Journal of Operator Theory
Volume 67, Issue 2, Spring 2012 pp. 495-510.
Additive derivations on algebras of measurable operatorsAuthors: 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 mix(M) of M. We show that 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 mix(M) is inner. In particular each derivation on the algebra LS(M), where M is a type I∞ or a type III von Neumann algebra, is inner.
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