Using a global symbol calculus for pseudodifferential operators on tori,
we build a canonical trace on classical pseudodifferential operators on non-
commutative tori in terms of a canonical discrete sum on the underlying
toroidal symbols. We characterise the canonical trace on operators on the
noncommutative torus as well as its underlying canonical discrete sum on
symbols of fixed (resp. any) non–integer order. On the grounds of this
uniqueness result, we prove that in the commutative setup, this canonical
trace on the noncommutative torus reduces to Kontsevich and Vishik’s
canonical trace which is thereby identified with a discrete sum. A similar
characterisation for the noncommutative residue on noncommutative tori
as the unique trace which vanishes on trace–class operators generalises
Fathizadeh and Wong’s characterisation in so far as it includes the case
of operators of fixed integer order.