On the index of nonlocal elliptic operators for compact Lie groups
type: Item
instance of: scholarly article
title: On the index of nonlocal elliptic operators for compact Lie groups
zbMATH Open document ID: 1248.58013
author: Anton Savin
published in: Central European Journal of Mathematics
publication date: 2012-01-16T00:00:00Z
review text: Let a compact Lie group \(G\) act on a closed smooth manifold \(M\). \textit{B. Yu. Sternin} [Cent. Eur. J. Math. 9, No. 4, 814--832 (2011; Zbl 1241.58012)] defined the notion of ellipticity for a nonlocal operator \(D\) of the form \(1 + \int _{G} D_{g} \circ T_{g} \, dg\) and showed that such elliptic operators are Fredholm. Here \(T_{g}\) represents the translation action of \(G\) on a space of functions on \(M\), and \(\{ D_{g} : g\in G\}\) is a smooth family of pseudodifferential operators on \(M\). To define ellipticity, Sternin lifted \(D\) to an operator \(\widehat{D}\) on \(M \times G\), defined the symbol of \(\widehat{D}\) as a function with values in bounded operators on \(L^{2}(G)\), and let ellipticity mean that this symbol was transversally elliptic in the sense that its values were invertible on the zero section's complement in the transverse cotangent bundle. The Fredholm index of \(D\) can be identified with the \(G\)-invariants of the index of this transversally elliptic operator \(\widehat{D}\). The paper under review gives a formula for this Fredholm index in the case that \(G\)'s action is locally free. This index formula recovers the results of \textit{V. E. Nazaikinskii, A. Yu. Savin} and \textit{B. Yu. Sternin} [Elliptic theory and noncommutative geometry. Nonlocal elliptic operators. Basel: Birkhäuser (2008; Zbl 1158.58013)] in the case that \(G\) is finite. Among the ingredients in the general compact \(G\) case is a connection on the \(L^{2}(G)\) bundle that can be used to define a Chern character with values in basic cohomology. Using an increasing sequence of \(G\)-invariant subspaces of \(L^{2}(G)\) (actually of vector-valued \(L^{2}\) functions on \(G\)), the author realizes his transversally elliptic operator as the norm limit of a sequence of transversally elliptic operators acting on sections of finite-dimensional bundles. To these the author applies the index theorems of \textit{T. Kawasaki} [Nagoya Math. J. 84, 135--157 (1981; Zb…
reviewed by: Peter Haskell
Mathematics Subject Classification ID: 58J20
Mathematics Subject Classification ID: 46L87
Mathematics Subject Classification ID: 19K56
Mathematics Subject Classification ID: 58J40
zbMATH DE Number: 5997980
zbMATH Keywords: nonlocal operator
zbMATH Keywords: basic cohomology
zbMATH Keywords: crossed product
zbMATH Keywords: transverse ellipticity
zbMATH Keywords: Chern character
zbMATH Keywords: index formula
zbMATH Keywords: Fredholm index
Wikidata QID: Q115227865
MaRDI profile type: Publication
full work available at URL: s11533-011-0028-9
OpenAlex ID: W2089202164
cites work: Q4765145
cites work: Elliptic operators and compact groups
cites work: The index of elliptic operators. II
cites work: Q3803830
cites work: Q4002484
cites work: The Chern character of a transversally elliptic symbol and the equivariant index
cites work: Equivariant cyclic homology and equivariant differential forms
cites work: Q5658847
cites work: Cyclic cohomology of etale groupoids
cites work: Q3791701
cites work: Non-commutative differential geometry
cites work: An equivariant noncommutative residue
cites work: Q3792859
cites work: The index of elliptic operators over V-manifolds
cites work: Orthogonal decompositions of direct sums and tensor products of algebras
cites work: Q3283462
cites work: Elliptic theory and noncommutative geometry. Nonlocal elliptic operators
cites work: Higher index theorems and the boundary map in cyclic cohomology
cites work: Q3205815
cites work: Superconnections and the Chern character
cites work: On a class of nonlocal elliptic operators for compact Lie groups. Uniformization and finiteness theorem
cites work: The homology of matrix Lie algebras over rings and the Hochschild homology
cites work: Equivariant index formulas for orbifolds
cites work: A De Rham Type Theorem for Orbit Spaces
DOI: 10.2478/S11533-011-0028-9
Recommended article: Nonlocal elliptic operators for compact Lie groups
Recommended article: A general Simonenko local principle and Fredholm condition for isotypical components
Recommended article: Q3343080
Recommended article: The analytic index for proper, Lie groupoid actions
Recommended article: On the index of pseudo-differential operators on compact Lie groups
Recommended article: Q4011092
Recommended article: The index of transversally elliptic operators for locally free actions
Recommended article: Q3468340
Recommended article: Q3137255
Recommended article: An index formula for nonlocal operators corresponding to a diffeomorphism of a manifold
description: scientific article; zbMATH DE number 5997980