LAUSR

For a commutative C*-algebra $${\mathcal {A}}$$A with unit e and a Hilbert $${\mathcal {A}}$$A-module $${\mathcal {M}}$$M, denote by End$$_{{\mathcal {A}}}({\mathcal {M}})$$A(M) the algebra of all bounded $${\mathcal {A}}$$A-linear mappings on $${\mathcal {M}}$$M, and by End$$^*_{{\mathcal {A}}}({\mathcal {M}})$$A∗(M) the algebra of all adjointable mappings on $${\mathcal {M}}$$M. We prove that if $${\mathcal {M}}$$M is full, then each derivation on End$$_{{\mathcal {A}}}({\mathcal {M}})$$A(M) is $${\mathcal {A}}$$A-linear, continuous, and inner, and each 2-local derivation on End$$_{{\mathcal {A}}}({\mathcal {M}})$$A(M) or End$$^{*}_{{\mathcal {A}}}({\mathcal {M}})$$A∗(M) is a derivation. If there exist $$x_0$$x0 in $${\mathcal {M}}$$M and $$f_0$$f0 in $${\mathcal {M}}^{'}$$M′, such that $$f_0(x_0)=e$$f0(x0)=e, where $${\mathcal {M}}^{'}$$M′ denotes the set of all bounded $${\mathcal {A}}$$A-linear mappings from $${\mathcal {M}}$$M to $${\mathcal {A}}$$A, then each $${\mathcal {A}}$$A-linear local derivation on End$$_{{\mathcal {A}}}({\mathcal {M}})$$A(M) is a derivation.