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On the product of two generalized derivations

Authors: Mohamed Barraa and Steen Pedersen
Journal: Proc. Amer. Math. Soc. 127 (1999), 2679-2683
MSC (1991): Primary 47B47, 46L40
Published electronically: April 15, 1999
MathSciNet review: 1610904
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Abstract: Two elements $A$ and $B$ in a ring $\mathfrak{R}$ determine a generalized derivation $\delta _{A,B}$ on $\mathfrak{R}$ by setting $\delta _{A,B}(X)$ $=AX-XA$ for any $X$ in $\mathfrak{R}$. We characterize when the product $\delta _{C,D}\delta _{A,B}$ is a generalized derivation in the cases when the ring $\mathfrak{R}$ is the algebra of all bounded operators on a Banach space $\mathcal{E}$, and when $\mathfrak{R}$ is a $C^{*}$-algebra $\mathfrak{A}$. We use these characterizations to compute the commutant of the range of $\delta _{A,B}$.

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Additional Information

Mohamed Barraa
Affiliation: Departement de Mathematiques, Faculte des Sciences–Semlalia, University Cadi Ayyad, B.P.: S. 15, 40000 Marrakech, Marocco

Steen Pedersen
Affiliation: Department of Mathematics, Wright State University, Dayton, Ohio 45435

Keywords: Derivation, generalized derivation, elementary operator, $C^{*}$--algebra
Received by editor(s): December 30, 1996
Received by editor(s) in revised form: November 20, 1997
Published electronically: April 15, 1999
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1999 American Mathematical Society

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