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We consider a smooth compact manifold with boundary, M,  embedded in a smooth manifold of the same dimension on which an amenable group \(\Gamma\) acts by isometries. We do not assume M to be invariant under \(\Gamma\). This results in a partial action of \(\Gamma\) on: For $g\in \Gamma$ we let \(M^\circ_g = g(M^\circ)\cap M^\circ\) and obtain diffeomorphisms \(g:M^\circ_{g^{-1}} \to

M^\circ_g

\).

We assume that any two images of \(\partial M\) under \(\Gamma\) either coincide or are disjoint and that only finitely many lie in M. The spherical blow-up of these images of \(\partial M\) in M yields a

manifold Y with  boundary consisting of finitely many components. Moreover, Y inherits a partial action of \(\Gamma\).

We can then define the C*-algebra \(\mathcal

A=\overline{\Psi_\Gamma(Y,\partial Y)}\) of operators on \(L^2(Y)\oplus

L^2(\partial Y)\), generated by the algebra \(\Psi(Y,\partial Y)\) of operators of order and type zero in Boutet de Monvel's calculus on Y and partial isometries associated with the partial action. Denote by

\(\Sigma=\overline{\Psi(Y,\partial Y)}/\mathcal K

\) the symbol space. If the partial action of \(\Gamma\) on Prim(\(\Sigma\)) is topologically

free, we find a criterion for the Fredholm property of the operators in \(\overline{\Psi_\Gamma(Y,\partial Y)}\).

Moreover, we obtain the classification of the elliptic elements in

\(\overline{\Psi_\Gamma(Y,\partial Y)}\) modulo stable homotopies: For \(\mathcal A_0= C(Y\sqcup \partial Y)\rtimes\Gamma\)

\(Ell(\mathcal A_0,\mathcal A)\cong K_0(C_0(T^*Y^\circ)\rtimes\Gamma

)\oplus K_0(C(\partial Y)\rtimes \Gamma).\)

If \(\Gamma\) is finitely generated and of polynomial growth, then the elements associated with the second summand do not contribute to the index.

Related project(s):
80Nonlocal boundary problems: Index theory and semiclassical asymptotics