2015/04/30 by Chris Heunen, Bert Lindenhovius · 7 citations
Computer Science · Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebra over a field #Algebraic number #Algebraic structures and combinatorial models #Commutative property #Isomorphism (crystallography) #Mathematical analysis #Mathematics #Order (exchange) #Pure mathematics #cs.LO #math.OA #msc:03G05 #msc:06B35 #msc:06C15 #msc:06E15 #msc:46L05
paper · pdf · open access · doi:10.1017/s0960129518000464
published in Mathematical Structures in Computer Science 29(7), 972-1006 (Cambridge University Press) · 42 pages
arxiv created 2019/02/27 · openalex publication_date 2019/03/21 · openalex created_date 2019/06/27 · arxiv updated 2020/12/03 · openalex updated_date 2026/08/05
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.