Non Commutative Arens Algebras and their Derivations

Details Non Commutative Arens Algebras and their Derivations Non Commutative Arens Algebras and their Derivations

2007-03-06

arxiv.org/abs/math/0703170v1

Mathematics

Functional Analysis

19 pages. Submitted to Journal of Functional analysis

Scientific paper

Given a von Neumann algebra $M$ with a faithful normal semi-finite trace $\tau,$ we consider the non commutative Arens algebra $L^{\omega}(M, \tau)=\bigcap\limits_{p\geq1}L^{p}(M, \tau)$ and the related algebras $L^{\omega}_2(M, \tau)=\bigcap\limits_{p\geq2}L^{p}(M, \tau)$ and $M+L^{\omega}_2(M, \tau)$ which are proved to be complete metrizable locally convex *-algebras. The main purpose of the present paper is to prove that any derivation of the algebra $M+L^{\omega}_2(M, \tau)$ is inner and all derivations of the algebras $L^{\omega}(M,\tau)$ and $L^{\omega}_2(M, \tau)$ are spatial and implemented by elements of $M+L^{\omega}_2(M, \tau).$

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