We study generalized derivations G defined on a complex Banach algebra A such that the spectrum σ(Gx) is finite for all x ∈ A. In particular, we show that if A is unital and semisimple, then G is inner and implemented by elements of the socle of A.
Let A be a Banach algebra, and let d: A → A be a continuous derivation such that each element in the range of d has a finite spectrum. In a series of papers it has been proved that such a derivation is an inner derivation implemented by an element from the socle modulo the radical of A (a precise formulation of this statement can be found in the Introduction). The aim of this paper is twofold: we extend this result to the case where d is not necessarily continuous, and we give a complete description...
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