Domains of commutative C*-subalgebras. (21st March 2019)

• Record Type: Journal Article
• Authors: Heunen, Chris; Lindenhovius, Bert
• Is Part Of: Mathematical structures in computer science. Volume 29:Number 7(2019)
• Page Start: 972
• Page End: 1006
• Publication Date: 2019-03-21
• Subjects: C*-algebra, -- commutative C*-subalgebra, -- domain, -- scattered; Computer science -- Mathematics -- Periodicals; 004.015105
• DOI: 10.1017/S0960129518000464
• Languages: English
• ISSNs: 0960-1295

Abstract

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.