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

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80 MATTHIAS LESCH, HENRI MOSCOVICI, AND MARKUS J. PFLAUM such that for all σ ∈ ∆ k , σ j > 0 ‖A 0 e −σ 0tD 2 (I − H)A 1 · . . . · A k e −σ ktD 2 (I − H)‖ 1 ( k∏ ) ≤ C(δ, ε) σ −a j/2 j t −a/2−(dim M)/2−ε e −tδ , for all 0 < t < ∞. j=0 (B.21) In particular, a j ≤ 1, j = 0, ..., k, then ‖(A 0 (I − H), ..., A k (I − H)) √ tD ‖ = O(t −a/2−(dim M)/2−0 e −tδ ), for all 0 < t < ∞. (B.22) Proof. We first reduce the problem to the case that all A j are compactly supported. To this end choose ϕ j0 −1, ϕ j0 ∈ C ∞ 0 (M) such that supp ϕ j0 ∩ supp(1 − ϕ j0 −1) = ∅ and such that ϕ j0 A j0 = A j0 ϕ j0 = A j0 . Decompose A j0 −1 = A j0 −1ϕ j0 −1 + A j0 −1(1 − ϕ j0 −1). First we show that the estimates (B.19), (B.21) hold if we replace A j0 −1 by A j0 −1(1− ϕ j0 −1): Case 1. Proposition B.5.3 gives ‖A j0 −1(1 − ϕ j0 −1)e −σ j 0 −1tD 2 ϕ j0 ‖ 1 ≤ C t0 σ N j 0 −1t N , for σ j0 −1t ≤ t 0 . (B.23) The operator norm of the other factors can be estimated using the Spectral Theorem, taking into account the D a j –boundedness of A j : Hence by the Hölder inequality ‖A j e −σ jtD 2 ‖ ≤ C t0 (σ j t) −a j/2 , for σ j t ≤ t 0 . (B.24) ‖A 0 e −σ 0tD 2 A 1 · . . . · A j0 −1(1 − ϕ j0 −1)e −σ j 0 −1tD 2 · A k e −σ ktD 2 ‖ 1 ( k∏ ≤ C t0 σ −a j/2 j )t − P k a j +N j=0 , 0 < t ≤ t0 , j=0 (B.25) which is even better than (B.19). Case 2 (D Fredholm). From (B.23), Proposition B.6 and the fact that H is a finite rank operator with e −ξD2 H = H we infer ‖A j0 −1(1 − ϕ j0 −1)e −σ j 0 −1tD 2 (I − H)ϕ j0 ‖ 1 ≤ C δ e −σ j 0 −1tδ , for all 0 < t < ∞. (B.26) To the other factors we apply Proposition B.6 with p = ∞: ‖A j e −σ jtD 2 (I − H)‖ ≤ C δ (σ j t) −a j/2 e −σ jtδ , 0 < t < ∞. (B.27) The Hölder inequality combined with (B.26),(B.27) gives (B.21). Altogether we are left to consider A 0 , ..., A j0 −1ϕ j0 −1, A j0 , ...A k where now A j0 −1ϕ j0 −1 and A j0 are compactly supported. Continuing this way, also to the right of j 0 , it remains to treat the case where each A j has compact support.
CONNES-CHERN CHARACTER AND ETA COCHAINS 81 Case 1. We apply Hölder’s inequality for Schatten norms and Proposition B.5: ‖A 0 e −σ 0tD 2 A 1 · . . . · A k e −σ ktD 2 ‖ 1 k∏ ≤ ‖A j e −σ jtD 2 ‖ σ −1 j j=0 ≤ C(t 0 , ε) ≤ C(t 0 , ε) k∏ (σ j t) −a j/2− j=0 ( k∏ j=0 σ −a j/2 j ) dim M+ε 2 σ j dim M+ε −a/2− t 2 , 0 < t ≤ t 0 , (B.28) dim M+ε − thanks to the fact that σ ↦→ σ 2 σ is bounded as σ → 0. Case 2. If D is Fredholm we estimate ‖A 0 e −σ 0tD 2 (I − H)A 1 · . . . · A k e −σ ktD 2 (I − H)‖ 1 using Hölder as in (B.28) and apply Proposition B.6 to the individual factors: ‖A j e −σ jtD 2 (I − H)‖ σ −1 j ≤ C δ (σ j t) −a dim M+ε j/2− σ 2 j e −σjtδ , 0 < t < ∞, (B.29) to reach the conclusion. Finally we remark that the inequalities (B.20), (B.22) follow by integrating the inequalities (B.19), (B.21) over the standard simplex ∆ k . Note that ∫ ∆ k (σ 0 · . . . · σ k ) −1/2 dσ < ∞. □ [APS75] [APS76] [ASS94] [BBFu98] References M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69. MR 0397797 (53 #1655a) , Spectral asymmetry and Riemannian geometry. III, Math. Proc. Cambridge Philos. Soc. 79 (1976), no. 1, 71–99. MR 0397799 (53 #1655c) J. Avron, R. Seiler, and B. Simon, The index of a pair of projections, J. Funct. Anal. 120 (1994), no. 1, 220–237. MR 1262254 (95b:47012) B. Booss-Bavnbek and K. Furutani, The Maslov index: a functional analytical definition and the spectral flow formula, Tokyo J. Math. 21 (1998), no. 1, 1–34. MR 1630119 (99e:58172) [BBLZ09] B. Booß-Bavnbek, M. Lesch, and C. Zhu, The Calderón projection: new definition and applications, J. Geom. Phys. 59 (2009), no. 7, 784–826. arXiv:0803.4160v1 [math.DG] , DOI: 10.1016/j.geomphys.2009.03.012, http://dx.doi.org/10.1016/j.geomphys.2009.03.012, MR 2536846 [BBWo93] B. Booß-Bavnbek and K. P. Wojciechowski, Elliptic boundary problems for Dirac operators, Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, 1993. MR 1233386 (94h:58168) [BDT89] P. Baum, R. G. Douglas, and M. E. Taylor, Cycles and relative cycles in analytic K-homology, J. Differential Geom. 30 (1989), no. 3, 761–804. MR 1021372 (91b:58244) [BGV92] N. Berline, E. Getzler, and M. Vergne, Heat kernels and Dirac operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 298, Springer-Verlag, Berlin, 1992. MR 1215720 (94e:58130)
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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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∫ This means that b M CONNES-CHER
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Page 53 and 54: Applying (5.9) to A 1 we get e −
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