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 σ-C*-algebra and E is a Hilbert A-bimodule then every derivation on L A (E) is zero, too.2010 Mathematics Subject Classification. Primary 46L08; Secondary 46L05, 46H25, 47B47.