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Noncommutative residue. I: Fundamentals. (English) Zbl 0649.58033
$$K$$-theory, arithmetic and geometry, Semin., Moscow Univ. 1984-86, Lect. Notes Math. 1289, 320-399 (1987).
[For the entire collection see Zbl 0621.00010.]
This is the first in a series of papers in which the author will develop a theory of “noncommutative residues” for pseudodifferential operators ($$\psi$$ DO’s). This theory has applications to the homological structure of the algebra of $$\psi$$ DO’s, to asymptotic expansions and to index formulae.
In this Chapter 1 the author defines the residue in the setting of symplectic geometry. Let $$Y = T^*_ 0 X$$ be the cotangent bundle of an n-dimensional manifold $$X$$, with the zero section deleted. $$Y$$ can be regarded as a principal $$\mathbb{R}^+$$-bundle over the sphere bundle $$Z$$ of $$X$$. Let omega be the standard symplectic form on $$Y$$ and $$Chi$$ the vector field which generates the $$\mathbb{R}^+$$ action. Define alpha to be the contraction $$i_\chi(\omega)$$. Let $$P_{-n}$$ be the set of functions on $$Y$$ which are homogeneous of degree $$-n$$ with respect to the $$\mathbb{R}^+$$ action. For $$f\in P_{-n}$$ the form $$f(\alpha \wedge (d\alpha)^{n-1})$$ can be lifted from a unique form $$\mu_ f$$ on $$\mathbb{Z}$$. The author defines a map: res$$_ 0: P_ n\to \Omega^{2n - 1}(\mathbb{Z})$$ by res$$_ 0(f) = \mu_ f$$. For forms with appropriate support the integrated residue is: Res$$(f) = \int_ \mathbb{Z} res_ 0(f)$$.
In more concrete terms, this can be written as $$\iint_{| xi | = 1} f(x,\xi)dx d \xi'$$, where $$d \xi'$$ denotes the standard measure on the sphere. Let P* denote the graded Lie algebra of homogeneous functions on Y, under the Poisson bracket. Then (if X is compact, say) the map Res defines a trace on P*, i.e. Res$$\{f,g\} = 0$$. The author also defines a ”total residue” mapping the standard chain complex associated to the algebra P* (and the adjoint representation) to the complex of forms on $$\mathbb{Z}$$. Let $$A$$ be a $$\psi$$ DO over X. In local coordinates $$A$$ is represented by: $As = (1/2\pi)^{n/2}\int p(x,\xi) s(\xi) e^{- ix\xi}d\xi,$ where the matrix value function p has an asymptotic expansion $$p \sim \sum_ r p_{\lambda - r}$$, and $$p_\mu$$ is homogeneous in $$\xi$$ of degree $$\mu$$. We put res$$(A) = \tau* (res_ 0(p_{-n}))$$, a matrix valued function of x. Here $$\tau*$$ is the operation of integration over the fibres of $$\mathbb{Z} \to X$$. The author shows, using the transformation formulae for $$\psi$$ DO’s under coordinated change, that this is an invariant definition, so res(A) is an n-form on $$X$$ intrinsically associated to $$A$$. (Here we assume for simplicity that $$X$$ is oriented.) Suppose $$A$$ is an elliptic operator for which the complex powers $$A^{-s}$$ can be defined. The (local) $$\zeta$$- function of$$A$$, associated to a point $$x \in X$$ is the meromorphic continuation of $$\zeta_ x(s) = Tr K_ s(x,x)$$, where $$K_ s$$ is the kernel of $$A^{-s}$$, which is continuous when Re(s) is large. The $$\zeta$$-function has poles at certain values $$s = s_ j$$. The author shows that the residues at these points can be expressed in terms of the construction above: res$$_{s = s_ j}(\zeta_ x(s)) = Tr(Res(A^{- s_ j}))$$. The $$\zeta$$-function is regular at $$s = 0$$ and there is a formula of a similar nature expressing $$\zeta_ x(0)$$ in terms of a ”logarithmic symbol” of the operator.
Reviewer: S.Donaldson

##### MSC:
 58J40 Pseudodifferential and Fourier integral operators on manifolds