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

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50 MATTHIAS LESCH, HENRI MOSCOVICI, AND MARKUS J. PFLAUM diagonal ∂ k t e −t∆ R (x, x) = ∂ k t (4πt) −1/2 =: c k t −1/2−k , we have ( f(x, p)P ∂ l xe −tD2) (x, p; x, p) = f(x, p) ( P e −tA2 ) (p, p)ck t −1/2−k ∼ t→0+ ∞ ∑ j=0 j−dim M−q f(x, p)a j (P, A)(p)c k t 2 . Comparing with (5.1) we find for the heat coefficients a j (Q, D) Furthermore, we have using Theorem (1.6) b Tr ( Qe −tD2 ) = (5.4) a j (Q, D)(x, p) = f(x, p)a j (P, A)(p)c k . (5.5) ∫ 0 −∞ ∫ ∂M tr x,p ( x∂ x f(x, p) ( P e −tA2 ) (p, p) )d vol ∂M (p)dx. (5.6) ( P e −tA 2 ) (p, p) has an x-independent asymptotic expansion as t → 0+. Since x∂ x f(x, p) = O(e (1−δ)x ), x → −∞, uniformly in p, we can plug the asymptotic expansion (5.4) into (5.6) and use (5.5) to find b Tr ( Qe −tD2 ) ∼ t→0+ ∼ t→0+ The claim is proved. ∑ ∞ ∫ 0 j=0 −∞ ∫ ∂M j−dim M−q · c k t 2 ∫ ∞ ∑ j=0 b (−∞,0)×∂M j−dim M−q · t 2 . ) tr x,p (x∂ x f(x, p)a j (P, A)(p) d vol ∂M (p)dx· tr x,p ( aj (Q, D)(x, p) ) d vol(x, p)· (5.7) 5.2. The b-trace of the JLO integrand. To ) extend Theorem 5.1 to expressions of the form b Tr (A 0 e −σ 0tD 2 A 1 e −σ 1tD 2 ...A k e −σ ktD 2 we use a trick which was already applied successfully in the proof of the local index formula in noncommutative geometry [CoMo95]. Namely, we successively commute A j e −σ jtD 2 and control the remainder. We will need the estimates proved in Subsections 3.3 and 3.4. We first need to introduce some notation (cf. [Les99, Lemma 4.2]). For a b-differential operator B ∈ b Diff(M; W ) we put inductively ∇ 0 DB := B, ∇ j+1 D B := [D2 , ∇ j DB]. (5.8) Note that since D 2 has scalar leading symbol we have ord(∇ j DB) ≤ j + ord B. □
CONNES-CHERN CHARACTER AND ETA COCHAINS 51 The following formula can easily be shown by induction. e −tD2 B = ∑n−1 j=0 (−t) j j! + (−t)n (n − 1)! ( ∇ j D B) e −tD2 + ∫ 1 0 (1 − s) n−1 e −stD2 ( ∇ n D B ) e −(1−s)tD2 ds. (5.9) The identity (5.9) easily allows to prove the following statement about local heat invariants, cf. [Wid79], [CoMo90], [BlFo90]: Proposition 5.2. Let A 0 , ..., A k ∈ b Diff(M; W ) of order a 0 , ..., a k . Then the Schwartz kernel of A 0 e −σ 0tD 2 A 1 e −σ 1tD 2 ...A k e −σ ktD 2 has a pointwise asymptotic expansion ( A 0 e −σ 0tD 2 A 1 e −σ 1tD 2 ...A k e −σ ktD 2) (p, p) = ∑ (−t) |α| σ α 1 0 (σ 0 + σ 1 ) α 2 ...(σ 0 + ... + σ k−1 ) αk· α! α∈Z k + ,|α|≤n =: · (A 0 ∇ α 1 D A 1...∇ α k D A ) ke −tD2 (p, p) + Op (t (n+1−a−dim M)/2 ), n∑ j−dim M−a a j (A 0 , ..., A k , D)(p) t 2 + O p (t (n+1−a−dim M)/2 ), j=0 (5.10) ∑ where a = k a j . The asymptotic expansion is locally uniformly in p. Furthermore, it j=0 is uniform for σ ∈ ∆ k . Again we are facing the problem explained before Theorem 5.1. Still we will be able to show that one obtains a correct formula by taking the b-trace on the left and partie finie integrals on the right of (5.10):
Page 1 and 2: CONNES-CHERN CHARACTER FOR MANIFOLD
Page 3 and 4: CONNES-CHERN CHARACTER AND ETA COCH
Page 7 and 8: ∫ b M CONNES-CHERN CHARACTER AND
Page 11 and 12: where CONNES-CHERN CHARACTER AND ET
Page 13 and 14: where the map ι(h) is defined by C
Page 21 and 22: ∫ This means that b M CONNES-CHER
Page 49: CONNES-CHERN CHARACTER AND ETA COCH
Page 53 and 54: Applying (5.9) to A 1 we get e −
Page 57 and 58: lim t→0+ CONNES-CHERN CHARACTER A

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