| Abstract: | For a commutative C*-algebra ({mathcal {A}}) with unit e and a Hilbert ({mathcal {A}})-module ({mathcal {M}}), denote by End(_{{mathcal {A}}}({mathcal {M}})) the algebra of all bounded ({mathcal {A}})-linear mappings on ({mathcal {M}}), and by End(^*_{{mathcal {A}}}({mathcal {M}})) the algebra of all adjointable mappings on ({mathcal {M}}). We prove that if ({mathcal {M}}) is full, then each derivation on End(_{{mathcal {A}}}({mathcal {M}})) is ({mathcal {A}})-linear, continuous, and inner, and each 2-local derivation on End(_{{mathcal {A}}}({mathcal {M}})) or End(^{*}_{{mathcal {A}}}({mathcal {M}})) is a derivation. If there exist (x_0) in ({mathcal {M}}) and (f_0) in ({mathcal {M}}^{'}), such that (f_0(x_0)=e), where ({mathcal {M}}^{'}) denotes the set of all bounded ({mathcal {A}})-linear mappings from ({mathcal {M}}) to ({mathcal {A}}), then each ({mathcal {A}})-linear local derivation on End(_{{mathcal {A}}}({mathcal {M}})) is a derivation. |
