This article provides a purely topological description of a certain index-theoretical map between equivariant K-theory groups of Roe C*-algebras of certain coarse spaces. This generalises results about the spectral properties of magnetic Laplacians, which are studied as an example in the article as well. The results about spectra say that sometimes a spectral gap for an operator is forced to close when it is restricted to a subspace.
The setup for the general theory is a coarse space \(X\) with an action of a discrete group \(G\), and a closed subset \(Z\subseteq X\) that is invariant under a subgroup \(K \subseteq G\). Assume, for simplicity, that the decomposition of \(X\) into \(Z\) and the closure of its complement is excisive, so that the coarse Mayer–Vietoris sequence applies; this contains a boundary map between the Roe C*-algebra K-theory for \(X\) and the boundary \(\partial Z\) of \(Z\). Assume that all this works \(K\)-equivariantly as well. Then there is a canonical map from the \(G\)-equivariant K-theory of the Roe C*-algebra of \(X\) to the \(K\)-equivariant K-theory of the Roe C*-algebra of \(\partial Z\). This map may be embedded in a long exact sequence in two different ways. One uses a general homotopy theoretic formalism for coarse cohomology theories. The other builds a suitable C*-algebra. These exact sequences are naturally isomorphic. In the C*-algebraic construction, the maps in the resulting exact sequence have index theoretic interpretations. This allows to prove results about the spectrum of selfadjoint extensions of restrictions of Dirac or Laplace type differential operators on \(X\) to \(Z\), provided these extensions are defined by \(K\)-invariant and local boundary conditions, such as Dirichlet or Neumann boundary conditions.