  It is established that the formation of rotationally supported disks during
the main accretion phase of star formation is suppressed by a moderately strong
magnetic field in the ideal MHD limit. Non-ideal MHD effects are expected to
weaken the magnetic braking, perhaps allowing the disk to reappear. We
concentrate on one such effect, ambipolar diffusion, which enables the field
lines to slip relative to the bulk neutral matter. We find that the slippage
does not sufficiently weaken the braking to allow rotationally supported disks
to form for realistic levels of cloud magnetization and cosmic ray ionization
rate; in some cases, the magnetic braking is even enhanced. Only in dense cores
with both exceptionally weak fields and unreasonably low ionization rate do
such disks start to form in our simulations. We conclude that additional
processes, such as Ohmic dissipation or Hall effect, are needed to enable disk
formation. Alternatively, the disk may form at late times when the massive
envelope that anchors the magnetic brake is dissipated, perhaps by a
protostellar wind.
