Bicyclic \(WJ^*\)-algebras in Pontryagin space of type \(\Pi_ 1\) (Q1324673)

A Pontryagin space of Type \(\text{II}_ 1\) is a Hilbert space \(H\) equipped with an indefinite inner product defined by \((\xi, \eta)= \langle J\xi, \eta \rangle\), where, for orthogonal projections \(P_ +\), of dimension one, and \(P_ -\), \(J= P_ +- P_ -\). A subalgebra \(M\) of \(B(H)\) is said to be a WJ*-algebra if, for \(a\) in \(M\), the adjoint \(a^*\) of \(a\) relative to the inner product \((.,.)\) also lies in \(M\). By carefully analyzing the structure of WJ*-algebras acting on a Pontryagin space of Type \(\text{II}_ 1\) the author is able to prove an analogue of the Tomita theorem for the case in which the WJ*-algebra possesses a cyclic and separating vector.