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

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

International Journal of Mathematics and Mathematical Sciences, 1998, vol. 21, 1-10

Abstract:

Let ( V , Γ ) and ( V ′ , Γ ′ ) be Gamma-Banach algebras over the fields F 1 and F 2 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 D 1 and D 2 are α - derivation and α ′ - derivation on ( V , Γ ) and ( V ′ , Γ ′ ) respectively and u = ∑ 1 x 1 ⊗ y 1 , 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 [ ( D 1 x 1 ) ⊗ y 1 + x 1 ⊗ ( D 2 y 1 ) ] and possessing many enlightening properties. The converse is also true under a certain restriction. Furthermore, the validity of the results ‖ D ‖ = ‖ D 1 ‖ + ‖ D 2 ‖ and sp ( D ) = sp ( D 1 ) + sp ( D 2 ) are fruitfully investigated.

Date: 1998
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:278960

DOI: 10.1155/S0161171298000490

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