On the product of two generalized derivations

Barraa, Mohamed; Pedersen, Steen

doi:10.1090/S0002-9939-99-04899-6

On the product of two generalized derivations
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by Mohamed Barraa and Steen Pedersen PDF

Proc. Amer. Math. Soc. 127 (1999), 2679-2683 Request permission

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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