Infinite Dimensional Analysis, Quantum Probability and Related Topics Vol. 13, No. 2 (2010) 305–337 c  World Scientific Publishing Company DOI: 10.1142/S0219025710004085 DERIVATIONS ON ALGEBRAS OF MEASURABLE OPERATORS SH. A. AYUPOV Institute of Mathematics and Information Technologies, Uzbekistan Academy of Sciences, Tashkent 100125, Uzbekistan sh ayupov@mail.ru K. K. KUDAYBERGENOV Karakalpak State University, Nukus 142012, Uzbekistan karim2006@mail.ru Received 1 December 2009 Communicated by L. Accardi The present paper is a survey of recent results concerning derivations on various algebras of measurable operators affiliated with von Neumann algebras. A complete description of derivation is obtained in the case of type I von Neumann algebras. A special section is devoted to the Abelian case, namely to the existence of nontrivial derivations on algebras of measurable function. Local derivations on the above algebras are also considered. Keywords : von Neumann algebras; regular algebra; measurable operator; locally mea- surable operator; central extensions of von Neumann algebras; noncommutative Arens algebras; derivation; inner derivation; spatial derivation; local derivation. AMS Subject Classification: 46L57, 46L50, 46L55, 46L60 1. Introduction This paper is devoted to a survey of recent results concerning derivations on certain classes of unbounded operator algebras. Let A be an algebra over the field complex number. A linear (additive) operator D : A→A is called a linear (additive) derivation if it satisfies the identity D(xy)= D(x)y + xD(y) for all x, y ∈A (Leibniz rule). Each element a ∈A defines a linear derivation D a on A given as D a (x)= ax - xa, x ∈A. Such derivations D a are said to be inner derivations. If the element a implementing the derivation D a on A, belongs to a larger algebra B, containing A (as a proper ideal as usual), then D a is called a spatial derivation. 305