  In a recent result by the authors (ref. [1]) it was proved that solutions of
the self-similar fragmentation equation converge to equilibrium exponentially
fast. This was done by showing a spectral gap in weighted $L^2$ spaces of the
operator defining the time evolution. In the present work we prove that there
is also a spectral gap in weighted $L^1$ spaces, thus extending exponential
convergence to a larger set of initial conditions. The main tool is an
extension result in ref. [4].
