Characterizations of categories of commutative C*-subalgebras

TL;DR

This paper characterizes the category of injective *-homomorphisms between commutative C*-subalgebras of a C*-algebra, identifying weakly terminal subalgebras for various classes, thus generalizing the Mackey-Piron programme.

Contribution

It introduces a method to find weakly terminal commutative subalgebras, extending the understanding of subalgebra categories in C*-algebras and their relation to lattice structures.

Findings

Identified weakly terminal commutative subalgebras for all commutative C*-algebras.

Extended the characterization to type I von Neumann algebras.

Provided a categorified perspective on the lattice of subalgebras.

Abstract

We aim to characterize the category of injective *-homomorphisms between commutative C*-subalgebras of a given C*-algebra A. We reduce this problem to finding a weakly terminal commutative subalgebra of A, and solve the latter for various C*-algebras, including all commutative ones and all type I von Neumann algebras. This addresses a natural generalization of the Mackey-Piron programme: which lattices are those of closed subspaces of Hilbert space? We also discuss the way this categorified generalization differs from the original question.