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

CONNES-CHERN CHARACTER AND ETA COCHAINS 9
B.2. Heat kernel estimates 75
B.3. Estimates for the JLO integrand 77
References 79
List of Figures
1 The compact manifold M with boundary. The picture of the collar does not
capture the b–metric. 12
2 Interior of M with cylindrical coordinates on the collar (0, 2) × ∂M ≃
(−∞, ln 2) × ∂M via (r, η) ↦→ (ln r, η). This picture correctly captures the
metric on the collar. 13
3 The cut-off functions ϕ and ψ. 36
4 By homotopy invariance the signed sum of the four spectral flows adds up to 0. 61
5 Contour of integration for calculating e −tD2 from the resolvent. 75
6 Contour of integration if bottom of the essential spectrum of D 2 is c. 76
1. Preliminaries
1.1. The general setup. Associated to a compact smooth manifold M with boundary
∂M, there is a commutative diagram of Fréchet algebras with exact rows
0 J ∞ (∂M, M) C ∞ (M)
Id
ϱ
E ∞ (∂M, M)
0 J (∂M, M) C ∞ (M) C ∞ (∂M) 0.
‖∂M
0
(1.1)
J (∂M, M) ⊂ C ∞ (M) is the closed ideal of smooth functions on M vanishing on ∂M,
J ∞ (∂M, M) ⊂ J (∂M, M) denotes the closed ideal of smooth functions on M vanishing
up to infinite order on ∂M, and E ∞ (∂M, M) is the algebra of Whitney functions over the
subset ∂M ⊂ M. More generally, for every closed subset X ⊂ M the ideal J ∞ (X, M) ⊂
C ∞ (M) is defined as being
:= { f ∈ C ∞ (M) | Df |X = 0 for every differential operator D on M } .
By Whitney’s extension theorem (cf. [Mal67, Tou72]), the algebra E ∞ (X, M) of Whitney
functions over X ⊂ M is naturally isomorphic to the quotient of C ∞ (M) by the
closed ideal J ∞ (X, M); we take this as a definition of E ∞ (X, M). The right vertical
arrow in diagram (1.1) is given by the map
E ∞ (X, M) → C ∞ (X),
F ↦→ F ‖X := F + J (X, M),
which is a surjection.
Let us check that the Fréchet algebra J ∞ := J ∞ (∂M, M) is a local C ∗ -algebra.
First, by Faà di Bruno’s formula the unitalization J ∞,+ of J ∞ is seen to be closed under
holomorphic calculus in the unitalization J + of the algebra J := C 0 (M \ ∂M). Since
J ∞,+ is also dense in J + , it follows that J ∞ := J ∞ (∂M, M) is indeed a local C ∗ -algebra