Journal of Operator Theory

Volume 67, Issue 2, Spring 2012 pp. 495-510.

Additive derivations on algebras of measurable operators

Authors: Sh.A. Ayupov (1) and K.K. Kudaybergenov (2)
Author institution: (1) Department of Algebras and Analysis, Inst. of Mathematics and Information Technologies, Tashkent, 100125, Uzbekistan and International Centre for Theoretical Physics, Trieste, Italy
(2) Dept. of Functional Analysis, Karakalpak State University, Nukus, 142012, Uzbekistan

Summary: Given a von Neumann algebra $M$ we introduce so called central extension $\mathrm{mix}(M)$ of $M$. We show that $\mathrm{mix}(M)$ is a $*$-subalgebra in the algebra $LS(M)$ of all locally measurable operators with respect to $M,$ and this algebra coincides with $LS(M)$ if and only if $M $ does not admit type II direct summands. We prove that if $M$ is a properly infinite von Neumann algebra then every additive derivation on the algebra $\mathrm{mix}(M)$ is inner. In particular each derivation on the algebra $LS(M)$, where $M$ is a type I$_\infty$ or a type III von Neumann algebra, is inner.

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