  We develop a novel a posteriori error estimator for the $L^2$ error committed
by the finite element discretization of the solution of the fractional
Laplacian. Our a posteriori error estimator takes advantage of the
semi-discretization scheme using rational approximations which allow to
reformulate the fractional problem into a family of non-fractional parametric
problems. The estimator involves applying the implicit Bank-Weiser error
estimation strategy to each parametric non-fractional problem and
reconstructing the fractional error through the same rational approximation
used to compute the solution to the original fractional problem. In addition we
propose an algorithm to adapt both the finite element mesh and the rational
scheme in order to balance the discretization errors. We provide several
numerical examples in both two and three-dimensions demonstrating the
effectivity of our estimator for varying fractional powers and its ability to
drive an adaptive mesh refinement strategy.
