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On a class of nonlocal elliptic operators for compact Lie groups. Uniformization and finiteness theorem. (English) Zbl 1241.58012
The author studies $$G$$-pseudodifferential operators of the form $P= \int_G P_g T_g dg,$ where $$P_g$$ is a smooth family of classical pseudodifferential operators on a compact manifold $$M$$, depending on the parameter $$g\in G$$, with $$G$$ a compact Lie group of diffeomorphisms acting on $$M$$, and $$dg$$ is the Haar measure on $$G$$. If $$G$$ is a discrete group, then the integral defining $$P$$ reduces to a finite sum; the corresponding $$G$$-pseudodifferential operators were already studied by many authors, see for example A. Connes [Noncommutative geometry. Transl. from the French by Sterling Berberian. San Diego, CA: Academic Press. xiii, 661 p. (1994; Zbl 0818.46076)]. Here the author studies the general case. In particular, the symbol of $$P$$ is defined as an element of the crossed product of the algebra of the classical symbols on $$S^*M$$ and the group $$G$$. Invertibility of such symbol implies the Fredholm property for $$P$$.

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