LAUSR

The first main theorem of this paper asserts that any $$(\sigma , \tau )$$(σ,τ)-derivation d, under certain conditions, either is a $$\sigma $$σ-derivation or is a scalar multiple of ($$\sigma - \tau $$σ-τ), i.e. $$d = \lambda (\sigma - \tau )$$d=λ(σ-τ) for some $$\lambda \in \mathbb {C} \backslash \{0\}$$λ∈C\{0}. By using this characterization, we achieve a result concerning the automatic continuity of $$(\sigma , \tau $$(σ,τ)-derivations on Banach algebras which reads as follows. Let $$\mathcal {A}$$A be a unital, commutative, semi-simple Banach algebra, and let $$\sigma , \tau : \mathcal {A} \rightarrow \mathcal {A}$$σ,τ:A→A be two distinct endomorphisms such that $$\varphi \sigma (\mathbf e )$$φσ(e) and $$\varphi \tau (\mathbf e )$$φτ(e) are non-zero complex numbers for all $$\varphi \in \Phi _\mathcal {A}$$φ∈ΦA. If $$d : \mathcal {A} \rightarrow \mathcal {A}$$d:A→A is a $$(\sigma , \tau )$$(σ,τ)-derivation such that $$\varphi d$$φd is a non-zero linear functional for every $$\varphi \in \Phi _\mathcal {A}$$φ∈ΦA, then d is automatically continuous. As another objective of this research, we prove that if $$\mathfrak {M}$$M is a commutative von Neumann algebra and $$\sigma :\mathfrak {M} \rightarrow \mathfrak {M}$$σ:M→M is an endomorphism, then every Jordan $$\sigma $$σ-derivation $$d:\mathfrak {M} \rightarrow \mathfrak {M}$$d:M→M is identically zero.