Derivations on the algebra of operators in hilbert C*-modules

Abstract

Let M be a full Hilbert C*-module over a C*-algebra A, and let End*A(M) be the algebra of adjointable operators on M. We show that if A is unital and commutative, then every derivation of End*A(M) is an inner derivation, and that if A is σ-unital and commutative, then innerness of derivations on “compact” operators completely decides innerness of derivations on End*A(M). If A is unital (no commutativity is assumed) such that every derivation of A is inner, then it is proved that every derivation of End*A(Ln(A)) is also inner, where Ln(A) denotes the direct sum of n copies of A. In addition, in case A is unital, commutative and there exist x0, y0 ∈ M such that <x0, y0〉 = 1, we characterize the linear A-module homomorphisms on End*A(M) which behave like derivations when acting on zero products.