On Axiomatizability of Non-Commutative $L_p$-Spaces

Canadian mathematical bulletin

关键词: graph decomposition;    resolvable designs;   

DOI  :  10.4153/CMB-2007-051-7

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

It is shown that Schatten $p$-classesof operators between Hilbert spaces of different (infinite)dimensions have ultrapowers which are (completely) isometric tonon-commutative $L_p$-spaces. On the other hand, these Schattenclasses are not themselves isomorphic to non-commutative $L_p$spaces. As a consequence, the class of non-commutative $L_p$-spacesis not axiomatizable in the first-order language developed byHenson and Iovino for normed space structures, neither in thesignature of Banach spaces, nor in that of operator spaces. Otherexamples of the same phenomenon are presented that belong to theclass of corners of non-commutative $L_p$-spaces. For $p=1$ thislast class, which is the same as the class of preduals of ternaryrings of operators, is itself axiomatizable in the signature ofoperator spaces.

【 授权许可】

Unknown