Cancellation and stable rank for direct limits of recursive subhomogeneous algebras

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dc.creatorPhillips, N. Christopher
dc.date2001-01-18
dc.date.accessioned2026-07-07T06:32:51Z
dc.date.available2026-07-07T06:32:51Z
dc.descriptionWe 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 $M_{\infty} (A)$ satisfy cancellation: if $e \oplus q \sim f \oplus q$, then $e \sim f$. (3) $A$ satisfies Blackadar's Second Fundamental Comparability Question: if $p, q \in M_{\infty} (A)$ 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) $K_0 (A)$ is unperforated for the strict order: if $η\in K_0 (A)$ 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, $K_0 (C^* (Z, X, h))$ is unperforated for the strict order, and satisfies the following K-theoretic version of Blackadar's Second Fundamental Comparability Question: if $η\in K_0 (A)$ satisfies $τ_* (\et) > 0$ for all normalized traces $τ$ on $A$, then there is a projection $p \in M_{\infty} (A)$ such that $η= [p]$.
dc.description27 pages, AMSLaTeX
dc.identifierhttps://arxiv.org/abs/math/0101157
dc.identifierhttp://arxiv.org/abs/math/0101157
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/98998
dc.subjectOperator Algebras
dc.subject19K14, 46L80, 46M40 (Primary) 19A13, 19B14, 54H20 (Secondary)
dc.titleCancellation and stable rank for direct limits of recursive subhomogeneous algebras
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