Complete orthogonal decomposition homomorphisms between matrix ordered Hilbert spaces (Q2701592)

For the Hilbert spaces \(H\) and \(H'\), put \({\mathcal H}_n= H_i\otimes M_n(\mathbb{C})\) and \({\mathcal H}_n'= H'\otimes M_n(\mathbb{C})\). Let \(h\) be a bounded linear map of \(H\) into \(H'\). A bijective linear map \(h\) is called an order isomorphism if \(hH^+= H^{\prime+}\), where \(H^+\) (resp. \(H^{\prime+})\) denotes a selfdual cone in \(H\) (resp. \(H^{\prime+}\)). If \(h_n{\mathcal H}^+_n={\mathcal H}^{\prime +}_n\), for every \(n\), then \(h\) is called a complete order isomorphism; \(h\) is called an orthogonal decomposition homomorphism if \(h\) is \(1\)-positive and \((h\xi, h\eta)= 0\), where \(\xi,\eta\in H^+\) with \((\xi,\eta)= 0\); \(h\) is called a complete orthogonal decomposition homomorphism if \(h_n\) is an orthogonal decomposition homomorphism for every \(n\) and \(h\) is called a complete orthogonal decomposition isomorphism if both \(h\) and \(h^{-1}\) are complete orthogonal decomposition homomorphisms.NEWLINENEWLINENEWLINEIn this paper it is shown that a complete order homomorphism and a complete orthogonal decomposition homomorphism between the noncommutative \(L^2\)-spaces induced respectively an isomorphism and a *-isomorphism between the associated reduced von Neumann algebras.