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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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