\(\alpha\)-derivations and their norm in projective tensor products of \(\Gamma\)-Banach algebras
Summary: Let \((V,\Gamma)\) and \((V', \Gamma')\) be Gamma-Banach algebras over the fields \(F_1\) and \(F_2\) isomorphic to a field \(F\) which possesses a real-valued valuation, and \((V,\Gamma)\otimes_p(V', \Gamma')\), their projective tensor product. It is shown that if \(D_1\) and \(D_2\) are \(\alpha\)-derivation and \(\alpha'\)-derivation on \((V, \Gamma)\) and \((V', \Gamma')\) respectively and \(u= \sum_i x_i\otimes y_i\) is an arbitrary element of \((V,\Gamma) \otimes_p(V', \Gamma')\), then there exists an \(\alpha\otimes \alpha'\)-derivation \(D\) on \((V,\Gamma)\otimes_p (V',\Gamma')\) satisfying the relation \[ D(u)= \sum_i[(D_1 x_i)\otimes y_i+ x_i\otimes (D_2 y_i)] \] 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 \(\text{sp}(D)= \text{sp}(D_1)+ \text{sp}(D_2)\) are fruitfully investigated.
scientific article; zbMATH DE number 1140280