Positive projections as generators of \(J\)-projections of type (B). (Q2470025)

In a Hilbert space \(H\), choose a self-adjoint symmetry operator~\(J\) (it satisfies \(J=J^*=J^{-1}\) and \(J\neq\pm I\)). The bilinear form \(\langle Jx,y\rangle\) defines an indefinite metric; the \(J\)-adjoint of an operator~\(A\) is defined by \(A^\#=JA^*J\). A von Neumann algebra~\(\mathcal{A}\) in \(B(H)\) is said to be of type~\((B)\) if its center contains two projections \(P\) and \(Q\) such that \(I=P+Q\) and \(Q=P^\#\). Finally then, put \(\mathcal{J}=P-Q\) and \(\mathcal{B}=P\,B(H)\,P+Q\,B(H)\,Q\). The \(J\)-projections of type~\((B)\) from the title are the \(J\)-self-adjoint projections from~\(\mathcal{B}\). \noindent The authors provide characterizations of such projections, a description of the \(J\)-projections that are in~\(\mathcal{A}\), a decomposition theorem for and further general properties of \(J\)-projections of type~\((B)\).