Lie groupoids, exact sequences, Connes-Thom elements, connecting maps and index maps
type: Item
instance of: scholarly article
title: Lie groupoids, exact sequences, Connes-Thom elements, connecting maps and index maps
zbMATH Open document ID: 1393.58013
author: Claire Debord
author: Georges Skandalis
published in: Journal of Geometry and Physics
publication date: 2018-05-09T00:00:00Z
review text: Let \(G\rightarrow [r]{s}M\) be a Lie groupoid with range and source \(M\), \(F\) a closed saturated submanifold \(M\), and \(W=M\setminus F\). Then \(G=G_W\sqcup G_F\), where \(G_W\) and \(G_F\) are subgroupoids of \(G\) having \(W\) and \(F\) as ranges and sources. From this partition, we have the exact sequence of \(C^\ast\) algebras \[ 0\to C^\ast(G_W)\to C^\ast(G)\to C^\ast(G_F)\to 0. \] In this paper, derived exact sequences of cohomology groups from this sequence mainly focusing on the associated index maps and full index maps are studied (cf. \textit{C. Debord} and \textit{G. Skandalis}, ``Blowup constructions for a Boutet de Monvel type calculus'', Preprint, \url{arXiv:1706.09588}]). To define index maps, the full symbol algebra \[ \Sigma^W(G)=C(\mathbb{S}\mathfrak{A}^\ast G)\times_{C(\mathbb{S}\mathfrak{A}^\ast G_F)}\Psi^\ast(G_F), \] where \(\mathbb{S}\mathfrak{A}^\ast G\) is the cosphere bundle of the algebroid \(\mathfrak{A}G\) of \(G\), and \(\Psi^\ast(G_F)\) is the \(C^\ast\)-algebra of order 0 pseudo differential operators on \(G_F\), is introduced, and the exact sequence \[ 0\to C^\ast(G_W)\to \Psi^\ast(G)\rightarrow{\sigma_{full}}\Sigma~W(G)\to 0 \] is presented (\S2.4). The authors remark the full symbol map has two components; the usual commutative symbol \(\Psi^\ast(G)\to C(\mathbb{S}\mathfrak{A}^\ast G)\), and the boundary (or non commutative) symbol \(\sigma_\partial: \Psi^\ast(G)\to \Psi^\ast(G_F)\). In \S2.4, the index map \(\mathrm{ind}^W_G\in KK(C^\ast(\mathfrak{A}^{nc}_W G), C^\ast(G_W))\) is defined. Also from (1) and (2), \(\partial^W_G\in KK^1(C^\ast(G_F),C^\ast(G_W))\) and \(\widetilde{\mathrm{ind}}^W_{full}(G)\in KK^1(\Sigma^W(G), C^\ast(G_W0)\) are defined (\S4.1. [\textit{G. G. Kasparov}, Izv. Akad. Nauk SSSR, Ser. Mat. 44, 571--636 (1980; Zbl 0448.46051)]), provided \(G_F\) is amenable. Then \(\widetilde{\mathrm{ind}}^W_{full}\) is shown to be the Kasparov product of the Connes-Thom element \([th]\in KK^1(\Sigma^W(G), \Sigma^W(G)\…
reviewed by: Akira Asada
Mathematics Subject Classification ID: 58H05
Mathematics Subject Classification ID: 19K35
Mathematics Subject Classification ID: 19K56
Mathematics Subject Classification ID: 22A22
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 58B34
Mathematics Subject Classification ID: 58J22
zbMATH DE Number: 6867194
zbMATH Keywords: Lie groupoid
zbMATH Keywords: operator algebras
zbMATH Keywords: operator K-theory
zbMATH Keywords: KK-theory
zbMATH Keywords: full symbol map
zbMATH Keywords: index map
MaRDI profile type: Publication
full work available at URL: j.geomphys.2018.03.012
OpenAlex ID: W2795786190
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DOI: 10.1016/J.GEOMPHYS.2018.03.012
description: scientific article; zbMATH DE number 6867194