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Uniqueness of multiplicative determinants on elliptic pseudodifferential operators. (English) Zbl 1193.58018

The authors consider classical pseudodifferential operators defined on a closed \(n\)-dimensional manifold \(M\), acting on smooth sections of a vector bundle \(E\) over \(M\). The authors first prove that traces on the algebra of the zero-order operator are linear combinations of the residue of M. Wodzicki [in: K-theory, arithmetic and geometry, Semin., Moscow Univ. 19840-86, Lect. Notes Math. 1289, 320–399 (1987; Zbl 0649.58033)] and of leading symbols traces given by any current \(\lambda\in (C^\infty(S^*M))'\). Attention is then fixed on multiplicative determinants on the pointwise connected component of identity, in the group of the invertible operators. Exhaustive results are presented.

MSC:

58J52 Determinants and determinant bundles, analytic torsion
58J42 Noncommutative global analysis, noncommutative residues
58J40 Pseudodifferential and Fourier integral operators on manifolds
35S05 Pseudodifferential operators as generalizations of partial differential operators

Citations:

Zbl 0649.58033
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References:

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