Hereditary C*-Subalgebra Lattices

Abstract

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

References in corpus (4)

• Annihilators and Type Decomposition in C*-Algebras
• Type Decomposition in Posets
• On the decomposition into Discrete, type II and type III $C^{\ast }$-algebras
• *-Annihilators in Proper *-Semigroups

Cited by in corpus (1)

• Locally Compact Stone Duality