  Galactic field black hole (BH) low-mass X-ray binaries (LMXBs) are believed
to form in situ via the evolution of isolated binaries. In the standard
formation channel, these systems survived a common envelope phase, after which
the remaining helium core of the primary star and the subsequently formed BH
are not expected to be highly spinning. However, the measured spins of BHs in
LMXBs cover the whole range of spin parameters. We propose here that the BH
spin in LMXBs is acquired through accretion onto the BH after its formation. In
order to test this hypothesis, we calculated extensive grids of detailed binary
mass-transfer sequences. For each sequence, we examined whether, at any point
in time, the calculated binary properties are in agreement with their
observationally inferred counterparts of 16 Galactic LMXBs. The "successful"
sequences give estimates of the mass that the BH has accreted since the onset
of Roche-Lobe overflow. We find that in all Galactic LMXBs with measured BH
spin, the origin of the spin can be accounted for by the accreted matter, and
we make predictions about the maximum BH spin in LMXBs where no measurement is
yet available. Furthermore, we derive limits on the maximum spin that any BH
can have depending on current properties of the binary it resides in. Finally
we discuss the implication that our findings have on the BH birth-mass
distribution, which is shifted by ~1.5 Msolar towards lower masses, compared to
the currently observed one.
