Hereditary C⁎-subalgebra lattices

We investigate the connections between order and algebra in the hereditary C*-subalgebra lattice H(A) and *-annihilator ortholattice P(A)⊥. In particular, we characterize ∨-distributive elements of H(A) as ideals, answering a 25 year old question, allowing the quantale structure of H(A) to be completely determined from its lattice structure. We also show that P(A)⊥ is separative, allowing for C*-algebra type decompositions which are completely consistent with the original von Neumann algebra type decompositions.