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Recursive subhomogeneous algebras

Author: N. Christopher Phillips
Journal: Trans. Amer. Math. Soc. 359 (2007), 4595-4623
MSC (2000): Primary 46L05; Secondary 19A13, 19B14, 19K14, 46L80
Published electronically: May 11, 2007
MathSciNet review: 2320643
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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
Article copyright: © Copyright 2007 American Mathematical Society

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