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Uniqueness of the Kontsevich-Vishik trace. (English) Zbl 1132.58018
Let $$M$$ be a closed $$n$$-dimensional manifold. M. Kontsevich and S. Vishik [Prog. Math. 131, 173–197 (1995; Zbl 0920.58061)] analyzed the properties of determinants of classical pseudodifferential operators. One important tool was the construction of a trace-like mapping TR defined on the set $$\mathcal D$$ of all classical pseudodifferential operators, whose complex orders are not integers greater than or equal to $$-n$$.
The authors prove that the Kontsevich-Vishik trace TR is the unique functional which (i) is linear on $$\mathcal D$$, (ii) has the trace property and (iii) coincides with the $$L^2$$-operator trace Tr on trace class pseudodifferential operators.
It is obvious that the Kontsevich-Vishik trace TR cannot be extended to a trace on the algebra of all pseudodifferential operators on $$M$$. G. Grubb [AMS Contemp. Math. 366, 67–93 (2005; Zbl 1073.58021)] defined even-even pseudodifferential operators of arbitrary integer order on odd-dimensional manifolds and even-odd pseudodifferential operators of arbitrary integer order on even-dimensional manifolds, and observed that TR is extended to a slightly larger domain $$\mathcal D^+$$ depending on $$n$$. It is shown that the extension on $$\mathcal D^+$$ of TR defined on $$\mathcal D$$ is unique.

##### MSC:
 58J40 Pseudodifferential and Fourier integral operators on manifolds 35S05 Pseudodifferential operators as generalizations of partial differential operators 58J42 Noncommutative global analysis, noncommutative residues
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##### References:
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