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Description of Derivations on Measurable Operator Algebras of Type I
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Description of Derivations on Measurable
Operator Algebras of Type I
S. Albeverio 1, Sh. A. Ayupov 2, K. K. Kudaybergenov 3
February 2, 2008
Abstract
Given a type I von Neumann algebra M with a faithful normal semi-finite
trace τ, let L(M, τ) be the algebra of all τ -measurable operators affiliated with
M. We give a complete description of all derivations on the algebra L(M, τ). In
particular, we prove that if M is of type I∞ then every derivation on L(M, τ) is
inner.
1 Institut fu?r Angewandte Mathematik, Universita?t Bonn, Wegelerstr. 6, D-
53115 Bonn (Germany); SFB 611, BiBoS; CERFIM (Locarno); Acc. Arch. (USI),
albeverio@uni-bonn.de
2 Institute of Mathematics and information technologies, Uzbekistan Academy of
Science, F. Hodjaev str. 29, 100125, Tashkent (Uzbekistan), e-mail: sh ayupov@mail.ru,
e ayupov@
3 Institute of Mathematics and information technologies, Uzbekistan Academy
of Science, F. Hodjaev str. 29, 100125, Tashkent (Uzbekistan), e-mail:
karim2006@mail.ru
AMS Subject Classifications (2000): 46L57, 46L50, 46L55, 46L60
Key words: von Neumann algebras, non commutative integration, measurable
operator, Kaplansky-Hilbert module, type I algebra, derivation, inner derivation.
1
1. Introduction
The present paper is devoted to a complete description of derivations on
the algebra of τ -measurable operators L(M, τ) affiliated with a type I
von Neumann algebra M and a normal faithful semi-finite trace τ.
Given a (complex) algebra A, a linear operator D : A → A is called a
derivation ifD(xy) = D(x)y+xD(y) for all x, y ∈ A. Each element a ∈ A
generates a derivation Da : A → A defined as Da(x) = ax ? xa, x ∈ A.
Such derivations are called inner derivations.
It is well known that all derivation on a von Neumann algebra are inner
and therefore are norm continuous. But the properties of derivations on
the unbounded operator algebra L(M, τ) seem to be very far from being
similar. Indeed, the re
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