Derivations, Local and 2-Local Derivations on Some Algebras of Operators on Hilbert C*-Modules

For a commutative C*-algebra A with unit e and a Hilbert A-module M, denote by End(A)(M) the algebra of all bounded A-linear mappings on M, and by End*(A)(M) the algebra of all adjointable mappings onM. We prove that ifMis full, then each derivation on End(A)(M) is A-linear, continuous, and inner, and each 2-local derivation on EndA(M) or End*(A)(M) is a derivation. If there exist x(0) in M and f(0) in M', such that f(0)(x(0)) = e, where M' denotes the set of all bounded A-linear mappings fromMto A, then each A-linear local derivation on End(A)(M) is a derivation.