2001/01/18 by N. Christopher Phillips, Phillips, N. Christopher
Mathematics · #19B14 #19K14 #46L80 #46M40 (Primary) 19A13 #54H20 (Secondary) #FOS: Mathematics #Operator Algebras (math.OA) #math.OA #msc:19A13 #msc:19B14 #msc:19K14 #msc:46L80 #msc:46M40 #msc:54H20
paper · pdf · doi:10.48550/arxiv.math/0101157
27 pages, AMSLaTeX
arxiv created 2001/01/18 · arxiv updated 2009/11/30
We prove the following results for a unital simple direct limit A of recursive subhomogeneous algebras with no dimension growth: (1) A has stable rank 1. (2) The projections in M∞ (A) satisfy cancellation: if e ⊕ q ∼ f ⊕ q, then e ∼ f. (3) A satisfies Blackadar's Second Fundamental Comparability Question: if p, q ∈ M∞ (A) are projections such that τ(p) < τ(q) for all normalized traces τ on A, then p is equivalent to a subprojection of q. (4) K0 (A) is unperforated for the strict order: if η∈ K0 (A) and there is n > 0 such that n η> 0, then η> 0. The last three of these results hold under certain weaker dimension growth conditions and without assuming simplicity. We use these results to obtain previously unknown information on the ordered K-theory of the crossed product C^* (Z, X, h) obtained from a minimal homeomorphism of an infinite finite dimensional compact metric space X. Specifically, K0 (C^* (Z, X, h)) is unperforated for the strict order, and satisfies the following K-theoretic version of Blackadar's Second Fundamental Comparability Question: if η∈ K0 (A) satisfies τ_* (\et) > 0 for all normalized traces τ on A, then there is a projection p ∈ M∞ (A) such that η= [p].