract: Using the theory of full and symmetric tensor norms on normed spaces, a theorem of Kürsten and Heinrich on ultrastability and maximality of normed operator ideals is extended to ideals of $n$-homogeneous polynomials and $n$-linear mappings--scalar-valued and vector-valued. The motivation for these results is the following important special case: the ``uniterated'' Aron-Berner extension $\overline{q}^{\mathfrak U}$: $ E'' \longrightarrow F''$ of an $n$-homogeneous polynomial $q: E\longrightarrow F$ to the bidual remains in certain ideals under preservation of the norm. Moreover, Lotz's characterization of maximal normed ideals of linear mappings through appropriate tensor norms is proved for ideals of $n$-homogeneous scalar-valued polynomials and ideals of $n$-linear mappings.

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Additional Information

Klaus Floret
Affiliation: Department of Mathematics, University of Oldenburg, D-26111 Oldenburg, Germany

Stephan Hunfeld
Affiliation: Werstener Dorfstrasse 209, D-40591 Düsseldorf, Germany

Keywords: Tensor products, symmetric tensor products, ideals of polynomials, ideals of $n$-linear mappings, ultraproducts
Received by editor(s): February 9, 1999
Received by editor(s) in revised form: November 20, 2000
Published electronically: December 27, 2001
Communicated by: Dale Alspach
Article copyright: © Copyright 2001 American Mathematical Society