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On Axiomatizability of Non-Commutative Lp-Spaces

Published online by Cambridge University Press:  20 November 2018

C. Ward Henson ,
Yves Raynaud  and
Andrew Rizzo
C. Ward Henson
Affiliation:
Mathematics Deptartment, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL 61801, U.S.A.
Yves Raynaud
Affiliation:
Institut de Mathématiques de Jussieu (CNRS), Projet Analyse Fonctionnelle, Case 186, 4 place Jussieu, 75252 Paris, Cedex 05 France
Andrew Rizzo
Affiliation:
Cross and Blue Shield of Illinois, 300 East Randolph Street, Chicago, IL 60601, U.S.A.
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Abstract

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It is shown that Schatten $p$-classes of operators between Hilbert spaces of different (infinite) dimensions have ultrapowers which are (completely) isometric to non-commutative ${{L}_{p}}$-spaces. On the other hand, these Schatten classes are not themselves isomorphic to non-commutative ${{L}_{p}}$ spaces. As a consequence, the class of non-commutative ${{L}_{p}}$-spaces is not axiomatizable in the first-order language developed by Henson and Iovino for normed space structures, neither in the signature of Banach spaces, nor in that of operator spaces. Other examples of the same phenomenon are presented that belong to the class of corners of non-commutative ${{L}_{p}}$-spaces. For $p\,=\,1$ this last class, which is the same as the class of preduals of ternary rings of operators, is itself axiomatizable in the signature of operator spaces.

Keywords

46L5203C6546B2046L0746M07
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 50 , Issue 4 , 01 December 2007 , pp. 519 - 534
Copyright
Copyright © Canadian Mathematical Society 2007

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