On plus-operators from von Neumann algebras (Q2387001)

The authors propose an original way to organize a von Neumann algebra \({\mathcal M}\) on a Hilbert space \((H,(.,.))\) as an indefinite inner product space. Let \({\mathcal Z}\) denote the center of \({\mathcal M}\) and \({\mathcal Z}^{pr}\) be the set of orthogonal projectors from \({\mathcal Z}\). On the set \({\mathcal Z}^+\) of nonnegative central operators, the order \({\mathcal Z}_1\leq{\mathcal Z}_2\) is defined by \(({\mathcal Z}_1 x,x)\leq ({\mathcal Z}_2x, x)\). Using some orthogonal projectors \(P^+\) and \(P^-= I- P^+\) in \({\mathcal M}\), the authors obtain a canonical symmetry \(J= P^++ P^-\), which leads to the indefinite inner product \([x, y]= (Jx,y)\). The operator \(V\in{\mathcal M}\) is non-contractive if \([Vx, Vx]\geq[x, x]\), \(\forall x\in H\). The support \(Q_x\) of a vector \(x\in H\) is the least projector in \({\mathcal Z}\) for which \(x= Q_x x\). Because each positive \(x\) generates a projector \(Q_{+x}\in{\mathcal Z}\), such that \([qx, x]> 0\) for all \(q\in{\mathcal Z}^{pr}\), \(q\leq Q_{+x}\), we may define \(J_{{\mathcal Z}}\)-positive (-nonnegative, etc.) vectors by \(Q_x= Q_{+x}\) (\(Q_{-x}= 0\), etc.). Finally, the \(J_{{\mathcal Z}}\)-plus-operators are defined by the condition \(V\beta^+\subseteq\beta^+\), where \(\beta^+\) is the set of all \(J_{{\mathcal Z}}\)-nonnegative vectors. These \(J_{{\mathcal Z}}\)-plus-operators and the non-contractive operators are studied in connection with other types of operators.