A metric approach to limit operators

Špakula, Ján; Willett, Rufus

doi:10.1090/tran/6660

A metric approach to limit operators
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by Ján Špakula and Rufus Willett PDF

Trans. Amer. Math. Soc. 369 (2017), 263-308 Request permission

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

Goulnara Arzhantseva, Erik Guentner, and Ján Špakula, Coarse non-amenability and coarse embeddings, Geom. Funct. Anal. 22 (2012), no. 1, 22–36. MR 2899681, DOI 10.1007/s00039-012-0145-z
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, DOI 10.1016/j.jfa.2008.10.018
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, DOI 10.4171/JNCG/128
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, DOI 10.1090/gsm/088
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, DOI 10.1016/j.aim.2008.03.016
Marius Dadarlat and Erik Guentner, Uniform embeddability of relatively hyperbolic groups, J. Reine Angew. Math. 612 (2007), 1–15. MR 2364071, DOI 10.1515/CRELLE.2007.081
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, DOI 10.1016/j.jfa.2010.12.025
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, DOI 10.1007/s10240-005-0030-5
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, DOI 10.1016/j.jfa.2014.03.002
Piotr W. Nowak and Guoliang Yu, Large scale geometry, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2012. MR 2986138, DOI 10.4171/112
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, DOI 10.1016/j.jfa.2009.04.017
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, DOI 10.1007/978-3-0348-7911-8
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$^\textrm {o}$ Colóquio Brasileiro de Matemática. [27th Brazilian Mathematics Colloquium]. MR 2536186
John Roe, Coarse cohomology and index theory on complete Riemannian manifolds, Mem. Amer. Math. Soc. 104 (1993), no. 497, x+90. MR 1147350, DOI 10.1090/memo/0497
John Roe, Lectures on coarse geometry, University Lecture Series, vol. 31, American Mathematical Society, Providence, RI, 2003. MR 2007488, DOI 10.1090/ulect/031
John Roe, Band-dominated Fredholm operators on discrete groups, Integral Equations Operator Theory 51 (2005), no. 3, 411–416. MR 2126819, DOI 10.1007/s00020-004-1326-4
John Roe, Hyperbolic groups have finite asymptotic dimension, Proc. Amer. Math. Soc. 133 (2005), no. 9, 2489–2490. MR 2146189, DOI 10.1090/S0002-9939-05-08138-4
John Roe and Rufus Willett, Ghostbusting and property A, J. Funct. Anal. 266 (2014), no. 3, 1674–1684. MR 3146831, DOI 10.1016/j.jfa.2013.07.004
Hiroki Sako, Property A and the operator norm localization property for discrete metric spaces, J. Reine Angew. Math. 690 (2014), 207–216. MR 3200343, DOI 10.1515/crelle-2012-0065
G. Skandalis, J. L. Tu, and G. Yu, The coarse Baum-Connes conjecture and groupoids, Topology 41 (2002), no. 4, 807–834. MR 1905840, DOI 10.1016/S0040-9383(01)00004-0
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
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, DOI 10.1007/s002229900032