2008/05/12 by Andrew S. Toms · 20 citations
Mathematics · #Advanced Operator Algebra Research #Banach space #Conjecture #Dimension (graph theory) #Hilbert space #Holomorphic and Operator Theory #Isomorphism (crystallography) #Semigroup #Separable space #Simple (philosophy) #Spectral Theory in Mathematical Physics #Uncountable set #Unitary state #math.FA #math.OA #msc:46L35 #msc:46L80
paper · pdf · doi:10.1007/s00220-008-0665-4
published in Communications in Mathematical Physics 289(2), 401-433 (Springer Science+Business Media) · 30 pages, no figures
arxiv created 2008/05/12 · openalex publication_date 2008/11/20 · arxiv updated 2015/05/12 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We prove that the C*-algebra of a minimal diffeomorphism satisfies Blackadar's Fundamental Comparability Property for positive elements. This leads to the classification, in terms of K-theory and traces, of the isomorphism classes of countably generated Hilbert modules over such algebras, and to a similar classification for the closures of unitary orbits of self-adjoint elements. We also obtain a structure theorem for the Cuntz semigroup in this setting, and prove a conjecture of Blackadar and Handelman: the lower semicontinuous dimension functions are weakly dense in the space of all dimension functions. These results continue to hold in the broader setting of unital simple ASH algebras with slow dimension growth and stable rank one. Our main tool is a sharp bound on the radius of comparison of a recursive subhomogeneous C*-algebra. This is also used to construct uncountably many non-Morita-equivalent simple separable amenable C*-algebras with the same K-theory and tracial state space, providing a C*-algebraic analogue of McDuff's uncountable family of II1 factors. We prove in passing that the range of the radius of comparison is exhausted by simple C*-algebras.