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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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