  We study the positivity and asymptotic behaviour of nonnegative solutions of
a general nonlocal fast diffusion equation,
  \[\partial_t u + \mathcal{L}\varphi(u) = 0,\] and the interplay between these
two properties. Here $\mathcal{L}$ is a stable-like operator and $\varphi$ is a
singular nonlinearity.
  We start by analysing positivity by means of a weak Harnack inequality
satisfied by a related elliptic (nonlocal) equation. Then we use this
positivity to establish the asymptotic behaviour: under certain hypotheses on
the nonlocal operator and nonlinearity, our solutions behave asymptotically as
the Barenblatt solution of the standard fractional fast diffusion equation.
  The main difficulty stems from the generality of the operator, which does not
allow the use of the methods that were available for the fractional Laplacian.
Our results are new even in the case where $\varphi$ is a power.
