  We derive a computable a posteriori error estimator for the $\alpha$-harmonic
extension problem, which localizes the fractional powers of elliptic operators
supplemented with Dirichlet boundary conditions. Our a posteriori error
estimator relies on the solution of small discrete problems on anisotropic
cylindrical stars. It exhibits built-in flux equilibration and is equivalent to
the energy error up to data oscillation, under suitable assumptions. We design
a simple adaptive algorithm and present numerical experiments which reveal a
competitive performance.
