Some remarks on derivations on the algebra of operators in Hilbert pro-\(C^*\)-bimodules (Q1714039)

For a Hilbert $C^*$-module $E$ over a $C^*$-algebra $A$, let $L_A(E)$ denote the $C^*$-algebra of all adjointable operators on $E$ and $K_A(E)$ be the closed two-sided $^*$-ideal of all compact operators on $E$. \par \textit{P. T. Li} et al. [Acta Math. Sin., Engl. Ser. 28, No. 8, 1615--1622 (2012; Zbl 1263.47042)] studied the relation between the innerness of derivations on $K_A(E)$ and $L_A(E)$ and proved that, if $A$ is a $\sigma$-unital commutative $C^*$-algebra and $E$ is a full Hilbert $A$-module, then every derivation on $L_A(E)$ is inner if every derivation on $K_A(E)$ is inner. In this paper, the authors show that the above result is valid in the context of pro-$C^*$-algebras and prove that the assumptions of $\sigma$-unitality and commutativity of $A$ are not required in this setting. Further, they consider derivations on the algebra of operators of Hilbert pro-$C^*$-bimodules and prove that, if $A$ is a commutative pro-$C^*$-algebra and $E$ is a Hilbert $A$-bimodule, then every derivation on $K_A(E)$ is zero. Furthermore, they prove that, if $A$ is a commutative $\sigma$-$C^*$-algebra and $E$ is a Hilbert $A$-bimodule, then every derivation on $L_A(E)$ is zero.