Derivations, local and 2-local derivations on some algebras of operators on Hilbert C*-modules

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${}_{\mathcal{A}}^{\ast }(\mathcal{M})$ 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${}_{\mathcal{A}}^{\ast }(\mathcal{M})$ is a derivation. If there exist $x_0$ in $\mathcal{M}$ and $f_0$ in ${\mathcal{M}}^{{}^{\prime }}$, such that $f_0(x_0)=e$, where ${\mathcal{M}}^{{}^{\prime }}$ 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.

12 pages