Domains of commutative C*-subalgebras

TL;DR

This paper explores the order-theoretic properties of the poset of commutative C*-subalgebras, revealing that many properties coincide with the algebra being scattered or finite-dimensional, and characterizing these via projections.

Contribution

It establishes equivalences between various order-theoretic properties of the dcpo of commutative subalgebras and the algebra's structural features, such as being scattered or finite-dimensional.

Findings

Many properties of the dcpo coincide for scattered C*-algebras.

Finite-dimensionality is characterized by properties of the dcpo for algebras with enough projections.

Approximately finite-dimensional elements correspond to Boolean subalgebras of projections.

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.