On a theorem of Kucerovsky for half-closed chains
Jens Kaad
June 12, 2017
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Polar decomposition
Definition
1
∂
Vertical data: (E , ri ∂θ
) : Cc∞ (R2 \ {0}) → C0 (0, ∞) .
Unbounded Kasparov module.
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Polar decomposition
Definition
1
2
∂
Vertical data: (E , ri ∂θ
) : Cc∞ (R2 \ {0}) → C0 (0, ∞) .
Unbounded Kasparov module.
d
Horizontal data: L2 (0, ∞) , i dr
: Cc∞ (0, ∞) → C. Half
closed chain.
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Polar decomposition
Definition
1
2
∂
Vertical data: (E , ri ∂θ
) : Cc∞ (R2 \ {0}) → C0 (0, ∞) .
Unbounded Kasparov module.
d
Horizontal data: L2 (0, ∞) , i dr
: Cc∞ (0, ∞) → C. Half
closed chain.
Question
Is it true that
in K 0
i ∂ b
ι∗ [R2 ] = E ,
⊗
r ∂θ C0
C0 (R2 \ {0}) ?
Jens Kaad
L2 (0, ∞) , i d
(0,∞)
dr
On a theorem of Kucerovsky for half-closed chains
KK -theory
Notation
1 A, B, C , etc. are separable C ∗ -algebras.
2
Kasparov’s bivariant K -theory denoted by KK (A, B).
3
Classes in KK -theory are given by Kasparov modules
(E , F ) : A → B up to homotopy.
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
KK -theory
Notation
1 A, B, C , etc. are separable C ∗ -algebras.
2
Kasparov’s bivariant K -theory denoted by KK (A, B).
3
Classes in KK -theory are given by Kasparov modules
(E , F ) : A → B up to homotopy.
Theorem (Kasparov)
There exists a bilinear and associative pairing
b B : KK (A, B) × KK (B, C ) → KK (A, C )
⊗
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
How to recognize a Kasparov product?
Theorem (Connes, Skandalis)
b B E2 , F ) : A → C , (E1 , F1 ) : A → B
Suppose that (E = E1 ⊗
and (E2 , F2 ) : B → C are Kasparov modules. Suppose
moreover that
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
How to recognize a Kasparov product?
Theorem (Connes, Skandalis)
b B E2 , F ) : A → C , (E1 , F1 ) : A → B
Suppose that (E = E1 ⊗
and (E2 , F2 ) : B → C are Kasparov modules. Suppose
moreover that
1
b B E2
Tξ F2 − (−1)∂ξ FTξ : E2 → E1 ⊗
is compact for all homogeneous ξ ∈ E1 .
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
How to recognize a Kasparov product?
Theorem (Connes, Skandalis)
b B E2 , F ) : A → C , (E1 , F1 ) : A → B
Suppose that (E = E1 ⊗
and (E2 , F2 ) : B → C are Kasparov modules. Suppose
moreover that
1
b B E2
Tξ F2 − (−1)∂ξ FTξ : E2 → E1 ⊗
is compact for all homogeneous ξ ∈ E1 .
2
b + (F1 ⊗1)F
b
b B E2 → E1 ⊗
b B E2
a∗ · F (F1 ⊗1)
· a : E1 ⊗
is positive modulo compacts for all a ∈ A.
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
How to recognize a Kasparov product?
Theorem (Connes, Skandalis)
b B E2 , F ) : A → C , (E1 , F1 ) : A → B
Suppose that (E = E1 ⊗
and (E2 , F2 ) : B → C are Kasparov modules. Suppose
moreover that
1
b B E2
Tξ F2 − (−1)∂ξ FTξ : E2 → E1 ⊗
is compact for all homogeneous ξ ∈ E1 .
2
b + (F1 ⊗1)F
b
b B E2 → E1 ⊗
b B E2
a∗ · F (F1 ⊗1)
· a : E1 ⊗
is positive modulo compacts for all a ∈ A.
b B E2 , F ] = [E1 , F1 ]⊗
b B [E2 , F2 ] in KK (A, C ).
Then [E1 ⊗
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Half closed chains (I)
Notation
A , B, C , etc. are ∗-algebras equipped with fixed C ∗ -norms and
C ∗ -completions A, B, C , etc.
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Half closed chains (I)
Notation
A , B, C , etc. are ∗-algebras equipped with fixed C ∗ -norms and
C ∗ -completions A, B, C , etc.
Definition
A half closed chain (E , D) : A → B consists of
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Half closed chains (I)
Notation
A , B, C , etc. are ∗-algebras equipped with fixed C ∗ -norms and
C ∗ -completions A, B, C , etc.
Definition
A half closed chain (E , D) : A → B consists of
1
A countably generated C ∗ -correspondence E from A to B;
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Half closed chains (I)
Notation
A , B, C , etc. are ∗-algebras equipped with fixed C ∗ -norms and
C ∗ -completions A, B, C , etc.
Definition
A half closed chain (E , D) : A → B consists of
1
2
A countably generated C ∗ -correspondence E from A to B;
A symmetric and regular unbounded operator
D : Dom(D) → E , such that:
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Half closed chains (II)
Definition (Continued)
1
a · (1 + D ∗ D)−1 : E → E is compact for all a ∈ A;
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Half closed chains (II)
Definition (Continued)
1
2
a · (1 + D ∗ D)−1 : E → E is compact for all a ∈ A;
a Dom(D ∗ ) ⊆ Dom(D) and the commutator
[D, a] : Dom(D) → E
is bounded for all a ∈ A .
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Half closed chains (II)
Definition (Continued)
1
2
a · (1 + D ∗ D)−1 : E → E is compact for all a ∈ A;
a Dom(D ∗ ) ⊆ Dom(D) and the commutator
[D, a] : Dom(D) → E
is bounded for all a ∈ A .
Remark
The unbounded Kasparov modules from A to B are exactly the
half closed chains (E , D) from A to B with D = D ∗ .
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Elliptic differential operators
Theorem (Baum, Douglas, Taylor)
Let M be any Riemannian manifold and let
2
D : Γ∞
c (M, E ) → L (M, E ) be any symmetric and elliptic first
order differential operator with closure D. Then
(L2 (M, E ), D) : Cc∞ (M) → C
is a half closed chain. Moreover, when M is complete and D has
finite propagation speed this half closed chain is a spectral triple.
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Bounded transforms of half closed chains
Theorem (Hilsum)
Suppose that (E , D) : A → B is a half closed chain from A to B.
Then (E , FD ) := (E , D(1 + D ∗ D)−1/2 ) : A → B is a Kasparov
module.
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Bounded transforms of half closed chains
Theorem (Hilsum)
Suppose that (E , D) : A → B is a half closed chain from A to B.
Then (E , FD ) := (E , D(1 + D ∗ D)−1/2 ) : A → B is a Kasparov
module.
Definition
The Baaj-Julg-bounded transform of a half closed chain
(E , D) : A → B is the class [E , FD ] ∈ KK (A, B).
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Kucerovsky’s theorem for unbounded Kasparov modules
Theorem
b B E2 , D) : A → C , (E1 , D1 ) : A → B and
Let (E1 ⊗
(E2 , D2 ) : B → C be unbounded Kasparov modules.
Suppose that
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Kucerovsky’s theorem for unbounded Kasparov modules
Theorem
b B E2 , D) : A → C , (E1 , D1 ) : A → B and
Let (E1 ⊗
(E2 , D2 ) : B → C be unbounded Kasparov modules.
Suppose that
1
There exists a dense B-submodule E1 ⊆ E such that
b B E2
DTξ − (−1)∂ξ Tξ D2 : Dom(D2 ) → E1 ⊗
is bounded adjointable for all homogeneous ξ ∈ E1 .
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Kucerovsky’s theorem for unbounded Kasparov modules
Theorem
b B E2 , D) : A → C , (E1 , D1 ) : A → B and
Let (E1 ⊗
(E2 , D2 ) : B → C be unbounded Kasparov modules.
Suppose that
1
There exists a dense B-submodule E1 ⊆ E such that
b B E2
DTξ − (−1)∂ξ Tξ D2 : Dom(D2 ) → E1 ⊗
2
is bounded adjointable for all homogeneous ξ ∈ E1 .
b and there exists κ ∈ (0, ∞) such that
Dom(D) ⊆ Dom(D1 ⊗1)
b
b
hDξ, (D1 ⊗1)ξi
+ h(D1 ⊗1)ξ,
Dξi ≥ −κhξ, ξi
for all ξ ∈ Dom(D).
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Kucerovsky’s theorem for unbounded Kasparov modules
Theorem
b B E2 , D) : A → C , (E1 , D1 ) : A → B and
Let (E1 ⊗
(E2 , D2 ) : B → C be unbounded Kasparov modules.
Suppose that
1
There exists a dense B-submodule E1 ⊆ E such that
b B E2
DTξ − (−1)∂ξ Tξ D2 : Dom(D2 ) → E1 ⊗
2
is bounded adjointable for all homogeneous ξ ∈ E1 .
b and there exists κ ∈ (0, ∞) such that
Dom(D) ⊆ Dom(D1 ⊗1)
b
b
hDξ, (D1 ⊗1)ξi
+ h(D1 ⊗1)ξ,
Dξi ≥ −κhξ, ξi
for all ξ ∈ Dom(D).
b B E2 , FD ] = [E1 , FD1 ]⊗
b B [E2 , FD2 ] in KK (A, C ).
Then [E1 ⊗
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Unbounded modular cycles (I)
Definition
An unbounded modular cycle (E , D, ∆) : A → B consists of
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Unbounded modular cycles (I)
Definition
An unbounded modular cycle (E , D, ∆) : A → B consists of
1
A countably generated C ∗ -correspondence E from A to B;
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Unbounded modular cycles (I)
Definition
An unbounded modular cycle (E , D, ∆) : A → B consists of
1
2
A countably generated C ∗ -correspondence E from A to B;
A selfadjoint and regular unbounded operator
D : D(D) → E ;
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Unbounded modular cycles (I)
Definition
An unbounded modular cycle (E , D, ∆) : A → B consists of
1
A countably generated C ∗ -correspondence E from A to B;
2
A selfadjoint and regular unbounded operator
D : D(D) → E ;
3
A bounded positive operator ∆ : E → E with dense image,
such that:
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Unbounded modular cycles (II)
Definition
1
a · (i + D)−1 : E → E is compact for all a ∈ A;
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Unbounded modular cycles (II)
Definition
1
a · (i + D)−1 : E → E is compact for all a ∈ A;
2
For each a ∈ A , λ ∈ C and each ξ ∈ Dom(D):
D(a + λ)∆(ξ) − ∆(a + λ)D(ξ) = ∆1/2 d∆ (a, λ)∆1/2 (ξ)
for some bounded operator d∆ (a, λ) : E → E .
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Unbounded modular cycles (II)
Definition
1
a · (i + D)−1 : E → E is compact for all a ∈ A;
2
For each a ∈ A , λ ∈ C and each ξ ∈ Dom(D):
D(a + λ)∆(ξ) − ∆(a + λ)D(ξ) = ∆1/2 d∆ (a, λ)∆1/2 (ξ)
3
for some bounded operator d∆ (a, λ) : E → E .
The sequence a · ∆(∆ + 1/n)−1 converges in operator
norm to a for all a ∈ A.
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Bounded transforms of unbounded modular cycles
Theorem (K.)
Suppose that (E , D, ∆) : A → B is an unbounded modular cycle.
Then (E , FD ) := (E , D(1 + D 2 )−1/2 ) : A → B is a Kasparov
module.
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Localization of half closed chains
Theorem (K., Suijlekom)
Let (E , D) : A → B be a half closed chain and let x ∈ A .
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Localization of half closed chains
Theorem (K., Suijlekom)
Let (E , D) : A → B be a half closed chain and let x ∈ A .
1
Define the C ∗ -correspondence xE = Im(x) ⊆ E from xAx ∗ to
B.
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Localization of half closed chains
Theorem (K., Suijlekom)
Let (E , D) : A → B be a half closed chain and let x ∈ A .
1
2
Define the C ∗ -correspondence xE = Im(x) ⊆ E from xAx ∗ to
B.
Define the unbounded operator Dx := xDx ∗ : D(Dx ) → Ex
with core D(D) ∩ Im(x).
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Localization of half closed chains
Theorem (K., Suijlekom)
Let (E , D) : A → B be a half closed chain and let x ∈ A .
1
2
Define the C ∗ -correspondence xE = Im(x) ⊆ E from xAx ∗ to
B.
Define the unbounded operator Dx := xDx ∗ : D(Dx ) → Ex
with core D(D) ∩ Im(x).
Then the localization (xE , Dx , xx ∗ ) : xA x ∗ → B is an
unbounded modular cycle. In particular, Dx is a selfadjoint
and regular unbounded operator.
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Localization as a Kasparov product
Theorem (K., Suijlekom)
Let (E , D) : A → B be a half closed chain and let x ∈ A . Then
b A [E , FD ]
[Ex , FDx ] = [xA, 0]⊗
in KK (xAx ∗ , B).
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Kucerovsky’s theorem for half closed chains
Theorem (K., Suijlekom)
b B E2 , D) : A → C , (E1 , D1 ) : A → B and
Let (E1 ⊗
(E2 , D2 ) : B → C be half closed chains. Suppose exist
dense B-submodule E1 ⊆ E1 and a countable generating
subset Λ ⊆ Asa :
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Kucerovsky’s theorem for half closed chains
Theorem (K., Suijlekom)
b B E2 , D) : A → C , (E1 , D1 ) : A → B and
Let (E1 ⊗
(E2 , D2 ) : B → C be half closed chains. Suppose exist
dense B-submodule E1 ⊆ E1 and a countable generating
subset Λ ⊆ Asa :
1
b B E2
DTξ − (−1)∂ξ Tξ D2 : Dom(D2 ) → E1 ⊗
is bounded adjointable for all homogeneous ξ ∈ E1 .
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Kucerovsky’s theorem for half closed chains
Theorem (K., Suijlekom)
b B E2 , D) : A → C , (E1 , D1 ) : A → B and
Let (E1 ⊗
(E2 , D2 ) : B → C be half closed chains. Suppose exist
dense B-submodule E1 ⊆ E1 and a countable generating
subset Λ ⊆ Asa :
1
b B E2
DTξ − (−1)∂ξ Tξ D2 : Dom(D2 ) → E1 ⊗
2
is bounded adjointable for all homogeneous ξ ∈ E1 .
b x] is trivial on Dom(D1 ⊗1).
b
For each x ∈ Λ, [D1 ⊗1,
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Kucerovsky’s theorem for half closed chains
Theorem (K., Suijlekom)
b B E2 , D) : A → C , (E1 , D1 ) : A → B and
Let (E1 ⊗
(E2 , D2 ) : B → C be half closed chains. Suppose exist
dense B-submodule E1 ⊆ E1 and a countable generating
subset Λ ⊆ Asa :
1
b B E2
DTξ − (−1)∂ξ Tξ D2 : Dom(D2 ) → E1 ⊗
2
3
is bounded adjointable for all homogeneous ξ ∈ E1 .
b x] is trivial on Dom(D1 ⊗1).
b
For each x ∈ Λ, [D1 ⊗1,
b and exists
For each x ∈ Λ, Dom(D) ∩ Im(x) ⊆ Dom(D1 ⊗1)
κx ∈ (0, ∞):
b
b
hDξ, (D1 ⊗1)ξi
+ h(D1 ⊗1)ξ,
Dξi ≥ −κx hξ, ξi
for all ξ ∈ Dom(D) ∩ Im(x).
Jens Kaad
On a theorem of Kucerovsky for half-closed chains
Kucerovsky’s theorem for half closed chains
Theorem (K., Suijlekom)
b B E2 , D) : A → C , (E1 , D1 ) : A → B and
Let (E1 ⊗
(E2 , D2 ) : B → C be half closed chains. Suppose exist
dense B-submodule E1 ⊆ E1 and a countable generating
subset Λ ⊆ Asa :
1
b B E2
DTξ − (−1)∂ξ Tξ D2 : Dom(D2 ) → E1 ⊗
2
3
is bounded adjointable for all homogeneous ξ ∈ E1 .
b x] is trivial on Dom(D1 ⊗1).
b
For each x ∈ Λ, [D1 ⊗1,
b and exists
For each x ∈ Λ, Dom(D) ∩ Im(x) ⊆ Dom(D1 ⊗1)
κx ∈ (0, ∞):
b
b
hDξ, (D1 ⊗1)ξi
+ h(D1 ⊗1)ξ,
Dξi ≥ −κx hξ, ξi
for all ξ ∈ Dom(D) ∩ Im(x).
b B E2 , FD ] = [E1 , FD1 ]⊗
b B [E2 , FD2 ] in KK (A, C ).
Then [E1 ⊗
Jens Kaad
On a theorem of Kucerovsky for half-closed chains