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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
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