×

Quasi-locality and property A. (English) Zbl 1444.46016

Summary: Let \(X\) be a metric space with bounded geometry, \(p \in \{0 \} \cup [1, \infty]\), and let \(E\) be a Banach space. The main result of this paper is that either if \(X\) has Yu’s Property A and \(p \in(1, \infty)\), or without any condition on \(X\) when \(p \in \{0, 1, \infty \}\), then quasi-local operators on \(\ell^p(X, E)\) belong to (the appropriate variant of) the Roe algebra of \(X\). This generalises the existing results of this type by B. V. Lange and V. S. Rabinovich [Mat. Zametki 37, No. 3, 407–421 (1985; Zbl 0569.47029)], A. Engel [“Index theory of uniform pseudodifferential operators”, Preprint (2015), arXiv:1502.00494; J. Noncommut. Geom. 13, No. 2, 617–666 (2019; Zbl 1436.58019)], A. Tikuisis and the first author [Commun. Math. Phys. 365, No. 3, 1019–1048 (2019; Zbl 1426.46037)], and Li, Wang and the second author [K. Li et al., J. Math. Anal. Appl. 474, No. 2, 1213–1237 (2019; Zbl 1420.46043)]. As consequences, we obtain that uniform \(\ell^p\)-Roe algebras (of spaces with Property A) are closed under taking inverses, and another condition characterising Property A, akin to the operator norm localisation for quasi-local operators.

MSC:

46B28 Spaces of operators; tensor products; approximation properties
46H35 Topological algebras of operators
20F65 Geometric group theory
46J40 Structure and classification of commutative topological algebras
47L10 Algebras of operators on Banach spaces and other topological linear spaces
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] Bell, G.; Dranishnikov, A., Asymptotic dimension, Topology Appl., 155, 12, 1265-1296 (2008) · Zbl 1149.54017
[2] Brodzki, Jacek; Niblo, Graham A.; Špakula, Ján; Willett, Rufus; Wright, Nick, Uniform local amenability, J. Noncommut. Geom., 7, 2, 583-603 (2013) · Zbl 1281.46023
[3] Chen, Xiaoman; Tessera, Romain; Wang, Xianjin; Yu, Guoliang, Metric sparsification and operator norm localization, Adv. Math., 218, 5, 1496-1511 (2008) · Zbl 1148.46040
[4] Chung, Yeong Chyuan; Li, Kang, Rigidity of \(\ell^p\) Roe-type algebras, Bull. Lond. Math. Soc., 50, 6, 1056-1070 (2018) · Zbl 1417.46037
[5] Dranishnikov, Alexander; Zarichnyi, Michael, Asymptotic dimension, decomposition complexity, and Haver’s property C, Topology Appl., 169, 99-107 (2014) · Zbl 1297.54064
[6] Engel, Alexander, Index theory of uniform pseudodifferential operators (2015), arXiv preprint
[7] Engel, Alexander, Rough index theory on spaces of polynomial growth and contractibility (2015), preprint · Zbl 1436.58019
[8] Georgescu, Vladimir, On the structure of the essential spectrum of elliptic operators on metric spaces, J. Funct. Anal., 260, 6, 1734-1765 (2011) · Zbl 1242.47052
[9] Georgescu, Vladimir, On the essential spectrum of elliptic differential operators, J. Math. Anal. Appl., 468, 2, 839-864 (2018) · Zbl 1400.35191
[10] Georgescu, Vladimir; Iftimovici, Andrei, Localizations at infinity and essential spectrum of quantum Hamiltonians. I. General theory, Rev. Math. Phys., 18, 4, 417-483 (2006) · Zbl 1109.47004
[11] Gromov, Mikhael, Groups of polynomial growth and expanding maps, Publ. Math. Inst. Hautes Études Sci., 53, 53-73 (1981) · Zbl 0474.20018
[12] Guentner, Erik; Tessera, Romain; Yu, Guoliang, A notion of geometric complexity and its application to topological rigidity, Invent. Math., 189, 2, 315-357 (2012) · Zbl 1257.57028
[13] Guentner, Erik; Tessera, Romain; Yu, Guoliang, Discrete groups with finite decomposition complexity, Groups Geom. Dyn., 7, 2, 377-402 (2013) · Zbl 1272.52041
[14] Hagger, Raffael; Lindner, Marko; Seidel, Markus, Essential pseudospectra and essential norms of band-dominated operators, J. Math. Anal. Appl., 437, 1, 255-291 (2016) · Zbl 1336.47036
[15] Hanke, Bernhard; Pape, Daniel; Schick, Thomas, Codimension two index obstructions to positive scalar curvature, Ann. Inst. Fourier (Grenoble), 65, 6, 2681-2710 (2015) · Zbl 1344.58012
[16] Lange, B. V.; Rabinovich, V. S., Noethericity of multidimensional discrete convolution operators, Mat. Zametki, 37, 3, 407-421 (1985), 462 · Zbl 0569.47029
[17] Li, Kang; Wang, Zhijie; Zhang, Jiawen, A quasi-local characterisation of \(L^p\)-Roe algebras, J. Math. Anal. Appl., 474, 2, 1213-1237 (2019) · Zbl 1420.46043
[18] Lindner, Marko, Infinite Matrices and Their Finite Sections, Frontiers in Mathematics (2006), Birkhäuser Verlag: Birkhäuser Verlag Basel, An introduction to the limit operator method · Zbl 1107.47001
[19] Lindner, Marko; Seidel, Markus, An affirmative answer to a core issue on limit operators, J. Funct. Anal., 267, 3, 901-917 (2014) · Zbl 1292.47020
[20] Phillips, N. Christopher, Crossed products of \(L^p\) operator algebras and the K-theory of Cuntz algebras on \(L^p\) spaces (2013), preprint
[21] Rabinovich, Vladimir; Roch, Steffen; Silbermann, Bernd, Limit Operators and Their Applications in Operator Theory, Operator Theory: Advances and Applications., vol. 150 (2004), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 1077.47002
[22] Roe, John, An index theorem on open manifolds. I, II, J. Differential Geom., 27, 1, 87-113 (1988), 115-136 · Zbl 0657.58041
[23] Roe, John, Index Theory, Coarse Geometry, and Topology of Manifolds, (Published for the Conference Board of the Mathematical Sciences, Washington, DC. Published for the Conference Board of the Mathematical Sciences, Washington, DC, CBMS Regional Conference Series in Mathematics, vol. 90 (1996), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 0853.58003
[24] Roe, John, Lectures on Coarse Geometry, University Lecture Series, vol. 31 (2003), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1042.53027
[25] Roe, John, Band-dominated Fredholm operators on discrete groups, Integral Equations Operator Theory, 51, 3, 411-416 (2005) · Zbl 1085.47061
[26] Sako, Hiroki, Property A and the operator norm localization property for discrete metric spaces, J. Reine Angew. Math., 690, 207-216 (2014) · Zbl 1295.46017
[27] Schick, Thomas, The topology of positive scalar curvature, (Proceedings of the International Congress of Mathematicians—Seoul 2014, Vol. II (2014), Kyung Moon Sa: Kyung Moon Sa Seoul), 1285-1307 · Zbl 1373.53053
[28] Seidel, Markus, Fredholm theory for band-dominated and related operators: a survey, Linear Algebra Appl., 445, 373-394 (2014) · Zbl 1301.47016
[29] Špakula, Ján; Tikuisis, Aaron, Relative commutant pictures of Roe algebras, Comm. Math. Phys., 365, 3, 1019-1048 (2019) · Zbl 1426.46037
[30] Špakula, Ján; Willett, Rufus, A metric approach to limit operators, Trans. Amer. Math. Soc., 369, 1, 263-308 (2017) · Zbl 1380.47024
[31] Stuart White, Rufus Willett, Cartan subalgebras in uniform Roe algebras, preprint, 2018. · Zbl 1478.46070
[32] Tu, Jean-Louis, Remarks on Yu’s property A for discrete metric spaces and groups, Bull. Soc. Math. France, 129, 1, 115-139 (2001) · Zbl 1036.58021
[33] Willett, Rufus, Some notes on property A, (Limits of Graphs in Group Theory and Computer Science (2009), EPFL Press: EPFL Press Lausanne), 191-281 · Zbl 1201.19002
[34] Yu, Guoliang, Zero-in-the-spectrum conjecture, positive scalar curvature and asymptotic dimension, Invent. Math., 127, 1, 99-126 (1997) · Zbl 0889.58082
[35] Yu, Guoliang, The Novikov conjecture for groups with finite asymptotic dimension, Ann. of Math. (2), 147, 2, 325-355 (1998) · Zbl 0911.19001
[36] Yu, Guoliang, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math., 139, 1, 201-240 (2000) · Zbl 0956.19004
[37] Zhang, Jiawen, Extreme cases of limit operator theory on metric spaces, Integral Equations Operator Theory, 90, 6, 73 (2018) · Zbl 07027299
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.