  The accretion process onto spinning objects in Kerr spacetimes is studied
with numerical simulations. Our results show that accretion onto compact
objects with Kerr parameter (characterizing the spin) $|a| < M$ and $|a| > M$
is very different. In the super-spinning case, for $|a|$ moderately larger than
$M$, the accretion onto the central object is extremely suppressed due to a
repulsive force at short distance. The accreting matter cannot reach the
central object, but instead is accumulated around it, forming a high density
cloud that continues to grow. The radiation emitted in the accretion process
will be harder and more intense than the one coming from standard black holes;
e.g. $\gamma$-rays could be produced as seen in some observations.
Gravitational collapse of this cloud might even give rise to violent bursts. As
$|a|$ increases, a larger amount of accreting matter reaches the central object
and the growth of the cloud becomes less efficient. Our simulations find that a
quasi-steady state of the accretion process exists for $|a|/M \gtrsim 1.4$,
independently of the mass accretion rate at large radii. For such high values
of the Kerr parameter, the accreting matter forms a thin disk at very small
radii. We provide some analytical arguments to strengthen the numerical
results; in particular, we estimate the radius where the gravitational force
changes from attractive to repulsive and the critical value $|a|/M \approx 1.4$
separating the two qualitatively different regimes of accretion. We briefly
discuss the observational signatures which could be used to look for such
exotic objects in the Galaxy and/or in the Universe.
