Derivations and 2-Local Derivations on Matrix Algebras and Algebras of Locally Measurable Operators

Abstract

Let $${\mathcal {A}}$$ be a unital algebra over $${\mathbb {C}}$$ and $${\mathcal {M}}$$ be a unital $${\mathcal {A}}$$-bimodule. We show that every derivation $$D: M_{n}({\mathcal {A}})\rightarrow M_{n}({\mathcal {M}}),$$$$n\ge 2,$$ can be represented as a sum $$D=D_{m}+\overline{\delta },$$ where $$D_{m}$$ is an inner derivation and $$\overline{\delta }$$ is a derivation induced by a derivation $$\delta $$ from $${\mathcal {A}}$$ into $${\mathcal {M}}.$$ If $${\mathcal {A}}$$ commutes with $${\mathcal {M}}$$, we prove that every 2-local inner derivation $$\Delta : M_{n}({\mathcal {A}}) \rightarrow M_{n}({\mathcal {M}})$$, $$n \ge 2$$, is an inner derivation. In addition, if $${\mathcal {A}}$$ is commutative and commutes with $${\mathcal {M}},$$ then every 2-local derivation $$\Delta : M_{n}({\mathcal {A}}) \rightarrow M_{n}({\mathcal {M}})$$, $$n \ge 2$$, is a derivation. Let $${\mathcal {R}}$$ be a finite von Neumann algebra of type $$\text {I}$$ with center $$\mathcal {Z}$$ and $$LS({\mathcal {R}})$$ be the algebra of locally measurable operators affiliated with $${\mathcal {R}}.$$ We also prove that if the lattice $$\mathcal {Z}_{\mathcal {P}}$$ of all projections in $$\mathcal {Z}$$ is atomic, then every derivation $$D:{\mathcal {R}}\rightarrow LS({\mathcal {R}})$$ is an inner derivation.