Publications

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

Let G be a compact Lie group that acts smoothly on a closed manifold M. Using a general Simonenko principle, we derive a novel criterion for the Fredholm property of G-pseudodifferential operators acting on Sobolev spaces of sections of vector bundles over M. In case the group is finite, we obtain a further characterization of the Fredholm property of G-pseudodifferential operators in terms of the invertibility of suitable symbols.

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

We study the Fredholm solvability for a new class of nonlocal boundary value problems associated with group actions on smooth manifolds. Namely, we consider the case in which the group action is defined on an ambient manifold without boundary and does not preserve the manifold with boundary on which the problem is stated. In particular, the group action does not map the boundary to itself. The orbits of the boundary under the group action split the manifold into subdomains, and this decomposition, being combined with the C*-algebra techniques, plays an important role in our approach to the analysis of the problem.

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