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Traces on pseudodifferential operators and sums of commutators. (English) Zbl 1194.58029
The paper contains proofs of known characterizations, Theorems 2.3 to 2.5, of traces on algebras of classical pseudodifferential operators. These characterizations state that, essentially, the traces on the algebras of integer and of non-integer order pseudodifferential operators are the noncommutative residue and the canonical trace, respectively. The canonical trace is the analytic continuation of the operator trace. The algebra of zeroth order operators possesses, in addition, traces given by linear functionals on the space of principal symbols. The aim of the paper is to give simple proofs of these uniqueness theorems for traces, using only basic pseudodifferential calculus and properties of the Schwartz kernels of pseudodifferential operators. The sums of commutators needed in the arguments are given directly. Finally, the onedimensional case is considered separately and shown to have special properties.

MSC:
58J40 Pseudodifferential and Fourier integral operators on manifolds
47G30 Pseudodifferential operators
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