  In this paper we establish uniqueness theorems for the noncommutative residue
and the canonical trace on pseudodifferential operators on noncommutative tori
of arbitrary dimension. The former is the unique trace up to constant multiple
on integer order pseudodifferential operators.The latter is the unique trace up
to constant multiple on non-integer order pseudodifferential operators. This
improves previous uniqueness results by Fathizadeh-Khalkhali, Fathizadeh-Wong,
and L\'evy-Neira-Paycha.
