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

68 MATTHIAS LESCH, HENRI MOSCOVICI, AND MARKUS J. PFLAUM
After having fixed these data for M, we choose the most essential ingredient for the
b-calculus, namely an exact b-metric for M. Following [Mel93], one understands by
this a Riemannian metric g b on M ◦ such that on Y ◦ , the metric can be written in the
form
g b |Y = 1 (dr ⊗ dr) ◦
r|Y 2 |Y ◦ + η|Y ∗ ◦g ∂,
(A.1)
◦
where g ∂ is a Riemannian metric on the boundary ∂M. Clearly, one then has in the
cylindrical coordinates (x, η)
g b |Y ◦ = (dx ⊗ dx) |Y ◦ + η∗ |Y ◦g ∂.
Now consider the cylinder R × ∂M := R × ∂M together with the product metric
g cyl = dx ⊗ dx + pr ∗ 2 g ∂ ,
(A.2)
(A.3)
where here (with a slight abuse of language), x denotes the first coordinate of the
cylinder, and pr 2 : R × ∂M → ∂M the projection onto the second factor.
Next we introduce various algebras of so-called b-functions on R × ∂M. For c ∈ R
define b C ( ∞ (−∞, c)×∂M ) resp. b C ( ∞ (c, ∞)×∂M ) as the algebra of smooth functions
f on (−∞, c)×∂M resp. on (c, ∞)×∂M for which there exist functions f0 − , f1 − , f2 − , . . . ∈
C ∞ (∂M) resp. f 0 + , f 1 + , f 2 + , . . . ∈ C ∞ (∂M) such that the following asymptotic expansions
hold true:
f ∼ x→−∞ f0 − + f1 − e x + f2 − e 2x + . . . resp.
(A.4)
f ∼ x→∞ f 0 + + f 1 + e −x + f 2 + e −2x + . . . .
More precisely, this means that there exists for every k, l ∈ N and every differential
operator D on ∂M a constant C > 0 such that
∣
∣
∣∂xDf(x, l p) − 0 l Df0 − (p) − . . . − k l Df − ∣∣
k (p)ekx ≤ Ce
(k+1)x
for all x ≤ c − 1 and p ∈ ∂M resp.
∣
∣
∣∂xDf(x, l p) − 0 l Df 0 + (p) − . . . − (−k) l Df + ∣∣
k (p)e−kx ≤ Ce
−(k+1)x
for all x ≥ c + 1 and p ∈ ∂M.
(A.5)
The asymptotic expansion guarantees that f ∈ b C ∞ ( (−∞, c) × ∂M ) if and only if the
transformed function [0, e c [×∂M ∋ (r, p) ↦→ f ( ln r, p ) is a smooth function on the collar
[0, e c [×∂M.
The algebra b C ∞ ( R × ∂M ) of b-functions on the full cylinder consists of all smooth
functions f on R × ∂M such that
f |(−∞,0)×∂M ∈ b C ∞ ( (−∞, 0) × ∂M ) and f |(0,∞)×∂M ∈ b C ∞ ( (0, ∞) × ∂M ) .
Next, we define the algebras of b-functions with compact support on the cylindrical
ends by b C ∞ cpt
(
(−∞, c) × ∂M
)
:=
{
f ∈ b C ∞ ( (−∞, c) × ∂M ) | f(x, p) = 0 for x ≥ c − ε, p ∈ ∂M and some ε > 0 }
resp. by b C ∞ cpt
(
(c, ∞) × ∂M
)
:=
{
f ∈ b C ∞ ( (c, ∞) × ∂M ) | f(x, p) = 0 for x ≤ c + ε, p ∈ ∂M and some ε > 0 } .