Characterizations of categories of commutative C*-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.

24 pages

References in corpus (6)

• A topos for algebraic quantum theory
• Active lattices determine AW*-algebras
• Noncommutativity as a colimit
• Diagonalizing matrices over AW*-algebras
• Complementarity in categorical quantum mechanics
• No-go theorems for functorial localic spectra of noncommutative rings

Cited by in corpus (4)

• Active lattices determine AW*-algebras
• Infinite-dimensional diagonalization and semisimplicity
• The many classical faces of quantum structures
• Classifying finite-dimensional C*-algebras by posets of their commutative C*-subalgebras