Positive operators on tensor products which are the tensor product of their restrictions (Q1120113)

Let E, F, \(E_ 1\), \(F_ 1\) be Banach lattices possessing quasi-interior positive elements e, u, \(e_ 1\), \(u_ 1\). Further on, let \(\psi\) : \(E\to {\mathbb{R}}\) be a positive linear functional, \(\delta\) : \(F\to {\mathbb{R}}^ a \)lattice homomorphism, \(S_ 0: E\to E_ 1\) a positive operator, \(T_ 0: F\to F_ 1\) a lattice homomorphism, such that \(\psi (e)=\delta (u)=1\), \(S_ 0(e)=e_ 1\) and \(T_ 0(u)=u_ 1\). We show that \(\psi\) \(\otimes \delta\) is the only positive linear functional on E\({\tilde \otimes}_{\ell}F\) coinciding with \(\psi\) on E and with \(\delta\) on F, resp. that \(S_ 0\otimes T_ 0\) is the only positive operator on E\({\tilde \otimes}_{\ell}F\) which coincides with \(S_ 0\) on E and with \(T_ 0\) on F. The proof uses the Kakutani-Krein theorem and an extreme point argument. A variant of this result for a Banach lattice F with the principal projection property is proved by elementary calculations with components. Some possible applications of the results are indicated. The paper finishes with analogous results for tensor products of states and pure states resp. tensor products of positive operators and pure completely positive operators on \(C^*\)-algebras, as well as with variants of this result for particular involutive Banach algebras.