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A metric approach to limit operators


Authors: Ján Špakula and Rufus Willett
Journal: Trans. Amer. Math. Soc. 369 (2017), 263-308
MSC (2010): Primary 47A53; Secondary 30Lxx, 46L85, 47B36
DOI: https://doi.org/10.1090/tran/6660
Published electronically: March 2, 2016
MathSciNet review: 3557774
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Abstract: We extend the limit operator machinery of Rabinovich, Roch, and Silbermann from $ \mathbb{Z}^N$ to (bounded geometry, strongly) discrete metric spaces. We do not assume the presence of any group structure or action on our metric spaces. Using this machinery and recent ideas of Lindner and Seidel, we show that if a metric space $ X$ has Yu's property A, then a band-dominated operator on $ X$ is Fredholm if and only if all of its limit operators are invertible. We also show that this always fails for metric spaces without property A.


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Additional Information

Ján Špakula
Affiliation: Mathematical Sciences, University of Southampton, SO17 1BJ, United Kingdom
Email: jan.spakula@soton.ac.uk

Rufus Willett
Affiliation: Department of Mathematics, 2565 McCarthy Mall, University of Hawai\kern.05em’\kern.05emi at Mānoa, Honolulu, Hawaii 96822
Email: rufus.willett@hawaii.edu

DOI: https://doi.org/10.1090/tran/6660
Received by editor(s): September 11, 2014
Received by editor(s) in revised form: December 18, 2014
Published electronically: March 2, 2016
Additional Notes: The second author was partially supported by the US NSF
Article copyright: © Copyright 2016 American Mathematical Society