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Uniformization of nonlocal elliptic operators and \(KK\)-theory. (English) Zbl 1291.58015
Let \(M\) be a smooth compact manifold. Let \(A\) be a (classical) pseudo-differential operator acting on \(C^\infty(M,\mathbb{C})\) and let \(G\) be a discrete group of diffeomorphisms of \(M\), that induce an operator \(T_g\) on \(C^\infty(M)\) by right composition. The authors consider the linear combinations of Fourier integral operators of the form \[ A= \sum_{g \in \Lambda } A_g T_g, \] where \(\Lambda\) is a finite subset of \(G\), and the operators \(A_g\) describe a family of pseudodifferential operators indexed by \(g \in \Lambda\). These operators are non local and non pseudolocal in general, and the problem of the index of \(A\) is a non trivial problem. Using classical tools coming from non-commutative geometry and especially \(K\)-theory, the authors describe a uniformization method to associate to \(A\) a pseudo-differential operator \(B\) with controlled index. The explicit technical feature is Kasparov’s \(KK\)-theory.
We have to remark that the group \(G\) is assumed to be discrete because of the necessity of amenability of \(G\), which suggests that the results of the paper cannot be extended straightway to \(G=\mathrm{Diff}(M)\), describing a map that would transform operators \(A\) into pseudo-differential operators with controlled index. This last result would be very interesting, and the present paper stands as a preliminary result.

58J40 Pseudodifferential and Fourier integral operators on manifolds
47G30 Pseudodifferential operators
35S05 Pseudodifferential operators as generalizations of partial differential operators
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