On Generalized p.q.-Baer $\ast$-rings

We introduced the class of weakly generalized p.q.-Baer $\ast$-rings. It is proved that under some assumptions every weakly generalized p.q.-Baer $\ast$-ring can be embedded in generalized p.q.-Baer $\ast$-ring. We proved that a generalized p.q.-Baer $\ast$-rings has partial comparability. If a generalized p.q.-Baer $\ast$-ring satisfies the parallelogram law then it is proved that every pair of projections has an orthogonal decomposition. A separation theorem for generalized p.q.-Baer $\ast$-rings is obtained. As an application of spectral theory, it is proved that generalized p.q.-Baer $\ast$-rings have a sheaf representation with injective sections.