  We present deep radio maps from the Very Large Array (VLA) for 16 sources
detected in a sub-millimeter (submm) survey of the distant Universe. Our deep
VLA 1.4-GHz maps allow us to identify radio counterparts or place stringent
limits (<20uJy in the source plane) on the radio flux of the submm sources. We
compare the spectral indices of our sources between 850um and 1.4GHz to
empirical and theoretical models for distant starburst galaxies and active
galactic nuclei (AGN) as a function of redshift. In this way we can derive
redshift limits for the submm sources, even in the absence of an optical or
near-infrared counterpart. We conclude that the submm population brighter than
\~1mJy has a median redshift of at least <z>~2, more probably <z>~2.5-3, with
almost all galaxies at z>>1. This estimate is a strong lower limit as both
misidentification of the radio counterparts and non-thermal emission from an
AGN will bias our redshift estimates to lower values. The high median redshift
means that the submm population, if predominately powered by starbursts,
contributes a substantial fraction of the total star formation density at high
redshifts. A comparison of the spectral index limits with spectroscopic
redshifts for proposed optical counterparts to individual submm galaxies
suggests that half of the submm sources remain unidentified and thus their
counterparts must be fainter than I>24.
