Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] Goulnara Arzhantseva, Erik Guentner, and Ján Špakula, Coarse non-amenability and coarse embeddings, Geom. Funct. Anal. 22 (2012), no. 1, 22-36. MR 2899681, https://doi.org/10.1007/s00039-012-0145-z
  • [2] J. Brodzki, S. J. Campbell, E. Guentner, G. A. Niblo, and N. J. Wright, Property A and $ \rm CAT(0)$ cube complexes, J. Funct. Anal. 256 (2009), no. 5, 1408-1431. MR 2490224 (2010i:20044), https://doi.org/10.1016/j.jfa.2008.10.018
  • [3] Jacek Brodzki, Graham A. Niblo, Ján Špakula, Rufus Willett, and Nick Wright, Uniform local amenability, J. Noncommut. Geom. 7 (2013), no. 2, 583-603. MR 3054308, https://doi.org/10.4171/JNCG/128
  • [4] Nathanial P. Brown and Narutaka Ozawa, $ C^*$-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, RI, 2008. MR 2391387 (2009h:46101)
  • [5] Xiaoman Chen, Romain Tessera, Xianjin Wang, and Guoliang Yu, Metric sparsification and operator norm localization, Adv. Math. 218 (2008), no. 5, 1496-1511. MR 2419930 (2010c:46150), https://doi.org/10.1016/j.aim.2008.03.016
  • [6] Marius Dadarlat and Erik Guentner, Uniform embeddability of relatively hyperbolic groups, J. Reine Angew. Math. 612 (2007), 1-15. MR 2364071 (2008h:20064), https://doi.org/10.1515/CRELLE.2007.081
  • [7] Vladimir Georgescu, On the structure of the essential spectrum of elliptic operators on metric spaces, J. Funct. Anal. 260 (2011), no. 6, 1734-1765. MR 2754891 (2012b:46154), https://doi.org/10.1016/j.jfa.2010.12.025
  • [8] Erik Guentner, Nigel Higson, and Shmuel Weinberger, The Novikov conjecture for linear groups, Publ. Math. Inst. Hautes Études Sci. 101 (2005), 243-268. MR 2217050 (2007c:19007), https://doi.org/10.1007/s10240-005-0030-5
  • [9] Marko Lindner and Markus Seidel, An affirmative answer to a core issue on limit operators, J. Funct. Anal. 267 (2014), no. 3, 901-917. MR 3212726, https://doi.org/10.1016/j.jfa.2014.03.002
  • [10] Piotr W. Nowak and Guoliang Yu, Large scale geometry, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2012. MR 2986138
  • [11] Hervé Oyono-Oyono and Guoliang Yu, $ K$-theory for the maximal Roe algebra of certain expanders, J. Funct. Anal. 257 (2009), no. 10, 3239-3292. MR 2568691 (2010h:46117), https://doi.org/10.1016/j.jfa.2009.04.017
  • [12] Vladimir Rabinovich, Steffen Roch, and Bernd Silbermann, Limit operators and their applications in operator theory, Operator Theory: Advances and Applications, vol. 150, Birkhäuser Verlag, Basel, 2004. MR 2075882 (2005e:47002)
  • [13] Jean Renault, $ C^\star $-algebras and dynamical systems, Publicações Matemáticas do IMPA. [IMPA Mathematical Publications], Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2009. 27$ ^{\rm o}$ Colóquio Brasileiro de Matemática. [27th Brazilian Mathematics Colloquium]. MR 2536186 (2011e:46110)
  • [14] John Roe, Coarse cohomology and index theory on complete Riemannian manifolds, Mem. Amer. Math. Soc. 104 (1993), no. 497, x+90. MR 1147350 (94a:58193), https://doi.org/10.1090/memo/0497
  • [15] John Roe, Lectures on coarse geometry, University Lecture Series, vol. 31, American Mathematical Society, Providence, RI, 2003. MR 2007488 (2004g:53050)
  • [16] John Roe, Band-dominated Fredholm operators on discrete groups, Integral Equations Operator Theory 51 (2005), no. 3, 411-416. MR 2126819 (2005k:46142), https://doi.org/10.1007/s00020-004-1326-4
  • [17] John Roe, Hyperbolic groups have finite asymptotic dimension, Proc. Amer. Math. Soc. 133 (2005), no. 9, 2489-2490 (electronic). MR 2146189 (2005m:20102), https://doi.org/10.1090/S0002-9939-05-08138-4
  • [18] John Roe and Rufus Willett, Ghostbusting and property A, J. Funct. Anal. 266 (2014), no. 3, 1674-1684. MR 3146831, https://doi.org/10.1016/j.jfa.2013.07.004
  • [19] Hiroki Sako, Property A and the operator norm localization property for discrete metric spaces, J. Reine Angew. Math. 690 (2014), 207-216. MR 3200343, https://doi.org/10.1515/crelle-2012-0065
  • [20] G. Skandalis, J. L. Tu, and G. Yu, The coarse Baum-Connes conjecture and groupoids, Topology 41 (2002), no. 4, 807-834. MR 1905840 (2003c:58020), https://doi.org/10.1016/S0040-9383(01)00004-0
  • [21] Rufus Willett, Some notes on property A, Limits of graphs in group theory and computer science, EPFL Press, Lausanne, 2009, pp. 191-281. MR 2562146 (2010i:22005)
  • [22] Guoliang Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math. 139 (2000), no. 1, 201-240. MR 1728880 (2000j:19005), https://doi.org/10.1007/s002229900032

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 47A53, 30Lxx, 46L85, 47B36

Retrieve articles in all journals with MSC (2010): 47A53, 30Lxx, 46L85, 47B36


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

American Mathematical Society