Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Cancellation and stable rank for direct limits of recursive subhomogeneous algebras
HTML articles powered by AMS MathViewer

by N. Christopher Phillips PDF
Trans. Amer. Math. Soc. 359 (2007), 4625-4652 Request permission

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 $\eta = [p].$
References
Similar Articles
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
  • © Copyright 2007 American Mathematical Society
  • 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
  • MathSciNet review: 2320644