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Are Non-Commutative Lp Spaces Really Non-Commutative?

Published online by Cambridge University Press:  20 November 2018

A. Katavolos
A. Katavolos*
Affiliation:
University of Crete, Herakleion, Crete
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1. The central objects in integration theory can be considered to be an abelian Von Neumann algebra, L∞, of the measure space, together with a (not necessarily finite-valued) positive linear functional on it, the integral (see [10]). It is natural, therefore, to attempt to construct a “non-commutative” integration theory starting with a non-abelian Von Neumann algebra. Segal [9] and Dixmier [2] have developed such a theory, and constructed the Non-Commutative Lp spaces associated with a Von Neumann algebra M and a normal, faithful, semifinite trace (i.e. a unitarily invariant weight) t on M. They show that there exists a unique ultra-weakly dense *-ideal J of M such that t (extends to) a positive linear form on J . A generalisation of the Hölder inequality then shows that, for 1 ≦ p < ∞, the function

is a norm on J, denoted by || • ||p.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 6 , 01 December 1981 , pp. 1319 - 1327
Copyright
Copyright © Canadian Mathematical Society 1981

References

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