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Trace asymptotics for $ C^{\ast}$-algebras from Smale spaces


Authors: D. B. Killough and I. F. Putnam
Journal: Proc. Amer. Math. Soc. 143 (2015), 317-325
MSC (2010): Primary 37D20; Secondary 46L55, 46L51
DOI: https://doi.org/10.1090/S0002-9939-2014-12221-0
Published electronically: September 16, 2014
MathSciNet review: 3272757
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Abstract: We consider $ C^{\ast }$-algebras associated with stable and unstable
equivalence in hyperbolic dynamical systems known as Smale spaces. These systems include shifts of finite type, in which case these $ C^{*}$-algebras are both AF-algebras. These algebras have fundamental representations on a single Hilbert space (subject to a choice of periodic points) which have a number of special properties. In particular, the product between any element of the first algebra with one from the second is compact. In addition, there is a single unitary operator which implements actions on both. Here, under the hypothesis that the system is mixing, we show that the (semi-finite) traces on these algebras may be obtained through a limiting process and the usual operator trace.


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Additional Information

D. B. Killough
Affiliation: Department of Mathematics, Physics, and Engineering, Mount Royal University, Calgary, Alberta, Canada T3E 6K6
Email: bkillough@mtroyal.ca

I. F. Putnam
Affiliation: Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada V8W 3R4
Email: ifputnam@uvic.ca

DOI: https://doi.org/10.1090/S0002-9939-2014-12221-0
Received by editor(s): August 24, 2012
Received by editor(s) in revised form: April 8, 2013
Published electronically: September 16, 2014
Additional Notes: The second author was supported in part by an NSERC Discovery Grant
Communicated by: Varghese Mathai
Article copyright: © Copyright 2014 American Mathematical Society