  In the frame of isogeometric analysis, we consider a Galerkin boundary
element discretization of the hyper-singular integral equation associated with
the 2D Laplacian. We propose and analyze an adaptive algorithm which locally
refines the boundary partition and, moreover, steers the smoothness of the
NURBS ansatz functions across elements. In particular and unlike prior work,
the algorithm can increase and decrease the local smoothness properties and
hence exploits the full potential of isogeometric analysis. We prove that the
new adaptive strategy leads to linear convergence with optimal algebraic rates.
Numerical experiments confirm the theoretical results. A short appendix
comments on analogous results for the weakly-singular integral equation.
