A Two Dimensional Adler-Manin Trace and Bi-singular Operators

Motivated by the theory of bi-singular pseudodifferential operators, we introduce a two dimensional version of the Adler-Manin trace. Our construction is rather general in the sense that it involves a twist afforded by an algebra automorphism. That is, starting from an algebra equipped with an automorphism, two twisted derivations, and a twisted invariant trace, we construct an algebra of formal twisted pseudodifferential symbols and define a noncommutative residue. Also, we provide related examples.