Kassel, Christian Le résidu non commutatif. (Noncommutative residue). (French) Zbl 0701.58054 Sémin. Bourbaki, Vol. 1988/89, 41e année, Exp. No. 708, Astérisque 177-178, 199-229 (1989). Let X be a compact n-dimensional differentiable manifold and E be a complex vector bundle over X. M. Wodzicki has defined in his Ph. D. thesis [Spectral asymmetry and noncommutative residue (in Russian), Steklov Inst. Math., Moscow (1984)] the noncommutative residue of a pseudo-differential operator A acting on the sections of E. In the first section one recalls the fundamental properties of the noncommutative residue. In § 2 an algebraic definition in one dimension is presented. V. Guillemin [Adv. Math. 55, 131-160 (1985; Zbl 0559.58025)] has defined the symplectic residue by using the symplectic cones (§ 3). From this geometric construction an intrinsic local form of the noncommutative residue is obtained. There exists a density \(\rho\) (A,B) such that \(res_ x[A,B]=d\rho (A,B)\). It is proved that \(\rho\) (A,B) is a cyclic l-cocycle in the sense of Connes. In the last sections, some applications also due to Wodzicki are given. The Hochschild homology and Connes’ spectral sequence are studied. A multiplicative residue is introduced and examples are mentioned.For the entire collection see [Zbl 0691.00001]. Reviewer: I.Mihai Cited in 8 Documents MSC: 58J40 Pseudodifferential and Fourier integral operators on manifolds Keywords:symplectic residue; noncommutative residue; density; Hochschild homology Citations:Zbl 0559.58025 PDFBibTeX XML Full Text: Numdam EuDML