2004/05/14 by N. Christopher Phillips, Phillips, N. Christopher · 3 citations
Mathematics · #19K14 (Secondary) #46L35 #46L80 #46M40 (Primary) 19A49 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA) #math.OA #msc:19A49 #msc:19K14 #msc:46L35 #msc:46L80 #msc:46M40
paper · pdf · doi:10.48550/arxiv.math/0405265
23 pages, AMSLaTeX
arxiv created 2004/05/14 · openalex publication_date 2004/05/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A be a unital simple direct limit of recursive subhomogeneous C*-algebras with no dimension growth. We give criteria which specify exactly when A has real rank zero, and exactly when A has the Property (SP): every nonzero hereditary subalgebra of A contains a nonzero projection. Specifically, A has real rank zero if and only if the natural map from K0 (A) to the continuous affine functions on the tracial state space has dense range, A has the Property (SP) if and only if the range of this map contains strictly positive functions with arbitrarily small norm. By comparison with results for unital simple direct limit of homogeneous C*-algebras with no dimension growth, one might hope that weaker conditions might suffice. We give examples to show that several plausible weaker conditions do not suffice for the results above. If A has real rank zero and at most countably many extreme tracial states, we apply results of H. Lin to show that A has tracial rank zero and is classifiable.