Some Remarks on Derivations on the Algebra of Operators in Hilbert Pro-C*-Bimodules

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.