Domains of commutative C*-subalgebras

Abstract

A C*-algebra is determined to a great extent by the partial order of its commutative C*-subalgebras. We study order-theoretic properties of this directed-complete partially ordered (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. Scattered C*-algebras are finitedimensional if and only if their dcpo is Lawson-scattered. General C*-algebras are finite-dimensional if and only if their dcpo is order-scattered.

The Many Classical Faces of Quantum Structures

“…This information is already enough to determine the piecewise structure of A, but as a Jordan algebra. (In fact, considering C(A) as a mere partially ordered set gives precisely the same information as considering it as a diagram [76]. This justifies Definition 2.1.)…”

“…The assignment A → C(C(A)) is not functorial, does not leave commutative C*-algebras invariant, and of course only works for scattered C*-algebras A in the first place [76]. Hence there is no contradiction with Theorem 2.5.…”

Orthogeometries and AW*-algebras

“…Posets of commutative subalgebras of operator algebras have been studied before, for instance in [4] where the poset V(M) of commutative von Neumann subalgebras of a von Neumann algebra M is considered. Since any von Neumann algebra is an AW*-algebra, and the AW*subalgebras of a von Neumann algebra M are the von Neumann subalgebras of M, we obtain [10,11,14,15,22,23] and is in general larger than A(M) in case M is an AW*-algebra. Suppose that ϕ : M → N is a normal Jordan homomorphism between AW*-algebras.…”

“…15. A proper orthogeometry morphism α : G 1 → G 2 is (finite) join preserving if it takes a minimal cone e of a (finite) set D of directions to a minimal cone f α (e) of f α[D].…”

Symmetries in Exact Bohrification

“…The intuition behind this criterion is that C ∈ C AF (A) if and only if it is generated by its projections, and any atom of C (A) is a C * -subalgebra of A that is generated by single proper projection in A. For details, we refer to Heunen & Lindenhovius (2015).…”