T. K. Dutta, H. K. Nath, and R. C. Kalita

Copyright © 1998 T. K. Dutta et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.