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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.

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