Some remarks on derivations on the algebra of operators in Hilbert pro-C*-bimodules

TL;DR

This paper investigates derivations on operator algebras over pro-C*-algebras, establishing conditions under which derivations are inner or trivial, especially in the context of commutative and bimodule structures.

Contribution

It proves that innerness of derivations on compact operators implies innerness on adjointable operators in full Hilbert modules and shows derivations are zero in certain commutative cases.

Findings

Innerness of derivations on $K_A(E)$ implies innerness on $L_A(E)$ for full modules.

Derivations on $K_A(E)$ are zero when $A$ is commutative.

Derivations on $L_A(E)$ are zero when $A$ is a commutative $\sigma$-C*-algebra.

Abstract

Suppose $A$ is a pro-C*-algebra. Let $L_{A}(E)$ be the pro-C*-algebra of adjointable operators on a Hilbert $A$-module $E$ and let $K_{A}(E)$ be the closed two sided $*$-ideal of all compact operators on $E$. We prove that if $E$ be a full Hilbert $A$-module, the innerness of derivations on $K_{A}(E)$ implies the innerness of derivations on $L_{A}(E)$. We show 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. Moreover, if $A$ is a commutative $\sigma$-C*-algebra and $E$ is a Hilbert $A$-bimodule then every derivation on $L_{A}(E)$ is zero, too.