Domains of commutative C*-subalgebras

Abstract

A C*-algebra is determined to a great extent by the partial order of its commutative C*-algebras. We study order-theoretic properties of this dcpo. Many properties coincide: the dcpo is, equivalently, algebraic, continuous, meet-continuous, atomistic, quasi-algebraic, or quasi-continuous, if and only if the C*-algebra is scattered. For C*-algebras with enough projections, these properties are equivalent to finite-dimensionality. Approximately finite-dimensional elements of the dcpo correspond to Boolean subalgebras of the projections of the C*-algebra, which determine the projections up to isomorphism. Scattered C*-algebras are finite-dimensional if and only if their dcpo is Lawson-scattered. General C*-algebras are finite-dimensional if and only if their dcpo is order-scattered.

42 pages

References in corpus (7)

• Semantics for a Quantum Programming Language by Operator Algebras
• Classifying finite-dimensional C*-algebras by posets of their commutative C*-subalgebras
• Boolean subalgebras of orthoalgebras
• Domains of commutative C*-subalgebras
• Piecewise Boolean algebras and their domains
• C*-Algebras With and Without $\ll$-Increasing Approximate Units
• Dye's Theorem and Gleason's Theorem for AW*-algebras

Cited by in corpus (2)

• Bohrification: From classical concepts to commutative algebras
• Domains of commutative C*-subalgebras