  We generalize some basic C*-algebra and von Neumann algebra theory on
hereditary C*-subalgebras and projections. In particular, we extend Murray-von
Neumann equivalence from projections to *-annihilators and show that several of
its important properties also extend from von Neumann algebras to proper
*-semigroups. Furthermore, we show how to obtain *-annihilator decompositions
of a proper *-semigroup that correspond to classical von Neumann algebra type
decompositions.
