Fredholm Lie groupoids were introduced by Carvalho, Nistor and Qiao as a tool for the study of partial differential equations on open manifolds. This article extends the definition to the setting of locally compact groupoids and proves that "the Fredholm property is local". Let G ⇒ X be a topological groupoid and (U i ) i∈I be an open cover of X. We show that G is a Fredholm groupoid if, and only if, its reductions G U i U i are Fredholm groupoids for all i ∈ I. We exploit this criterion to show that many groupoids encountered in practical applications are Fredholm. As an important intermediate result, we use an induction argument to show that the primitive spectrum of C * (G) can be written as the union of the primitive spectra of all C * (G| U i ), for i ∈ I.
Let G be a compact Lie group acting smoothly on a smooth, compact manifold M , let P ∈ ψ m (M ; E 0 , E 1 ) be a G-invariant, classical pseudodifferential operator acting between sections of two vector bundles E i → M , i = 0, 1, and let α be an irreducible representation of the group G. Then P induces a map πα(P ) : H s (M ; E 0 )α → H s−m (M ; E 1 )α between the α-isotypical components. We prove that the map πα(P ) is Fredholm if, and only if, P is transversally α-elliptic, a condition defined in terms of the principal symbol of P and the action of G on the vector bundles E i .
Let Γ be a finite abelian group acting on a smooth, compact manifold M without boundary and let P∈ψm(M;E0,E1) be a Γ-invariant, classical, pseudodifferential operator acting between sections of two Γ-equivariant vector bundles. Let α be an irreducible representation of Γ. We obtain necessary and sufficient conditions for the restriction πα(P):Hs(M;E0)α→Hs−m(M;E1)α of P between the α-isotypical components of Sobolev spaces to be Fredholm.
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