  To a domain with conical points \Omega, we associate a natural C*-algebra
that is motivated by the study of boundary value problems on \Omega, especially
using the method of layer potentials. In two dimensions, we allow \Omega to be
a domain with ramified cracks. We construct an explicit groupoid associated to
the boundary of \Omega and use the theory of pseudodifferential operators on
groupoids and its representations to obtain our layer potentials C*-algebra. We
study its structure, compute the associated K-groups, and prove Fredholm
conditions for the natural pseudodifferential operators affiliated to this
C*-algebra.
