Structure of derivations on various algebras of measurable operators for type I von Neumann algebras

Given a von Neumann algebra M denote by S(M) and LS(M) respectively the algebras of all measurable and locally measurable operators affiliated with M. For a faithful normal semi-finite trace τ on M let S(M, τ) (resp. S0(M, τ)) be the algebra of all τ -measurable (resp. τ -compact) operators from S(M). We give a complete description of all derivations on the above algebras of operators in the case of type I von Neumann algebra M. In particular, we prove that if M is of type I∞ then every derivation on LS(M) (resp. S(M) and S(M, τ)) is inner, and each derivation on S0(M, τ) is spatial and implemented by an element from S(M, τ). 1 Institut für Angewandte Mathematik, Universität Bonn, Wegelerstr. 6, D53115 Bonn (Germany); SFB 611, BiBoS; CERFIM (Locarno); Acc. Arch. (USI), [email protected] 2 Institute of Mathematics and information technologies, Uzbekistan Academy of Sciences, F. Hodjaev str. 29, 100125, Tashkent (Uzbekistan), e-mail: sh [email protected] 3 Karakalpak state university, Ch. Abdirov str. 1, 742012, Nukus (Uzbekistan), e-mail: [email protected] AMS Subject Classifications (2000): 46L57, 46L50, 46L55, 46L60