Connes-Chern Character for Manifolds with Boundary and ETA ...

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70 MATTHIAS LESCH, HENRI MOSCOVICI, AND MARKUS J. PFLAUM Next let us define the symbol spaces S m( T ∗ M 0 ; πT ∗ ∗ M E) . For fixed m ∈ R this space consists of all smooth sections a : T ∗ M 0 → πT ∗ ∗ M 0 E such that in local coordinates x : U → R dim M 0 of over U ⊂ M 0 open and vector bundle coordinates (x, η) : E |U → R dim M 0+dim R E the following estimate holds true for each compact K ⊂ U and appropriate C K > 0 depending on K: ∥ ∂ α x ∂ β ξ (η ◦ a)(ξ)∥ ∥ ≤ CK (1 + ‖T ∗ x(ξ)‖) m−|β| for all ξ ∈ T|KM ∗ 0 . (A.10) Given a symbol a ∈ S m( T ∗ M 0 ; πT ∗ ∗ M Hom(E, F )) one defines now a pseudodifferential operator Op(a) ∈ Ψ m( M 0 ; E, F ) by ( ) Op(a) u (p) := ∫ 1 := α (2π) dim M 0 (p, exp v) e −i〈v,ξ〉 a(p, ξ)τ E (u(exp v), p) dv dξ, (A.11) T pM 0 ×T ∗ p M 0 where u ∈ Γ ∞ cpt(E) and p ∈ M 0 . Moreover, there is a quasi-inverse, the symbol map σ : Ψ m( M 0 ; E, F ) → S m( T ∗ M 0 ; πT ∗ ∗ M Hom(E, F )) which is defined by ( ) (π(ξ) ) σ(A)(ξ)(e) := A α 0 (p, −) τ E (e, −) e i〈ξ,Φ(p,−)〉 , (A.12) where p ∈ M 0 , ξ ∈ Tp ∗ M 0 , e ∈ E p . It is a well-known result from global symbol calculus (cf. [Wid80, FuKe88, Pfl98]) that the map Op maps S −∞( T ∗ M 0 ; πT ∗ ∗ M Hom(E, F )) onto Ψ −∞( M 0 ; E, F ) and that up to these spaces, Op and σ are inverse to each other. A.3. Classical b-pseudodifferential operators. Let us explain in the following the basics of the (small) calculus of b-pseudodifferential operators on a manifold with boundary M. In our presentation, we lean on the approach [Loy05]. For more details on the original approach confer [Mel93]. In this section, we assume that M carries a b-metric denoted by g b . Furthermore, let π E : E → M and π F : F → M be two smooth Hermitian vector bundles over M, and fix metric connections ∇ E and ∇ F . Then observe that by the Schwartz Kernel Theorem there is an isomorphism between bounded linear maps A : J ∞( ∂M, M; E ) → J ∞( ∂M, M; F ′) ′ and the strong dual J ∞( ∂(M × M), M × M; E ⊠ F ′) ′ ( , where J ∞ ∂M, M; E ) := J ∞( ∂M, M ) · C ∞ (M; E). This isomorphism is given by ( A ↦→ K A := J ∞( ∂M, M; E ) ˆ⊗J ∞( ∂M, M; F ′) ) ∋ (u ⊗ v) ↦→ 〈Au, v〉 , (A.13) where we have used that J ∞( ∂(M × M), M × M; E ⊠ F ′) = J ∞( ∂M, M; E ) ˆ⊗J ∞( ∂M, M; F ′)
CONNES-CHERN CHARACTER AND ETA COCHAINS 71 with ˆ⊗ denoting the completed bornological tensor product. The b-volume form µ b associated to g b gives rise to an embedding M ∞( ∂M, M; E ′ ⊗ F ) ↩→ J ∞( ∂(M × M), M × M; E ⊠ F ′) ′ , ( ∫ k ↦→ u ⊗ v ↦→ 〈k(p, q), u(p) ⊗ v(q)〉 d ( ) ) µ b ⊗ µ b (p, q) , M×M (A.14) which we use implicitly throughout this work. In the formula for the embedding, 〈−, −〉 denotes the natural pairing of an element of a vector bundle with an element of the dual bundle over the same base point, u, v are elements of J ∞( ∂M, M; E ) and J ∞( ∂M, M; F ′) respectively, and M ∞( X, M; E) denotes for X ⊂ M closed the space of all sections u ∈ Γ ∞ (M \ X; E) such that in local coordinates (y, η) : π −1 E (U) → R dim M+dim E with U ⊂ M open one has for every compact K ⊂ U, p ∈ K \ X, and α ∈ N dim M an estimate of the form ∥ ∂ α y ( η ◦ u ) (p) ∥ ∥ ≤ C 1 (d ( y(p), y(X ∩ U) )) λ , where C > 0 and λ > 0 depend only on the local coordinate system, K, and α. The fundamental property of M ∞( X, M; E) is that J ∞ (X, M) · M ∞( X, M; E) ⊂ J ∞ (X, M; E). Note that the vector bundle E (and likewise the vector bundle F ) gives rise to a pull-back vector bundle pr ∗ ∂M E |∂M on the cylinder, where pr ∂M : R × ∂M → ∂M is the canonical projection. This pull-back vector bundle will be denoted by E (resp. F ), too. As further preparation we introduce two auxiliary functions ψ : M → [0, 1] and ϕ : M → [0, 1] on M which are smooth and satisfy supp ψ ⊂⊂ M 1 , ψ(p) = 1 for p ∈ M 3/2 , supp ϕ ⊂⊂ Y 1 , and finally ϕ(p) = 1 for p ∈ Y 1/2 . Such a pair of functions will be called a pair of auxiliary cut-off functions. By a b-pseudodifferential operator of order m ∈ R we now understand a continuous operator A : J ∞ (∂M, M; E) → J ∞ (∂M, M; F ) ′ such that for one (and hence for all) pair(s) of auxiliary cut-off functions the following is satisfied: ( b Ψ1) The operator (1 − ϕ)A(1 − ϕ) is a compactly supported pseudodifferential operator of order m in the interior M ◦ . ( b Ψ2) The operator ϕAψ is smoothing. Its integral kernel K ϕAψ has support in supp ϕ × M and lies in J ∞( ∂(M × M), M × M; E ′ ⊠ F ) . ( b Ψ3) The operator ψAϕ is smoothing. Its integral kernel K ψAϕ has support in M × supp ϕ and lies in J ∞( ∂(M × M), M × M; E ′ ⊠ F ) . ( b Ψ4) Consider the induced operator on the cylinder Ã : b S ( R × ∂M; E ) → b S ( R × ∂M; F ) ′ , ( u ↦→ (t, p) ↦→ [ (1 − ψ)A(1 − ψ) ( (x, η) ∗ u )]( (x, η) −1 (t, p) )) , where b S ( R × ∂M; E ) := b S (R) ˆ⊗C ∞( ∂M; E ) . Denote by ã := σ(Ã) ∈ S m( T ∗ (R×∂M); π ∗ Hom(E, F ) ) the complete symbol of Ã defined by Eq. (A.12)
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CONNES-CHERN CHARACTER FOR MANIFOLD
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CONNES-CHERN CHARACTER AND ETA COCH
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∫ b M CONNES-CHERN CHARACTER AND
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where CONNES-CHERN CHARACTER AND ET
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where the map ι(h) is defined by C
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