Hereditary C*-Subalgebra Lattices

TL;DR

This paper explores the lattice structure of hereditary C*-subalgebras and *-annihilators, providing new characterizations and decompositions that connect order-theoretic and algebraic properties in C*-algebras.

Contribution

It characterizes $igvee$-distributive elements as ideals and establishes the separativity of the *-annihilator ortholattice, enabling new algebraic decompositions.

Findings

Characterization of $igvee$-distributive elements as ideals

Proof that $ ext{P}(A)^ot$ is separative

Enabling C*-algebra type decompositions consistent with von Neumann theory

Abstract

We investigate the connections between order and algebra in the hereditary C*-subalgebra lattice $\mathcal{H}(A)$ and *-annihilator ortholattice $\mathscr{P}(A)^\perp$. 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 $\mathscr{P}(A)^\perp$ is separative, allowing for C*-algebra type decompositions which are completely consistent with the original von Neumann algebra type decompositions.