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

Noncommutative Residue and Canonical Trace on Noncommutative Tori. Uniqueness Results