α-Derivations and their norm in projective tensor products of Γ-Banach algebras

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.

Keywords

• Γ - Banach algebras
• projective tensor products
• α - derivations.