Structural Properties of Elementary Operators

Abstract

Let and denote complex Banach algebras and let b e a left Banach module and a right Banach -module. Ifwe define the bounded linear elementary operator R(A, B), acting on , byFor the case , elementary operators were introduced by Lumer and Rosenblum [19], who studied their spectral properties. In this setting many authors subsequently studied spectral, algebraic, metric, and structural properties of elementary operators, with particular attention devoted to the inner derivations δa (δa(x) = ax – xa) [25], generalized derivations τ(a, b) (τ(a, b)(x) = ax – xb) [9, 10], and elementary multiplications S(a, b) (S(a, b)(x) = axb), including left and right multiplications La and Rb [11].