Document Type Doctoral Thesis Author Van Zyl, Augustinus Johannes gusti.vanzyl@up.ac.za URN etd-07142009-180520 Document Title Metrical aspects of the complexification of tensor products and tensor norms Degree PhD Department Mathematics and Applied Mathematics Supervisor

Advisor Name Title Prof J Swart Committee Chair Prof J Diestel Committee Co-Chair Keywords

- tensor products
- Banach spaces
- complexification
Date 2009-07-08 Availability unrestricted AbstractWe study the relationship between real and complex tensor norms.The theory of tensor norms on tensor products of Banach spaces, was developed, by A. Grothendieck, in his

Resumé de la théorie métrique des produits tensoriels topologiques[3]. In this monograph he introduced a variety of ways to assign norms to tensor products of Banach spaces. As is usual in functional analysis, the real-scalar theory is very closely related to the complex-scalar theory. For example, there are, up to top ological equivalence, fourteen ``natural' tensor norms in each of the real-scalar and complex-scalar theories. This correspondence was remarked upon in theResumé, but without proving any formal relationships, although hinting at a certain injective relationship between real and complex (topological) equivalence classes of tensor norms.We make explicit connections between real and complex tensor norms in two different ways. This divides the dissertation into two parts.

In the first part, we consider the ``complexifications' of real Banach spaces and find tensor norms and complexification procedures, so that the complexification of the tensor product, which is itself a Banach space, is isometrically isomorphic to the tensor product of the complexifications. We have results for the injective tensor norm as well as the projective tensor norm.

In the second part we look for isomorphic results rather than isometric. We show that one can define the complexification of real tensor norm in a natural way. The main result is that the complexification of real topological equivalence classes that is induced by this definition, leads to an injective correspondence between the real and the complex tensor norm equivalence classes.

© 2009, University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria.

Van Zyl, AJ 2009,

Please cite as follows:Metrical aspects of the complexification of tensor products and tensor norms, PhD thesis, University of Pretoria, Pretoria, viewed yymmdd < http://upetd.up.ac.za/thesis/available/etd-07142009-180520/ >C206/ag

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