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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Cancellation and stable rank for direct limits of recursive subhomogeneous algebras


Author: N. Christopher Phillips
Journal: Trans. Amer. Math. Soc. 359 (2007), 4625-4652
MSC (2000): Primary 19K14, 46L80, 46M40; Secondary 19A13, 19B14, 54H20
DOI: https://doi.org/10.1090/S0002-9947-07-03849-4
Published electronically: May 11, 2007
MathSciNet review: 2320644
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Abstract: We prove the following results for a unital simple direct limit $A$ of recursive subhomogeneous algebras with no dimension growth: (1) $\operatorname {tsr}(A) = 1.$ (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 $\tau (p) < \tau (q)$ for all normalized traces $\tau$ on $A,$ then $p \precsim q.$ (4) $K_0 (A)$ is unperforated for the strict order: if $\eta \in K_0 (A)$ and there is $n > 0$ such that $n \eta > 0,$ then $\eta > 0.$ 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 $C^* (\mathbf {Z}, X, h)$ obtained from a minimal homeomorphism of a finite-dimensional infinite compact metric space $X.$ Specifically, $K_0 (C^* (\mathbf {Z}, X, h))$ is unperforated for the strict order, and satisfies the following K-theoretic version of Blackadar’s Second Fundamental Comparability Question: if $\eta \in K_0 (A)$ satisfies $\tau _* (\eta ) > 0$ for all normalized traces $\tau$ on $A,$ then there is a projection $p \in M_{\infty } (A)$ such that $\eta = [p].$


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Additional Information

N. Christopher Phillips
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222

Received by editor(s): January 22, 2001
Received by editor(s) in revised form: August 2, 2004
Published electronically: May 11, 2007
Additional Notes: This research was partially supported by NSF grants DMS 9400904 and DMS 9706850
Article copyright: © Copyright 2007 American Mathematical Society