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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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Recursive subhomogeneous algebras
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by N. Christopher Phillips PDF
Trans. Amer. Math. Soc. 359 (2007), 4595-4623 Request permission

Abstract:

We introduce and characterize a particularly tractable class of unital type 1 C*-algebras with bounded dimension of irreducible representations. Algebras in this class are called recursive subhomogeneous algebras, and they have an inductive description (through iterated pullbacks) which allows one to carry over from algebras of the form $C (X, M_n)$ many of the constructions relevant in the study of the stable rank and K-theory of simple direct limits of homogeneous C*-algebras. Our characterization implies, in particular, that if $A$ is a separable C*-algebra whose irreducible representations all have dimension at most $N < \infty ,$ and if for each $n$ the space of $n$-dimensional irreducible representations has finite covering dimension, then $A$ is a recursive subhomogeneous algebra. We demonstrate the good properties of this class by proving subprojection and cancellation theorems in it. Consequences for simple direct limits of recursive subhomogeneous algebras, with applications to the transformation group C*-algebras of minimal homeomorphisms, will be given in separate papers.
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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
  • © Copyright 2007 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 359 (2007), 4595-4623
  • MSC (2000): Primary 46L05; Secondary 19A13, 19B14, 19K14, 46L80
  • DOI: https://doi.org/10.1090/S0002-9947-07-03850-0
  • MathSciNet review: 2320643