The Balian-Low Theorem and Vector Bundles · TheBalian-LowTheoremandVectorBundles UlrikEnstad University of Oslo 07.12.2017 WorkinprogresswithFranzLuef Ulrik Enstad (UiO) The Balian-Low

The Balian-Low Theorem and Vector Bundles

Ulrik Enstad

University of Oslo

07.12.2017

Work in progress with Franz Luef

Ulrik Enstad (UiO) The Balian-Low Theorem and Vector Bundles 07.12.2017 1 / 15

Background

Gabor frames ⊆ harmonic analysis.Extension to locally compact (Hausdorff) Abelian groups G : Grepresents frequency, Haar measure gives us integration, Fouriertransform, etc.Balian-Low Theorem and vector bundles: Explored by R. Balan, 2001.M. Rieffel/F. Luef: Projective modules over noncommutative tori.


Background

Gabor frames ⊆ harmonic analysis.

Extension to locally compact (Hausdorff) Abelian groups G : Grepresents frequency, Haar measure gives us integration, Fouriertransform, etc.Balian-Low Theorem and vector bundles: Explored by R. Balan, 2001.M. Rieffel/F. Luef: Projective modules over noncommutative tori.


Background

Gabor frames ⊆ harmonic analysis.Extension to locally compact (Hausdorff) Abelian groups G : Grepresents frequency, Haar measure gives us integration, Fouriertransform, etc.

Balian-Low Theorem and vector bundles: Explored by R. Balan, 2001.M. Rieffel/F. Luef: Projective modules over noncommutative tori.


Background

Gabor frames ⊆ harmonic analysis.Extension to locally compact (Hausdorff) Abelian groups G : Grepresents frequency, Haar measure gives us integration, Fouriertransform, etc.Balian-Low Theorem and vector bundles: Explored by R. Balan, 2001.

M. Rieffel/F. Luef: Projective modules over noncommutative tori.


Background

Gabor frames ⊆ harmonic analysis.Extension to locally compact (Hausdorff) Abelian groups G : Grepresents frequency, Haar measure gives us integration, Fouriertransform, etc.Balian-Low Theorem and vector bundles: Explored by R. Balan, 2001.M. Rieffel/F. Luef: Projective modules over noncommutative tori.


Classical Gabor frames

Given η ∈ L2(R) and α, β > 0, let

ηk,l (x) = e2πiβlxη(x − αk).

Definition{ηk,l}k,l∈Z is called a Gabor frame if there exist constants C ,D ≥ 0 suchthat

C‖ξ‖2 ≤∑

k,l∈Z|〈ξ, ηk,l〉|2 ≤ D‖ξ‖2

for every ξ ∈ L2(R).

For every ξ ∈ L2(R), we have that

ξ =∑

k,l∈Z〈S−1ξ, ηk,l〉ηk,l .






C‖ξ‖2 ≤∑

k,l∈Z|〈ξ, ηk,l〉|2 ≤ D‖ξ‖2



ξ =∑







C‖ξ‖2 ≤∑

k,l∈Z|〈ξ, ηk,l〉|2 ≤ D‖ξ‖2



ξ =∑







C‖ξ‖2 ≤∑

k,l∈Z|〈ξ, ηk,l〉|2 ≤ D‖ξ‖2



ξ =∑



The Balian-Low Theorem

Let ξ(x) = e−πx2 and η = χ[0,1]. Then {ξk,l}k,l is a Gabor frame forall 0 < αβ < 1 but not αβ = 1. {ηk,l}k,l is a Gabor frame for αβ = 1.

Theorem (R. Balian, F. Low)Let η ∈ L2(R). Suppose that

1 αβ = 1.2 η is “nice”: η ∈ S0(R).

Then {ηk,l : k, l ∈ Z} is not a Gabor frame.

Moral: When working with Gabor frames at the critical densityαβ = 1, we cannot expect well-behaved Gabor atoms η.





1 αβ = 1.2 η is “nice”: η ∈ S0(R).







1 αβ = 1.2 η is “nice”: η ∈ S0(R).







1 αβ = 1.2 η is “nice”: η ∈ S0(R).




Gabor frames in L2(G)For the rest of the talk, G is a locally compact (Hausdorff) Abelian group.

For x ∈ G and ω ∈ G , define operators on L2(G) by

Txξ(t) = ξ(x−1t) Mωξ(t) = ω(t)ξ(t).

Set π(x , ω) = MωTx .Given a discrete set ∆ ⊆ G × G and η ∈ L2(G), we call

G(η,∆) = {π(z)η : z ∈ ∆}

the Gabor system over ∆ with atom η.If there exist C ,D ≥ 0 such that

C‖ξ‖2 ≤∑z∈∆|〈ξ, π(z)η〉|2 ≤ D‖ξ‖2

for all ξ ∈ L2(G), then G(η,∆) is called a Gabor frame.





Set π(x , ω) = MωTx .

Given a discrete set ∆ ⊆ G × G and η ∈ L2(G), we call

G(η,∆) = {π(z)η : z ∈ ∆}


C‖ξ‖2 ≤∑z∈∆|〈ξ, π(z)η〉|2 ≤ D‖ξ‖2







G(η,∆) = {π(z)η : z ∈ ∆}

the Gabor system over ∆ with atom η.

If there exist C ,D ≥ 0 such that

C‖ξ‖2 ≤∑z∈∆|〈ξ, π(z)η〉|2 ≤ D‖ξ‖2







G(η,∆) = {π(z)η : z ∈ ∆}


C‖ξ‖2 ≤∑z∈∆|〈ξ, π(z)η〉|2 ≤ D‖ξ‖2

for all ξ ∈ L2(G), then G(η,∆) is called a Gabor frame.Ulrik Enstad (UiO) The Balian-Low Theorem and Vector Bundles 07.12.2017 5 / 15

Balian-Low for L2(G)

A closed subgroup H of G is a uniform lattice in G is H is discreteand G/H is compact.The annihilator of H is the set H⊥ of ω ∈ G such that ω(h) = 1 forevery h ∈ H.

Theorem?Let G be a locally compact Abelian group, and let H be a uniform latticein G . If η ∈ S0(G), then G(η,H × H⊥) is not a Gabor frame for L2(G).

Example (Failure of general BL)Let D be a discrete group and set G = D × D, H = D × {e}. Setη = χ{e}×D. Then η ∈ S0(G), and G(η,H × H⊥) is an orthonormal basisfor L2(G).



A closed subgroup H of G is a uniform lattice in G is H is discreteand G/H is compact.

The annihilator of H is the set H⊥ of ω ∈ G such that ω(h) = 1 forevery h ∈ H.



















The Zak Transform

For ξ ∈ L2(G), the function ZHξ : G × G → C given by

ZHξ(x , ω) =∑h∈H

ξ(h−1x)ω(h).

is called the Zak transform of ξ.The Zak transform F = ZHξ satisfies the following quasiperiodicproperty:

F (xh, ωτ) = τ(h)Fξ(x , ω) ; x ∈ G , ω ∈ G , h ∈ H, τ ∈ H⊥. (1)

Gröchenig 1998: G(η,H × H⊥) is a Gabor frame if and only if thereexist 0 < A,B <∞ such that

A ≤ |ZHη(x , ω)| ≤ B

almost everywhere.If ξ ∈ S0(G), then ZHξ is continuous and bounded. Hence,G(η,H × H⊥) is NOT a Gabor frame if and only if ZHξ has a zero.


The Zak Transform



ξ(h−1x)ω(h).

is called the Zak transform of ξ.

The Zak transform F = ZHξ satisfies the following quasiperiodicproperty:



A ≤ |ZHη(x , ω)| ≤ B



The Zak Transform



ξ(h−1x)ω(h).




A ≤ |ZHη(x , ω)| ≤ B



The Zak Transform



ξ(h−1x)ω(h).




A ≤ |ZHη(x , ω)| ≤ B

almost everywhere.

If ξ ∈ S0(G), then ZHξ is continuous and bounded. Hence,G(η,H × H⊥) is NOT a Gabor frame if and only if ZHξ has a zero.


The Zak Transform



ξ(h−1x)ω(h).




A ≤ |ZHη(x , ω)| ≤ B

almost everywhere.If ξ ∈ S0(G), then ZHξ is continuous and bounded. Hence,G(η,H × H⊥) is NOT a Gabor frame if and only if ZHξ has a zero.Ulrik Enstad (UiO) The Balian-Low Theorem and Vector Bundles 07.12.2017 7 / 15

The vector bundle

Denote by E = EG,H the quotient of G × G × C by the equivalencerelation generated by

(x , ω, λ) ∼ (xh, ωτ, ω(h)λ) ; x ∈ G , ω ∈ G , h ∈ H, τ ∈ H⊥.

Get line bundle E over the compact space G/H × G/H⊥.Then Γ(E ) can be identified as the continuous maps F : G × G → Csatisfying (1).A line bundle is trivial if and only if it has a nonvanishing continuoussection.

TheoremLet G be a locally compact Abelian group, and let H be a uniform latticein G . Then the vector bundle E is nontrivial if and only if for everyη ∈ L2(G) the following implication holds: Whenever ZHη is continuous,then G(η,H × H⊥) is not a Gabor frame.


The vector bundle






The vector bundle






The vector bundle



Get line bundle E over the compact space G/H × G/H⊥.

Then Γ(E ) can be identified as the continuous maps F : G × G → Csatisfying (1).A line bundle is trivial if and only if it has a nonvanishing continuoussection.



The vector bundle



Get line bundle E over the compact space G/H × G/H⊥.Then Γ(E ) can be identified as the continuous maps F : G × G → Csatisfying (1).

A line bundle is trivial if and only if it has a nonvanishing continuoussection.



The vector bundle






The vector bundle






Twisted group C ∗-algebras

The map π(x , ω) = MωTx satisfies

π(x , ω)π(y , τ) = τ(x)π(xy , ωτ).

where c((x , ω), (y , τ)) = τ(x) is a 2-cocycle on G × G .Given a discrete subgroup ∆ ⊆ G × G , let A0 = L1(∆, c) be thec-twisted convolution algebra with operations

(f ∗ g)(z) =∑

w∈∆f (w)g(w−1z)c(w ,w−1z) f ∗(z) = c(z , z−1)f (z−1)

Let A = C∗(∆, c) be the corresponding c-twisted group C∗-algebra.We obtain the integrated representation Π : C∗(∆, c)→ B(L2(G))given by

Π(f ) =∑z∈∆

f (z)π(z).




π(x , ω)π(y , τ) = τ(x)π(xy , ωτ).

where c((x , ω), (y , τ)) = τ(x) is a 2-cocycle on G × G .

Given a discrete subgroup ∆ ⊆ G × G , let A0 = L1(∆, c) be thec-twisted convolution algebra with operations

(f ∗ g)(z) =∑



Π(f ) =∑z∈∆

f (z)π(z).




π(x , ω)π(y , τ) = τ(x)π(xy , ωτ).


(f ∗ g)(z) =∑



Π(f ) =∑z∈∆

f (z)π(z).




π(x , ω)π(y , τ) = τ(x)π(xy , ωτ).


(f ∗ g)(z) =∑


Let A = C∗(∆, c) be the corresponding c-twisted group C∗-algebra.

We obtain the integrated representation Π : C∗(∆, c)→ B(L2(G))given by

Π(f ) =∑z∈∆

f (z)π(z).




π(x , ω)π(y , τ) = τ(x)π(xy , ωτ).


(f ∗ g)(z) =∑



Π(f ) =∑z∈∆

f (z)π(z).


Modules over C ∗(∆, c)

M. Rieffel (1988)/F. Luef (2008): For f ∈ L1(∆, c) and ξ, η ∈ S0(G),the equations

f · ξ = Π(f )ξ•〈ξ, η〉 (z) = 〈ξ, π(z)η〉

give S0(G) the structure of a pre-inner product A0-module E0. Itscompletion E is a finitely generated projective left Hilbert A-module.Trace tr : A→ C given on A0 by tr(f ) = f (e).




f · ξ = Π(f )ξ•〈ξ, η〉 (z) = 〈ξ, π(z)η〉

give S0(G) the structure of a pre-inner product A0-module E0. Itscompletion E is a finitely generated projective left Hilbert A-module.

Trace tr : A→ C given on A0 by tr(f ) = f (e).




f · ξ = Π(f )ξ•〈ξ, η〉 (z) = 〈ξ, π(z)η〉

give S0(G) the structure of a pre-inner product A0-module E0. Itscompletion E is a finitely generated projective left Hilbert A-module.Trace tr : A→ C given on A0 by tr(f ) = f (e).


Module frames

DefinitionA module frame in E is a sequence (ηj)j∈J in E such that there existconstants C ,D ≥ 0 with

C•〈ξ, ξ〉 ≤∑j∈J•〈ξ, ηj〉∗ •〈ξ, ηj〉 ≤ D•〈ξ, ξ〉

for all ξ ∈ E .

Restrict to a one-element module frame {η}. Taking traces, we obtain

C‖ξ‖2 ≤∑z∈∆|〈ξ, π(z)η〉|2 ≤ D‖ξ‖2.


Module frames


C•〈ξ, ξ〉 ≤∑j∈J•〈ξ, ηj〉∗ •〈ξ, ηj〉 ≤ D•〈ξ, ξ〉

for all ξ ∈ E .


C‖ξ‖2 ≤∑z∈∆|〈ξ, π(z)η〉|2 ≤ D‖ξ‖2.


Module frames


C•〈ξ, ξ〉 ≤∑j∈J•〈ξ, ηj〉∗ •〈ξ, ηj〉 ≤ D•〈ξ, ξ〉

for all ξ ∈ E .


C‖ξ‖2 ≤∑z∈∆|〈ξ, π(z)η〉|2 ≤ D‖ξ‖2.


The critical density

When ∆ = H × H⊥, then c �H×H⊥= 1.Hence

C∗(∆, c) ∼= C(H × H⊥

)∼= C(G/H × G/H⊥).

Thus, E is a finitely generated projective module over the algebra ofcontinuous functions on G/H × G/H⊥.By the Serre-Swan theorem, there exists a Hermitian vector bundleE = EG,H → G/H × G/H⊥ such that E ∼= Γ(E ) as left HilbertC∗-modules.



When ∆ = H × H⊥, then c �H×H⊥= 1.

HenceC∗(∆, c) ∼= C

(H × H⊥

)∼= C(G/H × G/H⊥).





C∗(∆, c) ∼= C(H × H⊥

)∼= C(G/H × G/H⊥).





C∗(∆, c) ∼= C(H × H⊥

)∼= C(G/H × G/H⊥).

Thus, E is a finitely generated projective module over the algebra ofcontinuous functions on G/H × G/H⊥.

By the Serre-Swan theorem, there exists a Hermitian vector bundleE = EG,H → G/H × G/H⊥ such that E ∼= Γ(E ) as left HilbertC∗-modules.




C∗(∆, c) ∼= C(H × H⊥

)∼= C(G/H × G/H⊥).



The Zak transform as a map of modules

Equip Γ(E ) with the following C(G/H × G/H⊥)-module structureand inner product: [Rieffel 1983: The case G = R]

(f · F )(x , ω) = f ([x ], [ω])F (x , ω)

•〈F ,G〉 (x , ω) = F (x , ω)G(x , ω)

The Zak transform ZH : S0(G)→ Γ(E ) is A-linear, and extends to aninner product preserving A-module homomorphism

ZH : E → Γ(E ).

Module-theoretic proof of BL variantSuppose η ∈ S0(G), and let H be a uniform lattice in G . Then if EG,H isnontrivial, then {η} cannot be a module frame for EG,H .




(f · F )(x , ω) = f ([x ], [ω])F (x , ω)

•〈F ,G〉 (x , ω) = F (x , ω)G(x , ω)


ZH : E → Γ(E ).





(f · F )(x , ω) = f ([x ], [ω])F (x , ω)

•〈F ,G〉 (x , ω) = F (x , ω)G(x , ω)


ZH : E → Γ(E ).





(f · F )(x , ω) = f ([x ], [ω])F (x , ω)

•〈F ,G〉 (x , ω) = F (x , ω)G(x , ω)


ZH : E → Γ(E ).



Generalizations

Investigate the situation where H is not necessarily a uniform latticein G .Investigate more subgroups ∆ ⊂ G × G for which C∗(∆, c) iscommutative.


Thank you for your attention!


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The Balian-Low Theorem and Vector Bundles · TheBalian-LowTheoremandVectorBundles UlrikEnstad University of Oslo 07.12.2017 WorkinprogresswithFranzLuef Ulrik Enstad (UiO) The Balian-Low