Noncommutative Residue and Canonical Trace on Noncommutative Tori. Uniqueness Results

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 Lévy–Neira–Paycha.