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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
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

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