  The recent imaging of the M87 black hole at millimeter wavelengths by the
Event Horizon Telescope (EHT) collaboration has triggered a renewed interest in
numerical models for the accretion of magnetized plasma in the regime of
general relativistic magnetohydrodynamics (GRMHD). Here non-ideal simulations,
including both the resistive effects and, above all, the mean-field dynamo
action due to sub-scale, unresolved turbulence, are applied for the first time
to such systems in the fully nonlinear regime. Combined with the differential
rotation of the disk, the dynamo process is able to produce an exponential
growth of any initial seed magnetic field up to the values required to explain
the observations, when the instability tends to saturate even in the absence of
artificial quenching effects. Before reaching the final saturation stage we
observe a secondary regime of exponential growing, where the magnetic field
increases more slowly due to accretion, which is modifying the underlying
equilibrium. By varying the dynamo coefficient we obtain different growth
rates, though the field seems to saturate at approximately the same level, at
least for the limited range of parameters explored here, providing substantial
values for the MAD parameter for magnetized accretion. For reasonable values of
the central mass density and the commonly employed recipes for synchrotron
emission by relativistically hot electrons, our model is able to reproduce
naturally the observed flux of Sgr A*, the next target for EHT.
