  We consider subtorus actions on divisorial toric varieties. Here
divisoriality means that the variety has many Cartier divisors like
quasiprojective and smooth ones. We characterize when a subtorus action on such
a toric variety admits a categorical quotient in the category of divisorial
varieties. Our result generalizes previous statements for the quasiprojective
case. An important tool for the proof is a universal reduction of an arbitrary
toric variety to a divisorial one. This is done in terms of support maps, a
notion generalizing support functions on a polytopal fan. A further essential
step is the decomposition of a given subtorus invariant regular map to a
divisorial variety into an invariant toric part followed by a non-toric part.
