  For the discretization of the integral fractional Laplacian $(-\Delta)^s$, $0
< s < 1$, based on piecewise linear functions, we present and analyze a
reliable weighted residual a posteriori error estimator. In order to compensate
for a lack of $L^2$-regularity of the residual in the regime $3/4 < s < 1$,
this weighted residual error estimator includes as an additional weight a power
of the distance from the mesh skeleton. We prove optimal convergence rates for
an $h$-adaptive algorithm driven by this error estimator. Key to the analysis
of the adaptive algorithm are local inverse estimates for the fractional
Laplacian.
