  Chandra observations of distant elliptical galaxies have revealed large
numbers of Low Mass X-ray Binaries (LMXBs) accreting at rates in excess of
10^{-8} solar masses per year. The majority of these LMXBs reside in globular
clusters (GCs) and it has been suggested that many of the field LMXBs also
originated in GCs. We show here that ultracompact binaries with orbital periods
of 8-10 minutes and He or C/O donors of 0.06-0.08 solar masses naturally
provide the observed accretion rates from gravitational radiation losses alone.
Such systems are predicted to be formed in the dense GC environment, a
hypothesis supported by the 11.4 minute binary 4U 1820-30, the brightest
persistent LMXB in a Galactic GC. These binaries have short enough lifetimes
(less then 3 Myr) while bright that we calculate their luminosity function
under a steady-state approximation. This yields a luminosity function slope in
agreement with that observed for luminosities in the range of 6E37 ergs/sec to
5E38 ergs/sec, encouraging us to use the observed numbers of LMXBs per GC mass
to calculate the accumulated number of ultracompact binaries. For a constant
birthrate over 8 Gyrs, the number of ultracompact binaries which have evolved
through this bright phase is nearly 4000 in a 10 million solar mass GC,
consistent with dynamical interaction calculations. Perhaps most importantly,
if all ultracompacts become millisecond radio pulsars, then the observed
normalization agrees with the inferred number of millisecond radio pulsars in
47 Tuc and Galactic GCs in general.
