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On higher real and stable ranks for $ CCR$ $ C^*-$algebras


Author: Lawrence G. Brown
Journal: Trans. Amer. Math. Soc. 368 (2016), 7461-7475
MSC (2010): Primary 46L05; Secondary 46M20
Published electronically: January 27, 2016
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Abstract: We calculate the real rank and stable rank of $ CCR$ algebras which either have only finite dimensional irreducible representations or have finite topological dimension. We show that either rank of $ A$ is determined in a good way by the ranks of an ideal $ I$ and the quotient $ A/I$ in four cases: when $ A$ is $ CCR$; when $ I$ has only finite dimensional irreducible representations; when $ I$ is separable, of generalized continuous trace and finite topological dimension, and all irreducible representations of $ I$ are infinite dimensional; or when $ I$ is separable, stable, has an approximate identity consisting of projections, and has the corona factorization property. We also present a counterexample on higher ranks of $ M(A)$, $ A$ subhomogeneous, and a theorem of P. Green on generalized continuous trace algebras.


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Additional Information

Lawrence G. Brown
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: lgb@math.purdue.edu

DOI: https://doi.org/10.1090/tran/6616
Keywords: $C^*-$algebra, stable rank, real rank, $CCR$, generalized continuous trace
Received by editor(s): May 6, 2014
Received by editor(s) in revised form: October 21, 2014
Published electronically: January 27, 2016
Article copyright: © Copyright 2016 American Mathematical Society