Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Cocycle deformations of operator algebras and noncommutative
geometry
1 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Cocycle deformations of operator algebras and noncommutative
geometry
Deformation of C*-algebras by cocycles
1 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Cocycle deformations of operator algebras and noncommutative
geometry
Deformation of C*-algebras by cocycles
K-theory invariance under deformation: homotopy, bundles
1 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Cocycle deformations of operator algebras and noncommutative
geometry
Deformation of C*-algebras by cocycles
K-theory invariance under deformation: homotopy, bundles
Theta deformation: index theory, invariance of cyclic cohomology
1 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Cocycle deformations of operator algebras and noncommutative
geometry
Deformation of C*-algebras by cocycles
K-theory invariance under deformation: homotopy, bundles
Theta deformation: index theory, invariance of cyclic cohomology
Local index formula for theta deformations of manifolds
1 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Table of contents
1
Deformation quantization: background
2
Rieffel deformation
3
Rieffel deformation via crossed products
4
Deformation of C*-algebra by coaction
5
Some results on K-theory
2 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Deformation quantization: background
quantization: transitioning from classical mechanics −→ quantum mechanics
3 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Deformation quantization: background
quantization: transitioning from classical mechanics −→ quantum mechanics
classical mechanics
system determined by knowing positions and momenta at given time
3 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Deformation quantization: background
quantization: transitioning from classical mechanics −→ quantum mechanics
classical mechanics
system determined by knowing positions and momenta at given time
positions modelled by manifold, e.g. Rd , ”configuration space”
3 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Deformation quantization: background
quantization: transitioning from classical mechanics −→ quantum mechanics
classical mechanics
system determined by knowing positions and momenta at given time
positions modelled by manifold, e.g. Rd , ”configuration space”
momenta modelled by cotangent vectors
3 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Deformation quantization: background
quantization: transitioning from classical mechanics −→ quantum mechanics
classical mechanics
system determined by knowing positions and momenta at given time
positions modelled by manifold, e.g. Rd , ”configuration space”
momenta modelled by cotangent vectors
altogether a cotangent bundle, e.g. R2d , ”phase space”
3 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Deformation quantization: background
quantization: transitioning from classical mechanics −→ quantum mechanics
classical mechanics
system determined by knowing positions and momenta at given time
positions modelled by manifold, e.g. Rd , ”configuration space”
momenta modelled by cotangent vectors
altogether a cotangent bundle, e.g. R2d , ”phase space”
observables: functions on phase space, e.g. C ∞ (R2d )
3 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Deformation quantization: background
quantization: transitioning from classical mechanics −→ quantum mechanics
classical mechanics
system determined by knowing positions and momenta at given time
positions modelled by manifold, e.g. Rd , ”configuration space”
momenta modelled by cotangent vectors
altogether a cotangent bundle, e.g. R2d , ”phase space”
observables: functions on phase space, e.g. C ∞ (R2d )
3 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Deformation quantization: background
quantization: transitioning from classical mechanics −→ quantum mechanics
classical mechanics
system determined by knowing positions and momenta at given time
positions modelled by manifold, e.g. Rd , ”configuration space”
momenta modelled by cotangent vectors
altogether a cotangent bundle, e.g. R2d , ”phase space”
observables: functions on phase space, e.g. C ∞ (R2d )
quantum mechanics
positions and momenta may not be simultaneously observed
3 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Deformation quantization: background
quantization: transitioning from classical mechanics −→ quantum mechanics
classical mechanics
system determined by knowing positions and momenta at given time
positions modelled by manifold, e.g. Rd , ”configuration space”
momenta modelled by cotangent vectors
altogether a cotangent bundle, e.g. R2d , ”phase space”
observables: functions on phase space, e.g. C ∞ (R2d )
quantum mechanics
positions and momenta may not be simultaneously observed
observables: modelled as operators on Hilbert space
3 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Deformation quantization: background
quantization: transitioning from classical mechanics −→ quantum mechanics
classical mechanics
system determined by knowing positions and momenta at given time
positions modelled by manifold, e.g. Rd , ”configuration space”
momenta modelled by cotangent vectors
altogether a cotangent bundle, e.g. R2d , ”phase space”
observables: functions on phase space, e.g. C ∞ (R2d )
quantum mechanics
positions and momenta may not be simultaneously observed
observables: modelled as operators on Hilbert space
non-simultaneous observability encoded to non-commuting operators
3 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
quantization: classical −→ quantum
4 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
quantization: classical −→ quantum
commutative algebra of functions −→ noncommutative algebra of operators
4 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
quantization: classical −→ quantum
commutative algebra of functions −→ noncommutative algebra of operators
Classical: e.g. C ∞ (R2d ),
coordinate functions p1 , . . . , pd , q1 , . . . , qd
4 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
quantization: classical −→ quantum
commutative algebra of functions −→ noncommutative algebra of operators
Classical: e.g. C ∞ (R2d ),
coordinate functions p1 , . . . , pd , q1 , . . . , qd
Quantum: e.g. self-adjoint (unb.) operators P1 , . . . , Pd , Q1 , . . . , Qd such that
[Pj , Qj ] = i~I
4 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
quantization: classical −→ quantum
commutative algebra of functions −→ noncommutative algebra of operators
Classical: e.g. C ∞ (R2d ),
coordinate functions p1 , . . . , pd , q1 , . . . , qd
Quantum: e.g. self-adjoint (unb.) operators P1 , . . . , Pd , Q1 , . . . , Qd such that
[Pj , Qj ] = i~I
Seek an association: ”functions” −→ ”operators” such that
pj 7→ Pj ,
qj 7→ Qj
4 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Weyl quantization
The function (p, q) 7→ e i(x·p+y ·q) should correspond to the operator e i(x·P+y ·Q) .
5 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Weyl quantization
The function (p, q) 7→ e i(x·p+y ·q) should correspond to the operator e i(x·P+y ·Q) . By
expressing a function f using its Fourier transform b
f,
Z
b
f (x, y )e i(x·p+y ·q) dxdy
f (p, q) =
R2d
5 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Weyl quantization
The function (p, q) 7→ e i(x·p+y ·q) should correspond to the operator e i(x·P+y ·Q) . By
expressing a function f using its Fourier transform b
f,
Z
b
f (x, y )e i(x·p+y ·q) dxdy
f (p, q) =
R2d
we get a correspondence: function f −→ operator Lf where
Z
b
Lf =
f (x, y )e i(x·P+y ·Q) dxdy .
R2d
5 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Weyl quantization
The function (p, q) 7→ e i(x·p+y ·q) should correspond to the operator e i(x·P+y ·Q) . By
expressing a function f using its Fourier transform b
f,
Z
b
f (x, y )e i(x·p+y ·q) dxdy
f (p, q) =
R2d
we get a correspondence: function f −→ operator Lf where
Z
b
Lf =
f (x, y )e i(x·P+y ·Q) dxdy .
R2d
Interesting note: Lf Lg = Lbf ∗ω gb where b
f ∗ω gb is a convolution product twisted by
iJa·b
cocycle ω(a, b) = e
, where J is standard symplectic matrix of R2d multiplied by ~.
5 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Let V = R2d , and linear map J : V 0 −→ V s.t. J t = −J, i.e. real skew-symmetric
matrix (equivalent to a transl. inv. Poisson bracket {·, ·} on V ).
6 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Let V = R2d , and linear map J : V 0 −→ V s.t. J t = −J, i.e. real skew-symmetric
matrix (equivalent to a transl. inv. Poisson bracket {·, ·} on V ).
Then the twisted convolution product can be written (at least formally)
Z Z
(f ×J g )(y ) =
f (y + Jp)g (y + v )e 2πi p·v dv dp.
V0
V
6 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Let V = R2d , and linear map J : V 0 −→ V s.t. J t = −J, i.e. real skew-symmetric
matrix (equivalent to a transl. inv. Poisson bracket {·, ·} on V ).
Then the twisted convolution product can be written (at least formally)
Z Z
(f ×J g )(y ) =
f (y + Jp)g (y + v )e 2πi p·v dv dp.
V0
V
Moreover ×J is a new associative product for the functions C0 (V ). We get a
”deformed” function algebra C0 (V )J .
C0 (V ) −→ C0 (V )J
6 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Let V = R2d , and linear map J : V 0 −→ V s.t. J t = −J, i.e. real skew-symmetric
matrix (equivalent to a transl. inv. Poisson bracket {·, ·} on V ).
Then the twisted convolution product can be written (at least formally)
Z Z
(f ×J g )(y ) =
f (y + Jp)g (y + v )e 2πi p·v dv dp.
V0
V
Moreover ×J is a new associative product for the functions C0 (V ). We get a
”deformed” function algebra C0 (V )J .
C0 (V ) −→ C0 (V )J
Rieffel takes this further and builds a framework for deforming more general
C*-algebras equipped with an Rd -action
A −→ AJ .
6 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Let V = R2d , and linear map J : V 0 −→ V s.t. J t = −J, i.e. real skew-symmetric
matrix (equivalent to a transl. inv. Poisson bracket {·, ·} on V ).
Then the twisted convolution product can be written (at least formally)
Z Z
(f ×J g )(y ) =
f (y + Jp)g (y + v )e 2πi p·v dv dp.
V0
V
Moreover ×J is a new associative product for the functions C0 (V ). We get a
”deformed” function algebra C0 (V )J .
C0 (V ) −→ C0 (V )J
Rieffel takes this further and builds a framework for deforming more general
C*-algebras equipped with an Rd -action
A −→ AJ .
Continuous field of C*-algebras {A~J }~∈[0,1]
6 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Deformation of C*-algebra by action of Rd
Objective: Given C*-algebra A with action α : Rd −→ Aut(A), skew-symmetric
J ∈ Md (R),
−−−−→ obtain ”Deformed” C*-algebra AJ with product ×J .
7 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Deformation of C*-algebra by action of Rd
Objective: Given C*-algebra A with action α : Rd −→ Aut(A), skew-symmetric
J ∈ Md (R),
−−−−→ obtain ”Deformed” C*-algebra AJ with product ×J .
Let A∞ ⊆ A be dense *-subalgebra of smooth elements for the action α. Define the
deformed product ×J on A∞ as
Z
a ×J b =
αJx (a)αy (b)e 2πihx,y i dx dy
Rd ×Rd
where integral is understood in the oscillatory sense.
A∞ with the product ×J is a Frechet *-algebra, denoted A∞
J .
7 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Deformation of C*-algebra by action of Rd
Objective: Given C*-algebra A with action α : Rd −→ Aut(A), skew-symmetric
J ∈ Md (R),
−−−−→ obtain ”Deformed” C*-algebra AJ with product ×J .
Let A∞ ⊆ A be dense *-subalgebra of smooth elements for the action α. Define the
deformed product ×J on A∞ as
Z
a ×J b =
αJx (a)αy (b)e 2πihx,y i dx dy
Rd ×Rd
where integral is understood in the oscillatory sense.
A∞ with the product ×J is a Frechet *-algebra, denoted A∞
J .
Deformed C*-algebra AJ is obtained as a C*-completion of A∞
J via a representation
∞
2
d
πJ : AJ −→ L(L (R ) ⊗ A)
7 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Deformation of C*-algebra by action of abelian G
Kasprzak generalizes Rieffel deformation to allow action by abelian group G .
Objective: Given
C*-algebra A
action α : G −→ Aut(A) of abelian group G
b ×G
b −→ T, ψ(γ1 , γ2 + γ3 )ψ(γ2 , γ3 ) = ψ(γ1 + γ2 , γ3 )ψ(γ1 , γ2 )
2-cocycle ψ : G
deform the algebra A to an algebra Aψ .
8 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Deformation of C*-algebra by action of abelian G
Kasprzak generalizes Rieffel deformation to allow action by abelian group G .
Objective: Given
C*-algebra A
action α : G −→ Aut(A) of abelian group G
b ×G
b −→ T, ψ(γ1 , γ2 + γ3 )ψ(γ2 , γ3 ) = ψ(γ1 + γ2 , γ3 )ψ(γ1 , γ2 )
2-cocycle ψ : G
deform the algebra A to an algebra Aψ .
Idea: use crossed product A oα G and define Aψ as a fixed-point subalgebra w.r.t. a
twisted dual action α
bψ .
8 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Deformation of C*-algebra by action of abelian G
Kasprzak generalizes Rieffel deformation to allow action by abelian group G .
Objective: Given
C*-algebra A
action α : G −→ Aut(A) of abelian group G
b ×G
b −→ T, ψ(γ1 , γ2 + γ3 )ψ(γ2 , γ3 ) = ψ(γ1 + γ2 , γ3 )ψ(γ1 , γ2 )
2-cocycle ψ : G
deform the algebra A to an algebra Aψ .
Idea: use crossed product A oα G and define Aψ as a fixed-point subalgebra w.r.t. a
twisted dual action α
bψ .
By Landstad duality, A can be characterized as A = M(A oα G )αb where
b −→ Aut(A oα G ) is the dual action.
α
b:G
8 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Deformation of C*-algebra by action of abelian G
b ), ψγ (σ) = ψ(γ, σ). Define unitaries
Propose a twisted dual action α
bψ : let ψγ ∈ Cb (G
Uγ = λ(ψγ ) and set
α
bγψ := Uγ∗ α
bγ Uγ .
9 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Deformation of C*-algebra by action of abelian G
b ), ψγ (σ) = ψ(γ, σ). Define unitaries
Propose a twisted dual action α
bψ : let ψγ ∈ Cb (G
Uγ = λ(ψγ ) and set
α
bγψ := Uγ∗ α
bγ Uγ .
ψ
Get the deformed C*-algebra as Aψ := M(A oα G )αb .
9 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Deformation of C*-algebra by action of abelian G
b ), ψγ (σ) = ψ(γ, σ). Define unitaries
Propose a twisted dual action α
bψ : let ψγ ∈ Cb (G
Uγ = λ(ψγ ) and set
α
bγψ := Uγ∗ α
bγ Uγ .
ψ
Get the deformed C*-algebra as Aψ := M(A oα G )αb .
Rieffel’s AJ is obtained by G = Rd and ψ(x, y ) = e 2πihx,Jy i .
9 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Cocycle deformation - by coactions
Objective: Given
C*-algebra A
coaction δ : A −→ M(A ⊗ Cr∗ (G )) of locally compact group G
2-cocycle ω : G × G −→ T
deform the algebra A to an algebra Aω .
10 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Cocycle deformation - by coactions
Objective: Given
C*-algebra A
coaction δ : A −→ M(A ⊗ Cr∗ (G )) of locally compact group G
2-cocycle ω : G × G −→ T
deform the algebra A to an algebra Aω .
b )δb, where
Landstad-Quigg-Vaes duality: A can be characterized as A = M(A oδ G
b ) is the dual action.
δb = Ad(1 ⊗ ρ) : G −→ Aut(A oδ G
10 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Cocycle deformation - by coactions
Objective: Given
C*-algebra A
coaction δ : A −→ M(A ⊗ Cr∗ (G )) of locally compact group G
2-cocycle ω : G × G −→ T
deform the algebra A to an algebra Aω .
b )δb, where
Landstad-Quigg-Vaes duality: A can be characterized as A = M(A oδ G
b ) is the dual action.
δb = Ad(1 ⊗ ρ) : G −→ Aut(A oδ G
Again propose a twisted dual action δbgω := Ad(1 ⊗ ρωge ), where ρωge = ω
e (·, g )ρg .
This makes sense if ω is assumed to be continuous.
b )δbω
Define the deformed algebra Aω := M(A oδ G
10 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
But what about Borel cocycles? With ω Borel, the above δbω is not immediately
well-defined.
11 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
But what about Borel cocycles? With ω Borel, the above δbω is not immediately
well-defined. Instead look at Takesaki-Takai duality isomorphism
∗
b ⊗ K) ⊂ M(A ⊗ K ⊗ K), η(x) = W23 δ(x)W
b
b −→ M(A oδ G
η : A oδ G
23
11 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
But what about Borel cocycles? With ω Borel, the above δbω is not immediately
well-defined. Instead look at Takesaki-Takai duality isomorphism
∗
b ⊗ K) ⊂ M(A ⊗ K ⊗ K), η(x) = W23 δ(x)W
b
b −→ M(A oδ G
η : A oδ G
23
from which one recovers A as
b , ϕ ∈ K∗ } ⊂ M(A ⊗ K).
A = span{(ι ⊗ ι ⊗ ϕ)η(x) : x ∈ A oδ G
11 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
But what about Borel cocycles? With ω Borel, the above δbω is not immediately
well-defined. Instead look at Takesaki-Takai duality isomorphism
∗
b ⊗ K) ⊂ M(A ⊗ K ⊗ K), η(x) = W23 δ(x)W
b
b −→ M(A oδ G
η : A oδ G
23
from which one recovers A as
b , ϕ ∈ K∗ } ⊂ M(A ⊗ K).
A = span{(ι ⊗ ι ⊗ ϕ)η(x) : x ∈ A oδ G
Propose a twisted version
b −→ M(A ⊗ K ⊗ K), η ω (x) = W23 ω
b ω∗ W ∗
η ω : A oδ G
e23 δ(x)e
23 23
which is well-defined with ω Borel,
11 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
But what about Borel cocycles? With ω Borel, the above δbω is not immediately
well-defined. Instead look at Takesaki-Takai duality isomorphism
∗
b ⊗ K) ⊂ M(A ⊗ K ⊗ K), η(x) = W23 δ(x)W
b
b −→ M(A oδ G
η : A oδ G
23
from which one recovers A as
b , ϕ ∈ K∗ } ⊂ M(A ⊗ K).
A = span{(ι ⊗ ι ⊗ ϕ)η(x) : x ∈ A oδ G
Propose a twisted version
b −→ M(A ⊗ K ⊗ K), η ω (x) = W23 ω
b ω∗ W ∗
η ω : A oδ G
e23 δ(x)e
23 23
which is well-defined with ω Borel, and then define/recover Aω as
b , ϕ ∈ K∗ } ⊂ M(A ⊗ K).
Aω := span{(ι ⊗ ι ⊗ ϕ)η ω (x) : x ∈ A oδ G
11 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Some properties of Aω
12 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Some properties of Aω
exterior equivalent coactions δ1 ∼ δ2 =⇒ Aδ1 ,ω ⊗ K ∼
= Aδ2 ,ω ⊗ K
12 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Some properties of Aω
exterior equivalent coactions δ1 ∼ δ2 =⇒ Aδ1 ,ω ⊗ K ∼
= Aδ2 ,ω ⊗ K
(Aω1 )ω2 = Aω1 ω2
12 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Some properties of Aω
exterior equivalent coactions δ1 ∼ δ2 =⇒ Aδ1 ,ω ⊗ K ∼
= Aδ2 ,ω ⊗ K
(Aω1 )ω2 = Aω1 ω2
cohomologous 2-cocycles ω1 ∼ ω2 =⇒ Aω1 ∼
= Aω2
12 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Some properties of Aω
exterior equivalent coactions δ1 ∼ δ2 =⇒ Aδ1 ,ω ⊗ K ∼
= Aδ2 ,ω ⊗ K
(Aω1 )ω2 = Aω1 ω2
cohomologous 2-cocycles ω1 ∼ ω2 =⇒ Aω1 ∼
= Aω2
ω
∗
Aω carries a coaction of G , δ (x) = W23 (x ⊗ 1)W23
12 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Some properties of Aω
exterior equivalent coactions δ1 ∼ δ2 =⇒ Aδ1 ,ω ⊗ K ∼
= Aδ2 ,ω ⊗ K
(Aω1 )ω2 = Aω1 ω2
cohomologous 2-cocycles ω1 ∼ ω2 =⇒ Aω1 ∼
= Aω2
ω
∗
Aω carries a coaction of G , δ (x) = W23 (x ⊗ 1)W23
12 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Some properties of Aω
exterior equivalent coactions δ1 ∼ δ2 =⇒ Aδ1 ,ω ⊗ K ∼
= Aδ2 ,ω ⊗ K
(Aω1 )ω2 = Aω1 ω2
cohomologous 2-cocycles ω1 ∼ ω2 =⇒ Aω1 ∼
= Aω2
ω
∗
Aω carries a coaction of G , δ (x) = W23 (x ⊗ 1)W23
Relation to twisted crossed product
12 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Some properties of Aω
exterior equivalent coactions δ1 ∼ δ2 =⇒ Aδ1 ,ω ⊗ K ∼
= Aδ2 ,ω ⊗ K
(Aω1 )ω2 = Aω1 ω2
cohomologous 2-cocycles ω1 ∼ ω2 =⇒ Aω1 ∼
= Aω2
ω
∗
Aω carries a coaction of G , δ (x) = W23 (x ⊗ 1)W23
Relation to twisted crossed product
b , G , δ,
b ω),
Considering twisted dynamical system (A oδ G
12 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
Some properties of Aω
exterior equivalent coactions δ1 ∼ δ2 =⇒ Aδ1 ,ω ⊗ K ∼
= Aδ2 ,ω ⊗ K
(Aω1 )ω2 = Aω1 ω2
cohomologous 2-cocycles ω1 ∼ ω2 =⇒ Aω1 ∼
= Aω2
ω
∗
Aω carries a coaction of G , δ (x) = W23 (x ⊗ 1)W23
Relation to twisted crossed product
b , G , δ,
b ω),
Considering twisted dynamical system (A oδ G
b ob G .
Aω ⊗ K ∼
= A oδ G
δ,ω
12 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
K-theory: AJ and Aψ
Let AJ denote the Rieffel deformation of a C*-algebra A, with action
α : Rd −→ Aut(A) and skew-symmetric J ∈ Md (R).
Rieffel showed
K∗ (AJ ) ∼
= K∗ (A),
∗ = 0, 1
13 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
K-theory: AJ and Aψ
Let AJ denote the Rieffel deformation of a C*-algebra A, with action
α : Rd −→ Aut(A) and skew-symmetric J ∈ Md (R).
Rieffel showed
K∗ (AJ ) ∼
= K∗ (A),
∗ = 0, 1
using strong Morita equivalence between AJ and A ⊗ K ⊗ C0 (ker J) oβ Rd and the
Connes-Thom iso in K-theory.
13 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
K-theory: AJ and Aψ
Let AJ denote the Rieffel deformation of a C*-algebra A, with action
α : Rd −→ Aut(A) and skew-symmetric J ∈ Md (R).
Rieffel showed
K∗ (AJ ) ∼
= K∗ (A),
∗ = 0, 1
using strong Morita equivalence between AJ and A ⊗ K ⊗ C0 (ker J) oβ Rd and the
Connes-Thom iso in K-theory.
Let Aψ denote the deformation of A with action α : Rd −→ Aut(A) using 2-cocycle
ψ(x, y ) = e 2πihx,Jy i .
13 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
K-theory: AJ and Aψ
Let AJ denote the Rieffel deformation of a C*-algebra A, with action
α : Rd −→ Aut(A) and skew-symmetric J ∈ Md (R).
Rieffel showed
K∗ (AJ ) ∼
= K∗ (A),
∗ = 0, 1
using strong Morita equivalence between AJ and A ⊗ K ⊗ C0 (ker J) oβ Rd and the
Connes-Thom iso in K-theory.
Let Aψ denote the deformation of A with action α : Rd −→ Aut(A) using 2-cocycle
ψ(x, y ) = e 2πihx,Jy i .
Using the (Landstad duality) isomorphism Aψ oαψ Rd ∼
= A oα Rd and Connes-Thom
iso, one gets
K∗ (Aψ ) ∼
= K∗ (A), ∗ = 0, 1
13 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
K-theory: Aω
14 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
K-theory: Aω
Theorem (Echterhoff-Lück-Phillips-Walters)
Assume G satisfies B-C conj. with coeff. Then for two homotopic Borel 2-cocycles ω0 ,
ω1 ,
K∗ (Cr∗ (G , ω0 )) ∼
= K∗ (Cr∗ (G , ω1 ))
14 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
K-theory: Aω
Theorem (Echterhoff-Lück-Phillips-Walters)
Assume G satisfies B-C conj. with coeff. Then for two homotopic Borel 2-cocycles ω0 ,
ω1 ,
K∗ (Cr∗ (G , ω0 )) ∼
= K∗ (Cr∗ (G , ω1 ))
is generalized from considering twisted group C*-algebras to twisted crossed products
Theorem
Assume G satisfies the B-C conj. with coeff. Then for any action α of G on a
C*-algebra B and any two homotopic Borel 2-cocycles ω0 , ω1 ,
K∗ (B oα,ω0 G ) ∼
= K∗ (B oα,ω1 G ).
14 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
K-theory: Aω
Sketch of proof:
15 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
K-theory: Aω
Sketch of proof:
1. Start with Packer-Raeburn stabilization trick
K ⊗ (B oα,ω G ) ∼
= (K ⊗ B) oAdρω ⊗α G .
Let Ω be homotopy between cocycles ω0 and ω1 , AdρΩ : G −→ Aut(C ([0, 1]) ⊗ K),
ωt
AdρΩ
g (f )(t) = Adρg (f (t))
15 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
K-theory: Aω
Sketch of proof:
1. Start with Packer-Raeburn stabilization trick
K ⊗ (B oα,ω G ) ∼
= (K ⊗ B) oAdρω ⊗α G .
Let Ω be homotopy between cocycles ω0 and ω1 , AdρΩ : G −→ Aut(C ([0, 1]) ⊗ K),
ωt
AdρΩ
g (f )(t) = Adρg (f (t))
2. Show that
(evt o G )∗ : K∗ (C ([0, 1]) ⊗ K ⊗ B oAdρΩ ⊗α G ) −→ K∗ ((K ⊗ B) oAdρωt ⊗α G )
is isomorphism. It suffices ([ELP]) to show this for all compact subgroups H ⊂ G .
15 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
K-theory: Aω
Sketch of proof:
1. Start with Packer-Raeburn stabilization trick
K ⊗ (B oα,ω G ) ∼
= (K ⊗ B) oAdρω ⊗α G .
Let Ω be homotopy between cocycles ω0 and ω1 , AdρΩ : G −→ Aut(C ([0, 1]) ⊗ K),
ωt
AdρΩ
g (f )(t) = Adρg (f (t))
2. Show that
(evt o G )∗ : K∗ (C ([0, 1]) ⊗ K ⊗ B oAdρΩ ⊗α G ) −→ K∗ ((K ⊗ B) oAdρωt ⊗α G )
is isomorphism. It suffices ([ELP]) to show this for all compact subgroups H ⊂ G .
3. Restricted to compact H ⊂ G , the actions are exterior equivalent, and the resulting
isomorphism of crossed products coincides with the evaluation map
15 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
K-theory: Aω
Theorem
Assume G satisfies the Baum-Connes conj. with coeff. Then for any coaction δ of G
on a C*-algebra A and any two homotopic Borel 2-cocycles ω0 , ω1 ∈ Z 2 (G ; T), we have
K∗ (Aω0 ) ∼
= K∗ (Aω1 )
16 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
K-theory: Aω
Theorem
Assume G satisfies the Baum-Connes conj. with coeff. Then for any coaction δ of G
on a C*-algebra A and any two homotopic Borel 2-cocycles ω0 , ω1 ∈ Z 2 (G ; T), we have
K∗ (Aω0 ) ∼
= K∗ (Aω1 )
Proof.
K∗ (Aω0 ) ∼
= K∗ (Aω0 ⊗ K)
16 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
K-theory: Aω
Theorem
Assume G satisfies the Baum-Connes conj. with coeff. Then for any coaction δ of G
on a C*-algebra A and any two homotopic Borel 2-cocycles ω0 , ω1 ∈ Z 2 (G ; T), we have
K∗ (Aω0 ) ∼
= K∗ (Aω1 )
Proof.
K∗ (Aω0 ) ∼
= K∗ (Aω0 ⊗ K)
∼
b ob
= K∗ (A oδ G
δ,ω0
G)
16 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
K-theory: Aω
Theorem
Assume G satisfies the Baum-Connes conj. with coeff. Then for any coaction δ of G
on a C*-algebra A and any two homotopic Borel 2-cocycles ω0 , ω1 ∈ Z 2 (G ; T), we have
K∗ (Aω0 ) ∼
= K∗ (Aω1 )
Proof.
K∗ (Aω0 ) ∼
= K∗ (Aω0 ⊗ K)
∼
b ob
= K∗ (A oδ G
δ,ω0
G)
∼
b ob G )
= K∗ (A oδ G
δ,ω1
16 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
K-theory: Aω
Theorem
Assume G satisfies the Baum-Connes conj. with coeff. Then for any coaction δ of G
on a C*-algebra A and any two homotopic Borel 2-cocycles ω0 , ω1 ∈ Z 2 (G ; T), we have
K∗ (Aω0 ) ∼
= K∗ (Aω1 )
Proof.
K∗ (Aω0 ) ∼
= K∗ (Aω0 ⊗ K)
∼
b ob
= K∗ (A oδ G
δ,ω0
G)
∼
b ob G )
= K∗ (A oδ G
δ,ω1
∼
= K∗ (Aω ⊗ K)
1
16 / 16
Deformation quantization: background
Rieffel deformation
Rieffel deformation via crossed products
Deformation of C*-algebra by coaction
Some results on K-theory
K-theory: Aω
Theorem
Assume G satisfies the Baum-Connes conj. with coeff. Then for any coaction δ of G
on a C*-algebra A and any two homotopic Borel 2-cocycles ω0 , ω1 ∈ Z 2 (G ; T), we have
K∗ (Aω0 ) ∼
= K∗ (Aω1 )
Proof.
K∗ (Aω0 ) ∼
= K∗ (Aω0 ⊗ K)
∼
b ob
= K∗ (A oδ G
δ,ω0
G)
∼
b ob G )
= K∗ (A oδ G
δ,ω1
∼
= K∗ (Aω ⊗ K)
1
∼
= K∗ (Aω1 ).
16 / 16