Journal of Operator Theory

Volume 66, Issue 2, Fall 2011 pp. 261-300.

Derivations in algebras of operator-valued functions

Authors: A.F. Ber (1), B. de Pagter (2), and F.A. Sukochev (3)
Author institution: (1) Chief software developer, ISV "Solutions", Tashkent, Uzbekistan
(2) Delft Institute of Applied Mathematics, Faculty EEMCS, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands
(3) School of Mathematics and Statistics, University of New South Wales, Kensington, NSW 2052, Australia

Summary: In this paper we study derivations in subalgebras of $L_{0}^{\mathrm{wo}} ( \nu ;% \mathcal{L} ( X ) \! ) $, the algebra of all weak operator measurable functions $f:S\rightarrow \mathcal{L} ( X ) $, where $% \mathcal{L} ( X ) $ is the Banach algebra of all bounded linear operators on a Banach space $X$. It is shown, in particular, that all derivations on $L_{0}^{\mathrm{wo}} ( \nu ;\mathcal{L} ( X ) ) $ are inner whenever $X$ is separable and infinite dimensional. This contrasts strongly with the fact that $L_{0}^{\mathrm{wo}} ( \nu ;\mathcal{L} ( X ) ) $ admits non-trivial non-inner derivations whenever $X$ is finite dimensional and the measure $\nu $ is non-atomic. As an application of our approach, we study derivations in various algebras of measurable operators affiliated with von Neumann algebras.

Keywords: derivations, operator valued functions, non-commutative integration, measurable operators

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