A category of correspondences for KK -theory
B. Mesland1
joint with M.Lesch (Bonn)
1 Institut
für Analysis
Leibniz Universität Hannover, Germany
March 8, 2017
Mesland
KK -theory and correspondences
1 / 22
Kasparov’s KK -theory
To a pair of separable C ∗ -algebras, Kasparov associated a graded
abelian group KK∗ (A, B) in such a way that:
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KK -theory and correspondences
2 / 22
Kasparov’s KK -theory
To a pair of separable C ∗ -algebras, Kasparov associated a graded
abelian group KK∗ (A, B) in such a way that:
KK∗ (A, C) ' K ∗ (A), the K -homology of A;
Mesland
KK -theory and correspondences
2 / 22
Kasparov’s KK -theory
To a pair of separable C ∗ -algebras, Kasparov associated a graded
abelian group KK∗ (A, B) in such a way that:
KK∗ (A, C) ' K ∗ (A), the K -homology of A;
KK∗ (C, B) ' K∗ (B), the K -theory of B;
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KK -theory and correspondences
2 / 22
Kasparov’s KK -theory
To a pair of separable C ∗ -algebras, Kasparov associated a graded
abelian group KK∗ (A, B) in such a way that:
KK∗ (A, C) ' K ∗ (A), the K -homology of A;
KK∗ (C, B) ' K∗ (B), the K -theory of B;
there is an associative, bilinear product
KKi (A, B) × KKj (B, C ) → KKi+j (A, C ).
Mesland
KK -theory and correspondences
2 / 22
Kasparov’s KK -theory
To a pair of separable C ∗ -algebras, Kasparov associated a graded
abelian group KK∗ (A, B) in such a way that:
KK∗ (A, C) ' K ∗ (A), the K -homology of A;
KK∗ (C, B) ' K∗ (B), the K -theory of B;
there is an associative, bilinear product
KKi (A, B) × KKj (B, C ) → KKi+j (A, C ).
KK can be viewed as a category
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KK -theory and correspondences
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Unbounded picture of KK -theory
The cycles for KK0 (A, B) are given by pairs (X , S) consisting of
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KK -theory and correspondences
3 / 22
Unbounded picture of KK -theory
The cycles for KK0 (A, B) are given by pairs (X , S) consisting of
A Hilbert bimodule A XB
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KK -theory and correspondences
3 / 22
Unbounded picture of KK -theory
The cycles for KK0 (A, B) are given by pairs (X , S) consisting of
A Hilbert bimodule A XB
an operator S : DomS → A XB such that a(S ± i)−1 ∈ K(X )
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KK -theory and correspondences
3 / 22
Unbounded picture of KK -theory
The cycles for KK0 (A, B) are given by pairs (X , S) consisting of
A Hilbert bimodule A XB
an operator S : DomS → A XB such that a(S ± i)−1 ∈ K(X )
the ∗-subalgebra
LipS (A) := {a ∈ A : [S, a] extends boundedly to X → X }
is dense in A.
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KK -theory and correspondences
3 / 22
Question
Can we realize the Kasparov product as composition in a category
on the level of cycles?
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KK -theory and correspondences
4 / 22
Question
Can we realize the Kasparov product as composition in a category
on the level of cycles?
First step: leave the realm of C ∗ -algebras
Mesland
KK -theory and correspondences
4 / 22
Question
Can we realize the Kasparov product as composition in a category
on the level of cycles?
First step: leave the realm of C ∗ -algebras
For a KK -cycle (X , S) fix a norm dense ∗-subalgebra A ⊂ A which
is closed in the norm kak + k[S, a]k
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KK -theory and correspondences
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Question
Can we realize the Kasparov product as composition in a category
on the level of cycles?
First step: leave the realm of C ∗ -algebras
For a KK -cycle (X , S) fix a norm dense ∗-subalgebra A ⊂ A which
is closed in the norm kak + k[S, a]k
Notation: (A, X , S).
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KK -theory and correspondences
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The Hopf fibration
The fibre bundle S 1 → S 3 → S 2
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KK -theory and correspondences
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The Hopf fibration
The fibre bundle S 1 → S 3 → S 2
K -cycles (C 1 (S 3 ), L2 (S 3 , E), D) and (C 1 (S 2 ), L2 (S 2 , F), T )
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KK -theory and correspondences
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The Hopf fibration
The fibre bundle S 1 → S 3 → S 2
K -cycles (C 1 (S 3 ), L2 (S 3 , E), D) and (C 1 (S 2 ), L2 (S 2 , F), T )
Peter-Weyl decomposition C ∞ (S 3 ) '
Mesland
L
n∈Z C
∞ (S 2 , L )
n
KK -theory and correspondences
5 / 22
The Hopf fibration
The fibre bundle S 1 → S 3 → S 2
K -cycles (C 1 (S 3 ), L2 (S 3 , E), D) and (C 1 (S 2 ), L2 (S 2 , F), T )
Peter-Weyl decomposition C ∞ (S 3 ) '
L
n∈Z C
∞ (S 2 , L )
n
Grassmann connections ∇n : C ∞ (S 2 , Ln ) → C ∞ (S 2 , Ln ) ⊗ Ω1 (S 2 )
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KK -theory and correspondences
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The Hopf fibration
The fibre bundle S 1 → S 3 → S 2
K -cycles (C 1 (S 3 ), L2 (S 3 , E), D) and (C 1 (S 2 ), L2 (S 2 , F), T )
Peter-Weyl decomposition C ∞ (S 3 ) '
L
n∈Z C
∞ (S 2 , L )
n
Grassmann connections ∇n : C ∞ (S 2 , Ln ) → C ∞ (S 2 , Ln ) ⊗ Ω1 (S 2 )
Xn := C ∞ (S 2 , Ln ),
X :=
L
n∈Z Xn ,
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X := C ∞ (S 3 ) ⊂ X
KK -theory and correspondences
5 / 22
The Hopf fibration
The fibre bundle S 1 → S 3 → S 2
K -cycles (C 1 (S 3 ), L2 (S 3 , E), D) and (C 1 (S 2 ), L2 (S 2 , F), T )
Peter-Weyl decomposition C ∞ (S 3 ) '
L
n∈Z C
∞ (S 2 , L )
n
Grassmann connections ∇n : C ∞ (S 2 , Ln ) → C ∞ (S 2 , Ln ) ⊗ Ω1 (S 2 )
Xn := C ∞ (S 2 , Ln ),
X :=
L
n∈Z Xn ,
∇ := ⊕∇n : X → X ⊗ Ω1 (S 2 ),
Mesland
X := C ∞ (S 3 ) ⊂ X
S(xn ) := (nxn ) densely defined
KK -theory and correspondences
5 / 22
Theorem (Brain-Mesland-van Suijlekom)
There is an isomorphism L2 (S 3 , E) ' X ⊗C (S 2 ) L2 (S 2 , F) and an
equality D = S ⊗ 1 + 1 ⊗∇ T + 21 .
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KK -theory and correspondences
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Theorem (Brain-Mesland-van Suijlekom)
There is an isomorphism L2 (S 3 , E) ' X ⊗C (S 2 ) L2 (S 2 , F) and an
equality D = S ⊗ 1 + 1 ⊗∇ T + 21 .
The triple (X , S, ∇) is a correspondence:
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KK -theory and correspondences
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Theorem (Brain-Mesland-van Suijlekom)
There is an isomorphism L2 (S 3 , E) ' X ⊗C (S 2 ) L2 (S 2 , F) and an
equality D = S ⊗ 1 + 1 ⊗∇ T + 21 .
The triple (X , S, ∇) is a correspondence:
(X , S, ∇) ⊗ (L2 (S 2 , F), T ) = (X ⊗C (S 2 ) L2 (S 2 , F), S ⊗ 1 + 1 ⊗∇ T )
= (L2 (S 3 , E), D)
up to a "nice" bounded perturbation.
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KK -theory and correspondences
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More on connections
XB right Hilbert module,
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KK -theory and correspondences
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More on connections
XB right Hilbert module, with frame xi :
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KK -theory and correspondences
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More on connections
XB right Hilbert module, with frame xi :
P
i
xi hxi , xi = x for all x ∈ X
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KK -theory and correspondences
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More on connections
XB right Hilbert module, with frame xi :
P
i
xi hxi , xi = x for all x ∈ X
(B, Y , T ) KK -cycle for (B, C )
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KK -theory and correspondences
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More on connections
XB right Hilbert module, with frame xi :
P
i
xi hxi , xi = x for all x ∈ X
(B, Y , T ) KK -cycle for (B, C )
The frame xi is column finite for (B, Y , T ) if for all j
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KK -theory and correspondences
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More on connections
XB right Hilbert module, with frame xi :
P
i
xi hxi , xi = x for all x ∈ X
(B, Y , T ) KK -cycle for (B, C )
The frame xi is column finite for (B, Y , T ) if for all j
hxi , xj i ∈ B
and
∗
i [T , hxi , xj i] [T , hxi , xj i]
P
Mesland
<∞
KK -theory and correspondences
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More on connections
P
T -forms: Ω1T := { k bk0 [T , bk1 ] : bkj ∈ B} ⊂ End(Y )
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KK -theory and correspondences
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More on connections
P
T -forms: Ω1T := { k bk0 [T , bk1 ] : bkj ∈ B} ⊂ End(Y )
X := {x ∈ X :
B-submodule
P
i [T , hxi , xi]
∗ [T , hx
Mesland
i , xi]
< ∞} ⊂ X dense
KK -theory and correspondences
8 / 22
More on connections
P
T -forms: Ω1T := { k bk0 [T , bk1 ] : bkj ∈ B} ⊂ End(Y )
X := {x ∈ X :
B-submodule
P
i [T , hxi , xi]
∗ [T , hx
i , xi]
Connection ∇ : X → X ⊗B Ω1T , x 7→
Mesland
P
i
< ∞} ⊂ X dense
xi ⊗ [T , hxi , xi]
KK -theory and correspondences
8 / 22
More on connections
P
T -forms: Ω1T := { k bk0 [T , bk1 ] : bkj ∈ B} ⊂ End(Y )
X := {x ∈ X :
B-submodule
P
i [T , hxi , xi]
∗ [T , hx
i , xi]
Connection ∇ : X → X ⊗B Ω1T , x 7→
1 ⊗∇ T (x ⊗ y ) :=
P
i
P
i
< ∞} ⊂ X dense
xi ⊗ [T , hxi , xi]
xi ⊗ T hxi , xiy
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KK -theory and correspondences
8 / 22
More on connections
P
T -forms: Ω1T := { k bk0 [T , bk1 ] : bkj ∈ B} ⊂ End(Y )
X := {x ∈ X :
B-submodule
P
i [T , hxi , xi]
∗ [T , hx
i , xi]
Connection ∇ : X → X ⊗B Ω1T , x 7→
1 ⊗∇ T (x ⊗ y ) :=
P
i
P
i
< ∞} ⊂ X dense
xi ⊗ [T , hxi , xi]
xi ⊗ T hxi , xiy
Defined for x ∈ X , y ∈ DomT , symmetric operator
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KK -theory and correspondences
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More on connections
P
T -forms: Ω1T := { k bk0 [T , bk1 ] : bkj ∈ B} ⊂ End(Y )
X := {x ∈ X :
B-submodule
P
i [T , hxi , xi]
∗ [T , hx
i , xi]
Connection ∇ : X → X ⊗B Ω1T , x 7→
1 ⊗∇ T (x ⊗ y ) :=
P
i
P
i
< ∞} ⊂ X dense
xi ⊗ [T , hxi , xi]
xi ⊗ T hxi , xiy
Defined for x ∈ X , y ∈ DomT , symmetric operator
In general 1 ⊗∇ T is NOT essentially self-adjoint
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KK -theory and correspondences
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Self-adjointness
Definition
X is a complete projective module for (B, Y , T ) if there exists an
approximate unit wn for K(X ) contained in the algebraic convex
hull C (un ) of the approximate unit
un :=
X
|xi ihxi | ∈ K(X ),
|i|≤n
such that [1 ⊗∇ T , wn ] → 0 strictly on X ⊗B Y .
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KK -theory and correspondences
9 / 22
Self-adjointness
Definition
X is a complete projective module for (B, Y , T ) if there exists an
approximate unit wn for K(X ) contained in the algebraic convex
hull C (un ) of the approximate unit
un :=
X
|xi ihxi | ∈ K(X ),
|i|≤n
such that [1 ⊗∇ T , wn ] → 0 strictly on X ⊗B Y .
Theorem (Mesland-Rennie)
If X is a complete projective module for (B, Y , T ) the operator
1 ⊗∇ T is essentially self-adjoint and regular in X ⊗B Y .
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KK -theory and correspondences
9 / 22
Correspondence (X , S, ∇) for (B, Y , T )
(A, XB , S) KK -cycle
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KK -theory and correspondences
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Correspondence (X , S, ∇) for (B, Y , T )
(A, XB , S) KK -cycle
X ⊂ X complete projective submodule for (B, Y , T )
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KK -theory and correspondences
10 / 22
Correspondence (X , S, ∇) for (B, Y , T )
(A, XB , S) KK -cycle
X ⊂ X complete projective submodule for (B, Y , T )
X∇ closure wrt kxk + k∇(x)k
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KK -theory and correspondences
10 / 22
Correspondence (X , S, ∇) for (B, Y , T )
(A, XB , S) KK -cycle
X ⊂ X complete projective submodule for (B, Y , T )
X∇ closure wrt kxk + k∇(x)k
(S ± i)−1 X∇ ⊂ X∇ and aX∇ ⊂ X∇ for a ∈ A
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KK -theory and correspondences
10 / 22
Correspondence (X , S, ∇) for (B, Y , T )
(A, XB , S) KK -cycle
X ⊂ X complete projective submodule for (B, Y , T )
X∇ closure wrt kxk + k∇(x)k
(S ± i)−1 X∇ ⊂ X∇ and aX∇ ⊂ X∇ for a ∈ A
(∇S + S∇)(S ± i)−1 : X∇ ⊗B Ω1T → X ⊗B Ω1T bounded
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KK -theory and correspondences
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Correspondence (X , S, ∇) for (B, Y , T )
(A, XB , S) KK -cycle
X ⊂ X complete projective submodule for (B, Y , T )
X∇ closure wrt kxk + k∇(x)k
(S ± i)−1 X∇ ⊂ X∇ and aX∇ ⊂ X∇ for a ∈ A
(∇S + S∇)(S ± i)−1 : X∇ ⊗B Ω1T → X ⊗B Ω1T bounded
[∇, a] : X∇ ⊗B Ω1T → X ⊗B Ω1T bounded
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KK -theory and correspondences
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Self-adjointness and sums
Theorem (Kaad-Lesch)
Let s, t be a pair of self-adjoint regular operators in a Hilbert
module Z such that
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KK -theory and correspondences
11 / 22
Self-adjointness and sums
Theorem (Kaad-Lesch)
Let s, t be a pair of self-adjoint regular operators in a Hilbert
module Z such that
1
Z := (s ± i)−1 Domt ⊂ Domt is a core for t
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KK -theory and correspondences
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Self-adjointness and sums
Theorem (Kaad-Lesch)
Let s, t be a pair of self-adjoint regular operators in a Hilbert
module Z such that
1
2
Z := (s ± i)−1 Domt ⊂ Domt is a core for t
there exists C > 0 such that for z ∈ Z ,
k(st + ts)zk ≤ C (kzk + kszk).
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KK -theory and correspondences
11 / 22
Self-adjointness and sums
Theorem (Kaad-Lesch)
Let s, t be a pair of self-adjoint regular operators in a Hilbert
module Z such that
1
2
Z := (s ± i)−1 Domt ⊂ Domt is a core for t
there exists C > 0 such that for z ∈ Z ,
k(st + ts)zk ≤ C (kzk + kszk).
Then s + t : Doms ∩ Domt → Z is self-adjoint and regular.
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KK -theory and correspondences
11 / 22
Self-adjointness and sums
Theorem (Kaad-Lesch)
Let s, t be a pair of self-adjoint regular operators in a Hilbert
module Z such that
1
2
Z := (s ± i)−1 Domt ⊂ Domt is a core for t
there exists C > 0 such that for z ∈ Z ,
k(st + ts)zk ≤ C (kzk + kszk).
Then s + t : Doms ∩ Domt → Z is self-adjoint and regular.
We call such (s, t) a weakly anticommuting pair.
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KK -theory and correspondences
11 / 22
Kasparov product with a correspondence
Theorem (Mesland, Kaad-Lesch, Mesland-Rennie)
Let (X , S, ∇) be a correspondence for (B, Y , T ). Then
(X ⊗B Y , S ⊗ 1 + 1 ⊗∇ T )
is a KK -cycle representing the Kasparov product of (X , S) and
(Y , T ).
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KK -theory and correspondences
12 / 22
Kasparov product with a correspondence
Theorem (Mesland, Kaad-Lesch, Mesland-Rennie)
Let (X , S, ∇) be a correspondence for (B, Y , T ). Then
(X ⊗B Y , S ⊗ 1 + 1 ⊗∇ T )
is a KK -cycle representing the Kasparov product of (X , S) and
(Y , T ).
Main point: with s := S ⊗ 1 and t := 1 ⊗∇ T , the pair (s, t)
weakly anticommutes.
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KK -theory and correspondences
12 / 22
Surjectivity
Theorem (Mesland-Rennie)
Let x ∈ KK∗ (A, B) and y ∈ KK∗ (B, C ). There exist a cycle (Y , T )
representing y and a correspondence (X , S, ∇) for (B, Y , T )
representing x. Consequently
(X ⊗B Y , S ⊗ 1 + 1 ⊗∇ T )
represents the Kasparov product x ⊗ y ∈ KK∗ (A, C ).
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KK -theory and correspondences
13 / 22
Correspondence of spectral triples
Definition
Let (Ai , Hi , Di ) be spectral triples for i = 1, 2. A correspondence
from (A0 , H0 , D0 ) to (A1 , H1 , D1 ) is a correspondence (X , S, ∇)
for (A1 , H1 , D1 ) such that
(H0 , D0 + R) = (X ⊗A1 H1 , S ⊗ 1 + 1 ⊗∇ D1 )
for some R with aR = Ra bounded for a ∈ A1 .
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KK -theory and correspondences
14 / 22
Correspondence of spectral triples
Definition
Let (Ai , Hi , Di ) be spectral triples for i = 1, 2. A correspondence
from (A0 , H0 , D0 ) to (A1 , H1 , D1 ) is a correspondence (X , S, ∇)
for (A1 , H1 , D1 ) such that
(H0 , D0 + R) = (X ⊗A1 H1 , S ⊗ 1 + 1 ⊗∇ D1 )
for some R with aR = Ra bounded for a ∈ A1 .
The Hopf fibration is an example of this
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KK -theory and correspondences
14 / 22
Correspondence of spectral triples
Definition
Let (Ai , Hi , Di ) be spectral triples for i = 1, 2. A correspondence
from (A0 , H0 , D0 ) to (A1 , H1 , D1 ) is a correspondence (X , S, ∇)
for (A1 , H1 , D1 ) such that
(H0 , D0 + R) = (X ⊗A1 H1 , S ⊗ 1 + 1 ⊗∇ D1 )
for some R with aR = Ra bounded for a ∈ A1 .
The Hopf fibration is an example of this
General locally bounded perturbations studied by van den Dungen
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KK -theory and correspondences
14 / 22
Composition of correspondences
(X , S, ∇1 ) correspondence from (A0 , H0 , D0 ) to (A1 , H1 , D1 ) with
frame xi
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KK -theory and correspondences
15 / 22
Composition of correspondences
(X , S, ∇1 ) correspondence from (A0 , H0 , D0 ) to (A1 , H1 , D1 ) with
frame xi
(Y, T , ∇2 ) correspondence from (A1 , H1 , D1 ) to (A2 , H2 , D2 ) with
frame yk .
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KK -theory and correspondences
15 / 22
Composition of correspondences
(X , S, ∇1 ) correspondence from (A0 , H0 , D0 ) to (A1 , H1 , D1 ) with
frame xi
(Y, T , ∇2 ) correspondence from (A1 , H1 , D1 ) to (A2 , H2 , D2 ) with
frame yk .
Proposition
The Haagerup tensor product X ⊗hA1 Y ⊂ X ⊗A1 Y is a complete
projective module for (A2 , H2 , D2 ), with respect to the frame
xi ⊗ yk for X ⊗A1 Y .
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KK -theory and correspondences
15 / 22
Composition of correspondences
Proposition
The map
Ω1D1 → Ω1T ,
X
bi0 [D1 , bi1 ] 7→
i
X
bi0 [T ⊗ 1, bi1 ]
i
is well defined and completely contractive. Consequently
(X , S, ∇1 ) is a correspondence for (A1 , Y , T ) and
(X ⊗A1 Y , S ⊗ 1 + 1 ⊗∇1 T )
is a KK -cycle.
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KK -theory and correspondences
16 / 22
Smoothness of the sum
We wish to show that
Mesland
KK -theory and correspondences
17 / 22
Smoothness of the sum
We wish to show that
(X ⊗hA1 Y, S ⊗ 1 + 1 ⊗∇ T , 1 ⊗∇1 ∇2 )
is a correspondence for (A0 , H0 , D0 ) and (A2 , H2 , D2 ).
Mesland
KK -theory and correspondences
17 / 22
Smoothness of the sum
We wish to show that
(X ⊗hA1 Y, S ⊗ 1 + 1 ⊗∇ T , 1 ⊗∇1 ∇2 )
is a correspondence for (A0 , H0 , D0 ) and (A2 , H2 , D2 ). For this it
remains to show that
(S ⊗ 1 + 1 ⊗∇1 T ± i)−1 (X ⊗hA1 Y)1⊗∇1 ∇2 ⊂ (X ⊗hA1 Y)1⊗∇1 ∇2
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KK -theory and correspondences
17 / 22
Smoothness of the sum
We wish to show that
(X ⊗hA1 Y, S ⊗ 1 + 1 ⊗∇ T , 1 ⊗∇1 ∇2 )
is a correspondence for (A0 , H0 , D0 ) and (A2 , H2 , D2 ). For this it
remains to show that
(S ⊗ 1 + 1 ⊗∇1 T ± i)−1 (X ⊗hA1 Y)1⊗∇1 ∇2 ⊂ (X ⊗hA1 Y)1⊗∇1 ∇2
and that
[S ⊗ 1 + 1 ⊗∇1 T , 1 ⊗∇1 ∇2 ](S ⊗ 1 + 1 ⊗∇1 T ± i)−1
defines a bounded operator.
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KK -theory and correspondences
17 / 22
The last step...
Write s = S ⊗ 1 ⊗ 1, t = 1 ⊗∇1 T ⊗ 1 and ∂ = 1 ⊗∇1 1 ⊗∇2 D2
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KK -theory and correspondences
18 / 22
The last step...
Write s = S ⊗ 1 ⊗ 1, t = 1 ⊗∇1 T ⊗ 1 and ∂ = 1 ⊗∇1 1 ⊗∇2 D2
The pairs (s, t) and (s, ∂) weakly anticommute and
Mesland
KK -theory and correspondences
18 / 22
The last step...
Write s = S ⊗ 1 ⊗ 1, t = 1 ⊗∇1 T ⊗ 1 and ∂ = 1 ⊗∇1 1 ⊗∇2 D2
The pairs (s, t) and (s, ∂) weakly anticommute and
Theorem (Lesch)
Let (s, t), (t, ∂) and (s, ∂) be weakly anti-commuting pairs. Then
(s + t, ∂) is a weakly anticommuting pair.
Mesland
KK -theory and correspondences
18 / 22
The last step...
Write s = S ⊗ 1 ⊗ 1, t = 1 ⊗∇1 T ⊗ 1 and ∂ = 1 ⊗∇1 1 ⊗∇2 D2
The pairs (s, t) and (s, ∂) weakly anticommute and
Theorem (Lesch)
Let (s, t), (t, ∂) and (s, ∂) be weakly anti-commuting pairs. Then
(s + t, ∂) is a weakly anticommuting pair.
It thus remains to show that (t, ∂) weakly anticommute, to obtain:
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KK -theory and correspondences
18 / 22
The correspondence category
Theorem (Lesch-Mesland, in progress)
Composition of correspondences
(X , S, ∇1 ) ⊗ (Y, T , ∇2 ) := (X ⊗A1 Y, S ⊗ 1 + 1 ⊗∇1 T , 1 ⊗∇1 ∇2 )
yields a correspondence and correspondences form a category C .
The bounded transform C → KK defined by
(A, H, D) 7→ A,
(X , S, ∇) 7→ (X , S(1 + S 2 )−1/2 ),
is a functor that is surjective on both objects and morphisms.
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KK -theory and correspondences
19 / 22
Some consequences
KK -theory, including the product, can be defined directly and
without reference to the bounded picture
Mesland
KK -theory and correspondences
20 / 22
Some consequences
KK -theory, including the product, can be defined directly and
without reference to the bounded picture
Riemannian submersions (Kaad-v Suijlekom) are morphisms
Mesland
KK -theory and correspondences
20 / 22
Some consequences
KK -theory, including the product, can be defined directly and
without reference to the bounded picture
Riemannian submersions (Kaad-v Suijlekom) are morphisms
Morita equivalences of spectral triples (Kaad) are isomorphisms
Mesland
KK -theory and correspondences
20 / 22
Some consequences
KK -theory, including the product, can be defined directly and
without reference to the bounded picture
Riemannian submersions (Kaad-v Suijlekom) are morphisms
Morita equivalences of spectral triples (Kaad) are isomorphisms
the boundary map in the Cuntz-Pimsner sequence arises from
a morphism (Goffeng-Mesland-Rennie)
Mesland
KK -theory and correspondences
20 / 22
Some consequences
KK -theory, including the product, can be defined directly and
without reference to the bounded picture
Riemannian submersions (Kaad-v Suijlekom) are morphisms
Morita equivalences of spectral triples (Kaad) are isomorphisms
the boundary map in the Cuntz-Pimsner sequence arises from
a morphism (Goffeng-Mesland-Rennie)
Hecke operators and the boundary map in the Gysin sequence
for groups of hyperbolic isometries (Mesland-Sengun) arise
from morphisms
Mesland
KK -theory and correspondences
20 / 22
Future work
Product structure in the KK -bordism group
(Deeley-Goffeng-Mesland)
Mesland
KK -theory and correspondences
21 / 22
Future work
Product structure in the KK -bordism group
(Deeley-Goffeng-Mesland)
Gromov Hausdorff distance for spectral triples
(Cornelissen-Mesland)
Mesland
KK -theory and correspondences
21 / 22
Future work
Product structure in the KK -bordism group
(Deeley-Goffeng-Mesland)
Gromov Hausdorff distance for spectral triples
(Cornelissen-Mesland)
More spectral triples on purely infinite C ∗ -algebras (in
particular for boundary actions)
Mesland
KK -theory and correspondences
21 / 22
Summary
Complete projective submodule X ⊂ XB wrt to a KK-cycle
(B, Y , T ) (differentiable structure)
Mesland
KK -theory and correspondences
22 / 22
Summary
Complete projective submodule X ⊂ XB wrt to a KK-cycle
(B, Y , T ) (differentiable structure)
Connection ∇ : X → X ⊗B Ω1T and self-adjoint product
operator 1 ⊗∇ T in X ⊗B Y
Mesland
KK -theory and correspondences
22 / 22
Summary
Complete projective submodule X ⊂ XB wrt to a KK-cycle
(B, Y , T ) (differentiable structure)
Connection ∇ : X → X ⊗B Ω1T and self-adjoint product
operator 1 ⊗∇ T in X ⊗B Y
Correspondence (X , S, ∇) between spectral triples
Mesland
KK -theory and correspondences
22 / 22
Summary
Complete projective submodule X ⊂ XB wrt to a KK-cycle
(B, Y , T ) (differentiable structure)
Connection ∇ : X → X ⊗B Ω1T and self-adjoint product
operator 1 ⊗∇ T in X ⊗B Y
Correspondence (X , S, ∇) between spectral triples
Composition of correpondences
(X , S, ∇1 )⊗(Y, T , ∇2 ) := (X ⊗hB Y, S ⊗1+1⊗∇1 T , 1⊗∇1 ∇2 )
Mesland
KK -theory and correspondences
22 / 22
Summary
Complete projective submodule X ⊂ XB wrt to a KK-cycle
(B, Y , T ) (differentiable structure)
Connection ∇ : X → X ⊗B Ω1T and self-adjoint product
operator 1 ⊗∇ T in X ⊗B Y
Correspondence (X , S, ∇) between spectral triples
Composition of correpondences
(X , S, ∇1 )⊗(Y, T , ∇2 ) := (X ⊗hB Y, S ⊗1+1⊗∇1 T , 1⊗∇1 ∇2 )
Surjective functor C → KK
Mesland
KK -theory and correspondences
22 / 22
Summary
Complete projective submodule X ⊂ XB wrt to a KK-cycle
(B, Y , T ) (differentiable structure)
Connection ∇ : X → X ⊗B Ω1T and self-adjoint product
operator 1 ⊗∇ T in X ⊗B Y
Correspondence (X , S, ∇) between spectral triples
Composition of correpondences
(X , S, ∇1 )⊗(Y, T , ∇2 ) := (X ⊗hB Y, S ⊗1+1⊗∇1 T , 1⊗∇1 ∇2 )
Surjective functor C → KK
Commutative and noncommutative examples as elements of C
Mesland
KK -theory and correspondences
22 / 22