Abstract
Let (V,Γ) and (V′,Γ′) be Gamma-Banach algebras over the fields F1 and F2
isomorphic
to a field F
which possesses a real valued valuation, and (V,Γ)⊗p(V′,Γ′), their projective tensor product.
It is shown that if D1
and D2
are α - derivation and α′ - derivation on (V,Γ) and (V′,Γ′) respectively and
u=∑1x1⊗y1, is an arbitrary element of (V,Γ)⊗p(V′,Γ′), then there exists an α⊗α′- derivation
D on
(V,Γ)⊗p(V′,Γ′) satisfying the relation
D(u)=∑1[(D1x1)⊗y1+x1⊗(D2y1)]
and possessing many enlightening properties. The converse is also true under a certain restriction.
Furthermore, the validity of the results ‖D‖=‖D1‖+‖D2‖
and sp(D)=sp(D1)+sp(D2)
are fruitfully investigated.