Hilbert C*-modules in harmonicanalysisNordfjordeid 2019
Ulrik Enstad 4th July 2019
Hilbert C*-modules
Definition
Let A be a unital C*-algebra. A left Hilbert A-module is a left A-moduleE together with a map •〈·, ·〉 : E × E → A such that:
1 For all ξ, η, γ ∈ E and a,b ∈ A we have that
•〈aξ + bη, γ〉 = a•〈ξ, γ〉+ b•〈η, γ〉 .
2 For all ξ, η ∈ E we have that
•〈ξ, η〉∗ = •〈η, ξ〉 .
3 For all ξ ∈ E we have that •〈ξ, ξ〉 ≥ 0, and •〈ξ, ξ〉 = 0 impliesξ = 0.
4 E is complete with respect to the norm ‖ξ‖E := ‖•〈ξ, ξ〉 ‖1/2A .
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 1 / 14
Hilbert C*-modulesDefinition
Let A be a unital C*-algebra. A left Hilbert A-module is a left A-moduleE together with a map •〈·, ·〉 : E × E → A such that:
1 For all ξ, η, γ ∈ E and a,b ∈ A we have that
•〈aξ + bη, γ〉 = a•〈ξ, γ〉+ b•〈η, γ〉 .
2 For all ξ, η ∈ E we have that
•〈ξ, η〉∗ = •〈η, ξ〉 .
3 For all ξ ∈ E we have that •〈ξ, ξ〉 ≥ 0, and •〈ξ, ξ〉 = 0 impliesξ = 0.
4 E is complete with respect to the norm ‖ξ‖E := ‖•〈ξ, ξ〉 ‖1/2A .
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 1 / 14
Hilbert C*-modulesDefinition
Let A be a unital C*-algebra. A left Hilbert A-module is a left A-moduleE together with a map •〈·, ·〉 : E × E → A such that:
1 For all ξ, η, γ ∈ E and a,b ∈ A we have that
•〈aξ + bη, γ〉 = a•〈ξ, γ〉+ b•〈η, γ〉 .
2 For all ξ, η ∈ E we have that
•〈ξ, η〉∗ = •〈η, ξ〉 .
3 For all ξ ∈ E we have that •〈ξ, ξ〉 ≥ 0, and •〈ξ, ξ〉 = 0 impliesξ = 0.
4 E is complete with respect to the norm ‖ξ‖E := ‖•〈ξ, ξ〉 ‖1/2A .
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 1 / 14
Hilbert C*-modulesDefinition
Let A be a unital C*-algebra. A left Hilbert A-module is a left A-moduleE together with a map •〈·, ·〉 : E × E → A such that:
1 For all ξ, η, γ ∈ E and a,b ∈ A we have that
•〈aξ + bη, γ〉 = a•〈ξ, γ〉+ b•〈η, γ〉 .
2 For all ξ, η ∈ E we have that
•〈ξ, η〉∗ = •〈η, ξ〉 .
3 For all ξ ∈ E we have that •〈ξ, ξ〉 ≥ 0, and •〈ξ, ξ〉 = 0 impliesξ = 0.
4 E is complete with respect to the norm ‖ξ‖E := ‖•〈ξ, ξ〉 ‖1/2A .
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 1 / 14
Hilbert C*-modulesDefinition
Let A be a unital C*-algebra. A left Hilbert A-module is a left A-moduleE together with a map •〈·, ·〉 : E × E → A such that:
1 For all ξ, η, γ ∈ E and a,b ∈ A we have that
•〈aξ + bη, γ〉 = a•〈ξ, γ〉+ b•〈η, γ〉 .
2 For all ξ, η ∈ E we have that
•〈ξ, η〉∗ = •〈η, ξ〉 .
3 For all ξ ∈ E we have that •〈ξ, ξ〉 ≥ 0, and •〈ξ, ξ〉 = 0 impliesξ = 0.
4 E is complete with respect to the norm ‖ξ‖E := ‖•〈ξ, ξ〉 ‖1/2A .
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 1 / 14
Hilbert C*-modulesDefinition
Let A be a unital C*-algebra. A left Hilbert A-module is a left A-moduleE together with a map •〈·, ·〉 : E × E → A such that:
1 For all ξ, η, γ ∈ E and a,b ∈ A we have that
•〈aξ + bη, γ〉 = a•〈ξ, γ〉+ b•〈η, γ〉 .
2 For all ξ, η ∈ E we have that
•〈ξ, η〉∗ = •〈η, ξ〉 .
3 For all ξ ∈ E we have that •〈ξ, ξ〉 ≥ 0, and •〈ξ, ξ〉 = 0 impliesξ = 0.
4 E is complete with respect to the norm ‖ξ‖E := ‖•〈ξ, ξ〉 ‖1/2A .Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 1 / 14
Examples of Hilbert C*-modules
The left A-module Ak becomes a left Hilbert A-module with theinner product
•〈(a1, . . . , ak ), (b1, . . . , bk )〉 =k∑
j=1
ajb∗j .
For a countable index set J, define
`2(J ,A) = {(aj)j∈J ⊆ A :∑j∈J
aja∗j converges in A}.
This becomes a left Hilbert A-module with respect to
•⟨(aj), (bj)
⟩=∑j∈J
ajb∗j .
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 2 / 14
Examples of Hilbert C*-modulesThe left A-module Ak becomes a left Hilbert A-module with theinner product
•〈(a1, . . . , ak ), (b1, . . . , bk )〉 =k∑
j=1
ajb∗j .
For a countable index set J, define
`2(J ,A) = {(aj)j∈J ⊆ A :∑j∈J
aja∗j converges in A}.
This becomes a left Hilbert A-module with respect to
•⟨(aj), (bj)
⟩=∑j∈J
ajb∗j .
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 2 / 14
Examples of Hilbert C*-modulesThe left A-module Ak becomes a left Hilbert A-module with theinner product
•〈(a1, . . . , ak ), (b1, . . . , bk )〉 =k∑
j=1
ajb∗j .
For a countable index set J, define
`2(J ,A) = {(aj)j∈J ⊆ A :∑j∈J
aja∗j converges in A}.
This becomes a left Hilbert A-module with respect to
•⟨(aj), (bj)
⟩=∑j∈J
ajb∗j .
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 2 / 14
Sections of vector bundles
If A is commutative, then A ∼= C(X ) for a compact Hausdorffspace X .For a Hermitian vector bundle E → X , denote by Γ(E) the set ofcontinuous sections of E .Γ(E) becomes a left Hilbert C(X )-module with
(f · s)(x) = f (x)s(x)
•〈s, t〉 (x) = 〈s(x), t(x)〉xfor s, t ∈ Γ(E), f ∈ C(X ).
Theorem (Serre–Swan)
If E is a finitely generated Hilbert C(X )-module, then there exists aunique Hermitian vector bundle E → X such that E ∼= Γ(E).
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 3 / 14
Sections of vector bundlesIf A is commutative, then A ∼= C(X ) for a compact Hausdorffspace X .
For a Hermitian vector bundle E → X , denote by Γ(E) the set ofcontinuous sections of E .Γ(E) becomes a left Hilbert C(X )-module with
(f · s)(x) = f (x)s(x)
•〈s, t〉 (x) = 〈s(x), t(x)〉xfor s, t ∈ Γ(E), f ∈ C(X ).
Theorem (Serre–Swan)
If E is a finitely generated Hilbert C(X )-module, then there exists aunique Hermitian vector bundle E → X such that E ∼= Γ(E).
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 3 / 14
Sections of vector bundlesIf A is commutative, then A ∼= C(X ) for a compact Hausdorffspace X .For a Hermitian vector bundle E → X , denote by Γ(E) the set ofcontinuous sections of E .
Γ(E) becomes a left Hilbert C(X )-module with
(f · s)(x) = f (x)s(x)
•〈s, t〉 (x) = 〈s(x), t(x)〉xfor s, t ∈ Γ(E), f ∈ C(X ).
Theorem (Serre–Swan)
If E is a finitely generated Hilbert C(X )-module, then there exists aunique Hermitian vector bundle E → X such that E ∼= Γ(E).
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 3 / 14
Sections of vector bundlesIf A is commutative, then A ∼= C(X ) for a compact Hausdorffspace X .For a Hermitian vector bundle E → X , denote by Γ(E) the set ofcontinuous sections of E .Γ(E) becomes a left Hilbert C(X )-module with
(f · s)(x) = f (x)s(x)
•〈s, t〉 (x) = 〈s(x), t(x)〉xfor s, t ∈ Γ(E), f ∈ C(X ).
Theorem (Serre–Swan)
If E is a finitely generated Hilbert C(X )-module, then there exists aunique Hermitian vector bundle E → X such that E ∼= Γ(E).
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 3 / 14
Sections of vector bundlesIf A is commutative, then A ∼= C(X ) for a compact Hausdorffspace X .For a Hermitian vector bundle E → X , denote by Γ(E) the set ofcontinuous sections of E .Γ(E) becomes a left Hilbert C(X )-module with
(f · s)(x) = f (x)s(x)
•〈s, t〉 (x) = 〈s(x), t(x)〉xfor s, t ∈ Γ(E), f ∈ C(X ).
Theorem (Serre–Swan)
If E is a finitely generated Hilbert C(X )-module, then there exists aunique Hermitian vector bundle E → X such that E ∼= Γ(E).
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 3 / 14
Orthonormal bases in HilbertC*-modules?
Any separable Hilbert space H admits a countable orthonormalbasis.A left Hilbert A-module E is called countably generated if thereexists a countable set S ⊆ E such that∑
j;finite
ajsj : aj ∈ A, sj ∈ E
= E.
Does any countably generated Hilbert C∗-module admit acountable “orthonormal basis”?No!
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 4 / 14
Orthonormal bases in HilbertC*-modules?
Any separable Hilbert space H admits a countable orthonormalbasis.
A left Hilbert A-module E is called countably generated if thereexists a countable set S ⊆ E such that∑
j;finite
ajsj : aj ∈ A, sj ∈ E
= E.
Does any countably generated Hilbert C∗-module admit acountable “orthonormal basis”?No!
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 4 / 14
Orthonormal bases in HilbertC*-modules?
Any separable Hilbert space H admits a countable orthonormalbasis.
A left Hilbert A-module E is called countably generated if thereexists a countable set S ⊆ E such that∑
j;finite
ajsj : aj ∈ A, sj ∈ E
= E.
Does any countably generated Hilbert C∗-module admit acountable “orthonormal basis”?No!
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 4 / 14
Orthonormal bases in HilbertC*-modules?
Any separable Hilbert space H admits a countable orthonormalbasis.A left Hilbert A-module E is called countably generated if thereexists a countable set S ⊆ E such that∑
j;finite
ajsj : aj ∈ A, sj ∈ E
= E.
Does any countably generated Hilbert C∗-module admit acountable “orthonormal basis”?No!
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 4 / 14
Orthonormal bases in HilbertC*-modules?
Any separable Hilbert space H admits a countable orthonormalbasis.A left Hilbert A-module E is called countably generated if thereexists a countable set S ⊆ E such that∑
j;finite
ajsj : aj ∈ A, sj ∈ E
= E.
Does any countably generated Hilbert C∗-module admit acountable “orthonormal basis”?
No!
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 4 / 14
Orthonormal bases in HilbertC*-modules?
Any separable Hilbert space H admits a countable orthonormalbasis.A left Hilbert A-module E is called countably generated if thereexists a countable set S ⊆ E such that∑
j;finite
ajsj : aj ∈ A, sj ∈ E
= E.
Does any countably generated Hilbert C∗-module admit acountable “orthonormal basis”?No!
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 4 / 14
Frames
Definition (Frank–Larson)
Let J be a countable index set. A sequence (ηj)j∈J ⊆ E is called aframe if there exists C,D >0 such that
C•〈ξ, ξ〉 ≤∑j∈J•⟨ξ, ηj
⟩•⟨ξ, ηj
⟩∗ ≤ D•〈ξ, ξ〉
for all ξ ∈ E .
The frame is normalized tight if one can choose C = D = 1.
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 5 / 14
FramesDefinition (Frank–Larson)
Let J be a countable index set. A sequence (ηj)j∈J ⊆ E is called aframe if there exists C,D >0 such that
C•〈ξ, ξ〉 ≤∑j∈J•⟨ξ, ηj
⟩•⟨ξ, ηj
⟩∗ ≤ D•〈ξ, ξ〉
for all ξ ∈ E .
The frame is normalized tight if one can choose C = D = 1.
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 5 / 14
FramesDefinition (Frank–Larson)
Let J be a countable index set. A sequence (ηj)j∈J ⊆ E is called aframe if there exists C,D >0 such that
C•〈ξ, ξ〉 ≤∑j∈J•⟨ξ, ηj
⟩•⟨ξ, ηj
⟩∗ ≤ D•〈ξ, ξ〉
for all ξ ∈ E .
The frame is normalized tight if one can choose C = D = 1.
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 5 / 14
Dual frames
Theorem (Frank–Larson)
If (ηj)j∈J is a frame in E , there exists another frame (γj)j∈J such that
ξ =∑j∈J•⟨ξ, ηj
⟩γj =
∑j∈J•⟨xi , γj
⟩ηj
for all ξ ∈ E .
Theorem (Frank–Larson)
If E is a countably generated Hilbert A-module, then it admits a (count-able) frame.
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 6 / 14
Dual framesTheorem (Frank–Larson)
If (ηj)j∈J is a frame in E , there exists another frame (γj)j∈J such that
ξ =∑j∈J•⟨ξ, ηj
⟩γj =
∑j∈J•⟨xi , γj
⟩ηj
for all ξ ∈ E .
Theorem (Frank–Larson)
If E is a countably generated Hilbert A-module, then it admits a (count-able) frame.
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 6 / 14
Dual framesTheorem (Frank–Larson)
If (ηj)j∈J is a frame in E , there exists another frame (γj)j∈J such that
ξ =∑j∈J•⟨ξ, ηj
⟩γj =
∑j∈J•⟨xi , γj
⟩ηj
for all ξ ∈ E .
Theorem (Frank–Larson)
If E is a countably generated Hilbert A-module, then it admits a (count-able) frame.
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 6 / 14
Properties of frames
A finite set {η1, . . . , ηk} ⊆ E is a frame if and only if it is an(algebraic) generating set.If E → X is a Hermitian vector bundle over a compactHausdorff space X , a set of sections {s1, . . . , sk} ⊆ Γ(E) is aframe if and only if for every x ∈ X ,
span{s1(x), . . . , sk (x)} = Ex .
In particular, if E has rank k , a frame of k elements in Γ(E) isexactly a global trivialization of E .
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 7 / 14
Properties of framesA finite set {η1, . . . , ηk} ⊆ E is a frame if and only if it is an(algebraic) generating set.
If E → X is a Hermitian vector bundle over a compactHausdorff space X , a set of sections {s1, . . . , sk} ⊆ Γ(E) is aframe if and only if for every x ∈ X ,
span{s1(x), . . . , sk (x)} = Ex .
In particular, if E has rank k , a frame of k elements in Γ(E) isexactly a global trivialization of E .
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 7 / 14
Properties of framesA finite set {η1, . . . , ηk} ⊆ E is a frame if and only if it is an(algebraic) generating set.If E → X is a Hermitian vector bundle over a compactHausdorff space X , a set of sections {s1, . . . , sk} ⊆ Γ(E) is aframe if and only if for every x ∈ X ,
span{s1(x), . . . , sk (x)} = Ex .
In particular, if E has rank k , a frame of k elements in Γ(E) isexactly a global trivialization of E .
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 7 / 14
Properties of framesA finite set {η1, . . . , ηk} ⊆ E is a frame if and only if it is an(algebraic) generating set.If E → X is a Hermitian vector bundle over a compactHausdorff space X , a set of sections {s1, . . . , sk} ⊆ Γ(E) is aframe if and only if for every x ∈ X ,
span{s1(x), . . . , sk (x)} = Ex .
In particular, if E has rank k , a frame of k elements in Γ(E) isexactly a global trivialization of E .
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 7 / 14
Projective modules
Recall that a module E over A is called projective if there existsanother module F such that E ⊕ F ∼= Ak for some k ∈ N.Equivalently, there exists a projection P ∈ Mk (A) for some ksuch that E ∼= AkP.If (η1, . . . , ηk ) is a frame for E , then the matrix
P = (•⟨ηi , ηj
⟩)ki ,j=1
is a projection in Mk (A), and E ∼= AkP.In particular, the K -theory class of E is represented by P.If τ is a trace on A, then the induced trace τ̃ : K0(A)→ C
τ̃([E ]) =k∑
j=1
τ(•⟨ηj , ηj
⟩).
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 8 / 14
Projective modulesRecall that a module E over A is called projective if there existsanother module F such that E ⊕ F ∼= Ak for some k ∈ N.
Equivalently, there exists a projection P ∈ Mk (A) for some ksuch that E ∼= AkP.If (η1, . . . , ηk ) is a frame for E , then the matrix
P = (•⟨ηi , ηj
⟩)ki ,j=1
is a projection in Mk (A), and E ∼= AkP.In particular, the K -theory class of E is represented by P.If τ is a trace on A, then the induced trace τ̃ : K0(A)→ C
τ̃([E ]) =k∑
j=1
τ(•⟨ηj , ηj
⟩).
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 8 / 14
Projective modulesRecall that a module E over A is called projective if there existsanother module F such that E ⊕ F ∼= Ak for some k ∈ N.Equivalently, there exists a projection P ∈ Mk (A) for some ksuch that E ∼= AkP.
If (η1, . . . , ηk ) is a frame for E , then the matrix
P = (•⟨ηi , ηj
⟩)ki ,j=1
is a projection in Mk (A), and E ∼= AkP.In particular, the K -theory class of E is represented by P.If τ is a trace on A, then the induced trace τ̃ : K0(A)→ C
τ̃([E ]) =k∑
j=1
τ(•⟨ηj , ηj
⟩).
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 8 / 14
Projective modulesRecall that a module E over A is called projective if there existsanother module F such that E ⊕ F ∼= Ak for some k ∈ N.Equivalently, there exists a projection P ∈ Mk (A) for some ksuch that E ∼= AkP.If (η1, . . . , ηk ) is a frame for E , then the matrix
P = (•⟨ηi , ηj
⟩)ki ,j=1
is a projection in Mk (A), and E ∼= AkP.
In particular, the K -theory class of E is represented by P.If τ is a trace on A, then the induced trace τ̃ : K0(A)→ C
τ̃([E ]) =k∑
j=1
τ(•⟨ηj , ηj
⟩).
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 8 / 14
Projective modulesRecall that a module E over A is called projective if there existsanother module F such that E ⊕ F ∼= Ak for some k ∈ N.Equivalently, there exists a projection P ∈ Mk (A) for some ksuch that E ∼= AkP.If (η1, . . . , ηk ) is a frame for E , then the matrix
P = (•⟨ηi , ηj
⟩)ki ,j=1
is a projection in Mk (A), and E ∼= AkP.In particular, the K -theory class of E is represented by P.
If τ is a trace on A, then the induced trace τ̃ : K0(A)→ C
τ̃([E ]) =k∑
j=1
τ(•⟨ηj , ηj
⟩).
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 8 / 14
Projective modulesRecall that a module E over A is called projective if there existsanother module F such that E ⊕ F ∼= Ak for some k ∈ N.Equivalently, there exists a projection P ∈ Mk (A) for some ksuch that E ∼= AkP.If (η1, . . . , ηk ) is a frame for E , then the matrix
P = (•⟨ηi , ηj
⟩)ki ,j=1
is a projection in Mk (A), and E ∼= AkP.In particular, the K -theory class of E is represented by P.If τ is a trace on A, then the induced trace τ̃ : K0(A)→ C
τ̃([E ]) =k∑
j=1
τ(•⟨ηj , ηj
⟩).
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 8 / 14
Gabor frames
Given θ ∈ R \ {0}, define operators Tθ,M : L2(R)→ L2(R) by
Tθξ(t) = ξ(t − θ)
Mξ(t) = e2πitξ(t).
These two operators satisfy
MTθ = e2πiθTθM.
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 9 / 14
Gabor framesGiven θ ∈ R \ {0}, define operators Tθ,M : L2(R)→ L2(R) by
Tθξ(t) = ξ(t − θ)
Mξ(t) = e2πitξ(t).
These two operators satisfy
MTθ = e2πiθTθM.
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 9 / 14
Gabor framesGiven θ ∈ R \ {0}, define operators Tθ,M : L2(R)→ L2(R) by
Tθξ(t) = ξ(t − θ)
Mξ(t) = e2πitξ(t).
These two operators satisfy
MTθ = e2πiθTθM.
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 9 / 14
Problem
Given η1, . . . , ηk ∈ L2(R) and θ ∈ R \ {0}, when is the set
{MnT mθ ηj : 1 ≤ j ≤ k ,m,n ∈ Z}
an orthonormal basis for L2(R), or more generally a frame for L2(R)?
A Balian–Low Theorem (Battle)
If η ∈ S(R), then{MnT m
θ η : m,n ∈ Z}
cannot be an orthonormal basis for L2(R).
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 10 / 14
Problem
Given η1, . . . , ηk ∈ L2(R) and θ ∈ R \ {0}, when is the set
{MnT mθ ηj : 1 ≤ j ≤ k ,m,n ∈ Z}
an orthonormal basis for L2(R), or more generally a frame for L2(R)?
A Balian–Low Theorem (Battle)
If η ∈ S(R), then{MnT m
θ η : m,n ∈ Z}
cannot be an orthonormal basis for L2(R).
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 10 / 14
Problem
Given η1, . . . , ηk ∈ L2(R) and θ ∈ R \ {0}, when is the set
{MnT mθ ηj : 1 ≤ j ≤ k ,m,n ∈ Z}
an orthonormal basis for L2(R), or more generally a frame for L2(R)?
A Balian–Low Theorem (Battle)
If η ∈ S(R), then{MnT m
θ η : m,n ∈ Z}
cannot be an orthonormal basis for L2(R).
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 10 / 14
The noncommutative torus
Define
Aθ = span{MnT mθ : m,n ∈ Z} ⊆ B(L2(R)).
The C∗-algebra Aθ is called the noncommutative 2-torus withparameter θ and is the universal C∗-algebra generated by twounitaries u and v satisfying vu = e2πiθuv .Aθ is equipped with a trace τ : Aθ → C given by
τ(∑
m,nam,nMnT m
θ
)= a0,0.
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 11 / 14
The noncommutative torusDefine
Aθ = span{MnT mθ : m,n ∈ Z} ⊆ B(L2(R)).
The C∗-algebra Aθ is called the noncommutative 2-torus withparameter θ and is the universal C∗-algebra generated by twounitaries u and v satisfying vu = e2πiθuv .Aθ is equipped with a trace τ : Aθ → C given by
τ(∑
m,nam,nMnT m
θ
)= a0,0.
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 11 / 14
The noncommutative torusDefine
Aθ = span{MnT mθ : m,n ∈ Z} ⊆ B(L2(R)).
The C∗-algebra Aθ is called the noncommutative 2-torus withparameter θ and is the universal C∗-algebra generated by twounitaries u and v satisfying vu = e2πiθuv .
Aθ is equipped with a trace τ : Aθ → C given by
τ(∑
m,nam,nMnT m
θ
)= a0,0.
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 11 / 14
The noncommutative torusDefine
Aθ = span{MnT mθ : m,n ∈ Z} ⊆ B(L2(R)).
The C∗-algebra Aθ is called the noncommutative 2-torus withparameter θ and is the universal C∗-algebra generated by twounitaries u and v satisfying vu = e2πiθuv .Aθ is equipped with a trace τ : Aθ → C given by
τ(∑
m,nam,nMnT m
θ
)= a0,0.
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 11 / 14
The Heisenberg module
for a =∑
m,n am,nMnT nθ ∈ Aθ and ξ ∈ S(R), define(∑
m,namnMnT n
θ
)ξ =
∑m,n
amnMnT nθ ξ.
For ξ, η ∈ S(R), define
•〈ξ, η〉 =∑m,n〈ξ,MnT m
θ η〉MnT mθ ∈ Aθ.
We call the resulting completed left Hilbert Aθ-module aHeisenberg module and denote it by Eθ.
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 12 / 14
The Heisenberg modulefor a =
∑m,n am,nMnT n
θ ∈ Aθ and ξ ∈ S(R), define(∑m,n
amnMnT nθ
)ξ =
∑m,n
amnMnT nθ ξ.
For ξ, η ∈ S(R), define
•〈ξ, η〉 =∑m,n〈ξ,MnT m
θ η〉MnT mθ ∈ Aθ.
We call the resulting completed left Hilbert Aθ-module aHeisenberg module and denote it by Eθ.
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 12 / 14
The Heisenberg modulefor a =
∑m,n am,nMnT n
θ ∈ Aθ and ξ ∈ S(R), define(∑m,n
amnMnT nθ
)ξ =
∑m,n
amnMnT nθ ξ.
For ξ, η ∈ S(R), define
•〈ξ, η〉 =∑m,n〈ξ,MnT m
θ η〉MnT mθ ∈ Aθ.
We call the resulting completed left Hilbert Aθ-module aHeisenberg module and denote it by Eθ.
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 12 / 14
The Heisenberg modulefor a =
∑m,n am,nMnT n
θ ∈ Aθ and ξ ∈ S(R), define(∑m,n
amnMnT nθ
)ξ =
∑m,n
amnMnT nθ ξ.
For ξ, η ∈ S(R), define
•〈ξ, η〉 =∑m,n〈ξ,MnT m
θ η〉MnT mθ ∈ Aθ.
We call the resulting completed left Hilbert Aθ-module aHeisenberg module and denote it by Eθ.
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 12 / 14
Theorem (Luef)
A set {η1, . . . , ηk} of functions in S(R) is a (normalized tight) frame forEθ if and only if
{MnT mθ ηj : 1 ≤ j ≤ k ,m,n ∈ Z}
is a (normalized tight) frame for L2(R).
If (η1, . . . , ηk ) is a normalized tight frame for Eθ, then
τ̃([Eθ]) =k∑
j=1
τ(•⟨ηj , ηj
⟩) =
k∑j=1
‖ηj‖2.
If in addition {MnT mθ ηj : 1 ≤ j ≤ k ,m,n ∈ Z} is an orthonormal
basis, thenτ̃([Eθ]) = k .
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 13 / 14
Theorem (Luef)
A set {η1, . . . , ηk} of functions in S(R) is a (normalized tight) frame forEθ if and only if
{MnT mθ ηj : 1 ≤ j ≤ k ,m,n ∈ Z}
is a (normalized tight) frame for L2(R).
If (η1, . . . , ηk ) is a normalized tight frame for Eθ, then
τ̃([Eθ]) =k∑
j=1
τ(•⟨ηj , ηj
⟩) =
k∑j=1
‖ηj‖2.
If in addition {MnT mθ ηj : 1 ≤ j ≤ k ,m,n ∈ Z} is an orthonormal
basis, thenτ̃([Eθ]) = k .
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 13 / 14
Theorem (Luef)
A set {η1, . . . , ηk} of functions in S(R) is a (normalized tight) frame forEθ if and only if
{MnT mθ ηj : 1 ≤ j ≤ k ,m,n ∈ Z}
is a (normalized tight) frame for L2(R).
If (η1, . . . , ηk ) is a normalized tight frame for Eθ, then
τ̃([Eθ]) =k∑
j=1
τ(•⟨ηj , ηj
⟩) =
k∑j=1
‖ηj‖2.
If in addition {MnT mθ ηj : 1 ≤ j ≤ k ,m,n ∈ Z} is an orthonormal
basis, thenτ̃([Eθ]) = k .
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 13 / 14
Theorem (Luef)
A set {η1, . . . , ηk} of functions in S(R) is a (normalized tight) frame forEθ if and only if
{MnT mθ ηj : 1 ≤ j ≤ k ,m,n ∈ Z}
is a (normalized tight) frame for L2(R).
If (η1, . . . , ηk ) is a normalized tight frame for Eθ, then
τ̃([Eθ]) =k∑
j=1
τ(•⟨ηj , ηj
⟩) =
k∑j=1
‖ηj‖2.
If in addition {MnT mθ ηj : 1 ≤ j ≤ k ,m,n ∈ Z} is an orthonormal
basis, thenτ̃([Eθ]) = k .
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 13 / 14
Theorem (Rieffel)
Let τ̃ : K0(Aθ)→ C denote the extension of the trace τ. Then
τ̃([Eθ]) = θ.
If {MnT mθ η : m,n ∈ Z} is an orthormal basis for L2(R), then
θ = 1.Consequently Aθ ∼= C(T2).By the Serre–Swan theorem, there exists a unique vectorbundle E → T2 such that
Eθ ∼= Γ(E).
One can show that this is a line bundle with Chern class −1.
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 14 / 14
Theorem (Rieffel)
Let τ̃ : K0(Aθ)→ C denote the extension of the trace τ. Then
τ̃([Eθ]) = θ.
If {MnT mθ η : m,n ∈ Z} is an orthormal basis for L2(R), then
θ = 1.Consequently Aθ ∼= C(T2).By the Serre–Swan theorem, there exists a unique vectorbundle E → T2 such that
Eθ ∼= Γ(E).
One can show that this is a line bundle with Chern class −1.
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 14 / 14
Theorem (Rieffel)
Let τ̃ : K0(Aθ)→ C denote the extension of the trace τ. Then
τ̃([Eθ]) = θ.
If {MnT mθ η : m,n ∈ Z} is an orthormal basis for L2(R), then
θ = 1.
Consequently Aθ ∼= C(T2).By the Serre–Swan theorem, there exists a unique vectorbundle E → T2 such that
Eθ ∼= Γ(E).
One can show that this is a line bundle with Chern class −1.
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 14 / 14
Theorem (Rieffel)
Let τ̃ : K0(Aθ)→ C denote the extension of the trace τ. Then
τ̃([Eθ]) = θ.
If {MnT mθ η : m,n ∈ Z} is an orthormal basis for L2(R), then
θ = 1.Consequently Aθ ∼= C(T2).
By the Serre–Swan theorem, there exists a unique vectorbundle E → T2 such that
Eθ ∼= Γ(E).
One can show that this is a line bundle with Chern class −1.
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 14 / 14
Theorem (Rieffel)
Let τ̃ : K0(Aθ)→ C denote the extension of the trace τ. Then
τ̃([Eθ]) = θ.
If {MnT mθ η : m,n ∈ Z} is an orthormal basis for L2(R), then
θ = 1.Consequently Aθ ∼= C(T2).By the Serre–Swan theorem, there exists a unique vectorbundle E → T2 such that
Eθ ∼= Γ(E).
One can show that this is a line bundle with Chern class −1.
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 14 / 14
Theorem (Rieffel)
Let τ̃ : K0(Aθ)→ C denote the extension of the trace τ. Then
τ̃([Eθ]) = θ.
If {MnT mθ η : m,n ∈ Z} is an orthormal basis for L2(R), then
θ = 1.Consequently Aθ ∼= C(T2).By the Serre–Swan theorem, there exists a unique vectorbundle E → T2 such that
Eθ ∼= Γ(E).
One can show that this is a line bundle with Chern class −1.
Ulrik Enstad Hilbert C*-modules in harmonic analysis 4th July 2019 14 / 14
Ulrik Enstad
Hilbert C*-modules inharmonic analysisNordfjordeid 2019
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