Nonlocal elliptic boundary value problems and the theory of $C^*$-algebras

Given a group action on a manifold, we consider the induced representation of the group on the space of functions by shift operators. We consider the elliptic theory for the elements of operator algebras generated by shift operators and pseudodifferential operators. Especially interesting is the case of manifolds with boundary in which the considered operator algebras enable one to study nonlocal boundary value problems.
We explain the main results in this field. To illustrate the general theory, we present an example of nonlocal boundary value problems for operators with shear mappings of a finite cylinder. We write out the ellipticity conditions for this class of nonlocal boundary value problems.