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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.

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