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

CONNES-CHERN CHARACTER AND ETA COCHAINS 47
Proof. Changing variables u j = t 2 σ j , j = 1, ..., q; u j = σ j , j = q + 1, ..., n we find
t
∫∆ 2q f(t 2 σ 1 , ..., t 2 σ j )dσ
n
∫
=
{t −2 (u 1 +...+u q)+u q+1 +...+u n)≤1}
f(u 1 , ..., u q )du
∫
∫
(4.6)
= f(u 1 , ..., u q ) (
du
t 2 ∆ q 1−t −2 (u 1 +...+u q)
)∆ n−q
∫
1 (
=
1 − t −2 (u 1 + ... + u q ) ) n−q
f(u)du.
(n − q)! t 2 ∆ q
By assumption f is rapidly decreasing, hence we may apply the Dominated Convergence
Theorem to reach the conclusion.
□
4.2.1. Estimating the transgressed b-Chern character.
Proposition 4.4. For k ≥ 1 and a 0 , ..., a k ∈ b C ∞ (M) we have
{
b /ch k O(t −2 ), k even ,
(tD, D)(a 0 , ..., a k ) =
t → ∞. (4.7)
O(t −3 ), k odd ,
Proof. /ch k (tD, D)(a 0 , ..., a k ) is a sum of terms of the form
T = b 〈a 0 , [tD, a 1 ], ..., [tD, a i−1 ], D, [tD, a i ], ..., [tD, a k ]〉 tD . (4.8)
Writing A 0 = a 0 , A j = [D, a j ], j = 1, ..., i − 1, A j = [D, a j−1 ], j = i + 1, ..., k + 1, A i := D
we find
∑
T = t k
b 〈A 0 H 0 , ..., A k+1 H k+1 〉 tD , (4.9)
H j ∈{H,I−H}
where the sum runs over all sequences H 0 , ..., H k+1 with H j ∈ {H, I − H}. Since
H[D, a j ]H = 0, HD = DH = 0 (note A i = D !) only terms containing no more than
[k/2] + 1 copies of H can give a non-zero contribution.
Consider such a nonzero summand with at least one index j with H j = H and denote
by q the number of indices l with H l = I − H. Then q ≥ [ k+1 ] + 1 and we infer from
2
Lemmas 4.2,4.3
{
t k b 〈A 0 H 0 , ..., A k+1 H k+1 〉 tD = O(t k−2q O(t −2 ), k even,
) =
(4.10)
O(t −3 ), k odd .
We infer from Theorem 3.9 that the remaining summand with H j = I − H for all j
decays exponentially as t → ∞ and we are done.
□
4.2.2. The limit as t → ∞ of the b-Chern character. As in [CoMo93] we put
and
ϱ H (A) := HAH, (4.11)
ω H (A, B) := ϱ H (AB) − ϱ H (A)ϱ H (B). (4.12)