· Frames for Hilbert and Banach spaces A sequence g j : j 2J in a Hilbert space His a frame if...

Hilbert C ∗-modules over noncommutative toriTexas A & M

Luef, FranzUC Berkeley

16.07.2012

Luef, Franz UC Berkeley Hilbert C∗-modules over noncommutative tori

Frames for Hilbert and Banach spaces

A sequence gj : j ∈ J in a Hilbert space H is a frame ifthere exists positive constants A,B such that for all f ∈ H

A‖f ‖2H ≤

∑j∈J|〈f , gj〉|2 ≤ B‖f ‖2

H. (1)

analysis operator: Cf = 〈f , gj〉 : j ∈ J

synthesis operator: Dc =∑

j∈J cjgj for c = (cj)j∈J

frame operator: Sf = DCf =∑

j∈J〈f , gj〉gjBasic fact: gj : j ∈ J is a frame for H ⇔, then the frameoperator S is a positive invertible operator on H.

Luef, Franz Hilbert C∗-modules over noncommutative tori



A‖f ‖2H ≤

∑j∈J|〈f , gj〉|2 ≤ B‖f ‖2

H. (1)

analysis operator: Cf = 〈f , gj〉 : j ∈ Jsynthesis operator: Dc =

∑j∈J cjgj for c = (cj)j∈J






A‖f ‖2H ≤

∑j∈J|〈f , gj〉|2 ≤ B‖f ‖2

H. (1)




j∈J〈f , gj〉gj

Basic fact: gj : j ∈ J is a frame for H ⇔, then the frameoperator S is a positive invertible operator on H.




A‖f ‖2H ≤

∑j∈J|〈f , gj〉|2 ≤ B‖f ‖2

H. (1)




j∈J〈f , gj〉gj

Basic fact: gj : j ∈ J is a frame for H ⇔, then the frameoperator S is a positive invertible operator on H.




A‖f ‖2H ≤

∑j∈J|〈f , gj〉|2 ≤ B‖f ‖2

H. (1)








A‖f ‖2H ≤

∑j∈J|〈f , gj〉|2 ≤ B‖f ‖2

H. (1)







Another useful reformulation of the notion of frames:Let g ⊗ h be the rank one operator defined by(g ⊗ h)f = 〈f , g〉h. Then

A · IH ≤∑j∈J

gj ⊗ gj ≤ B · IH,

the series converges in the strong operator topology.Frames are of relevance because they allow the construction of(non-orthogonal) expansions.

If gj : j ∈ J is a frame for H, then S−1gj : j ∈ J is aframe with frame bounds B−1,A−1 f ∈ H and

f =∑j∈J〈f ,S−1gj〉gj . (2)



Another useful reformulation of the notion of frames:Let g ⊗ h be the rank one operator defined by(g ⊗ h)f = 〈f , g〉h. Then

A · IH ≤∑j∈J

gj ⊗ gj ≤ B · IH,

the series converges in the strong operator topology.Frames are of relevance because they allow the construction of(non-orthogonal) expansions.

If gj : j ∈ J is a frame for H, then S−1gj : j ∈ J is aframe with frame bounds B−1,A−1 f ∈ H and

f =∑j∈J〈f ,S−1gj〉gj . (2)


Banach frames

Grochenig introduced Banach frames:

A sequence gj : j ∈ J of a Banach space B is called aBanach frame if there exists an associated sequencespace Bd(J) and a continuous reconstruction operatorR : Bd(J)→ B such that for all f ∈ B

R((〈f , gj〉)j∈J)) = f ,

C−1‖f ‖B ≤ ‖(〈f , gj〉)j∈J)‖Bd ≤ C‖f ‖Bfor some constant C ≥ 1.

Coorbit spaces provide a natural class of Banach framesfor a wide class of function spaces.

Banach-Gelfand triples

Casazza, Han and Larson introduced the notion of aframing for a Banach space.


Banach frames

Grochenig introduced Banach frames:

A sequence gj : j ∈ J of a Banach space B is called aBanach frame if there exists an associated sequencespace Bd(J) and a continuous reconstruction operatorR : Bd(J)→ B such that for all f ∈ B

R((〈f , gj〉)j∈J)) = f ,

C−1‖f ‖B ≤ ‖(〈f , gj〉)j∈J)‖Bd ≤ C‖f ‖Bfor some constant C ≥ 1.

Coorbit spaces provide a natural class of Banach framesfor a wide class of function spaces.

Banach-Gelfand triples

Casazza, Han and Larson introduced the notion of aframing for a Banach space.


Frames for Hilbert C ∗-modules

Theory of frames for Hilbert C ∗-modules due to Frankand Larson.

Relevance of Hilbert C ∗-modules for wavelets wasdemonstrated by Packer and Rieffel (projectivemultiresolution analysis).

Construction of equivalence bimodules betweennoncommutative tori is Gabor analysis.












Left Hilbert C ∗-modules

In the 1970’s Paschke and Rieffel independently generalized thenotion of Hilbert spaces to so-called Hilbert C ∗-modules.

Defintion:

Let A be a unital C ∗-algebra. Then a vector space V is a leftHilbert A-module, i.e. (A, g) 7→ A · g is a map fromX ×A → A, with a pairing A〈., .〉 such that for all f , g , h ∈ V :

(a) A〈λg + µh, k〉 = λA〈g , k〉+ µA〈h, k〉 for all λ, µ ∈ C;

(b) A〈A · f , g〉 = A · A〈f , g〉 for all A ∈ A;

(c) A〈f , g〉 = A〈g , f 〉∗;(d) A〈f , f 〉 ≥ 0.

(e) V is complete with respect to the norm

A‖f ‖ := ‖A〈f , f 〉‖1/2. If the idealspanA〈f , g〉 : f , g ∈ V is dense in A, then V is calledfull.




Defintion:




(c) A〈f , g〉 = A〈g , f 〉∗;(d) A〈f , f 〉 ≥ 0.






Defintion:




(c) A〈f , g〉 = A〈g , f 〉∗;(d) A〈f , f 〉 ≥ 0.






Defintion:




(c) A〈f , g〉 = A〈g , f 〉∗;

(d) A〈f , f 〉 ≥ 0.






Defintion:




(c) A〈f , g〉 = A〈g , f 〉∗;(d) A〈f , f 〉 ≥ 0.






Defintion:




(c) A〈f , g〉 = A〈g , f 〉∗;(d) A〈f , f 〉 ≥ 0.






Defintion:




(c) A〈f , g〉 = A〈g , f 〉∗;(d) A〈f , f 〉 ≥ 0.




Right Hilbert C ∗-modules

Defintion:

Let B be a unital C ∗-algebra. Then a vector space V is a rightHilbert B-module, i.e. (g ,B) 7→ g · B is a map fromB × V → B, with a pairing 〈., .〉B which is linear in the secondvariable, such that for all f , g , h ∈ V :

(a) 〈f , λg + µh〉B = λ〈f , g〉B + µ〈f , h〉B for all λ, µ ∈ C;

(b) 〈·f , g · B〉 = 〈f , g〉B B for all B ∈ B;

(c) 〈f , g〉B = 〈g , f 〉∗B;

(d) 〈f , f 〉B ≥ 0.

(e) V is complete with respect to the norm‖f ‖B := ‖〈f , f 〉B‖1/2. If the idealspan〈f , g〉B : f , g ∈ V is dense in B, then V is calledfull.



Defintion:




(c) 〈f , g〉B = 〈g , f 〉∗B;

(d) 〈f , f 〉B ≥ 0.




Defintion:




(c) 〈f , g〉B = 〈g , f 〉∗B;

(d) 〈f , f 〉B ≥ 0.




Defintion:




(c) 〈f , g〉B = 〈g , f 〉∗B;

(d) 〈f , f 〉B ≥ 0.




Defintion:




(c) 〈f , g〉B = 〈g , f 〉∗B;

(d) 〈f , f 〉B ≥ 0.




Defintion:




(c) 〈f , g〉B = 〈g , f 〉∗B;

(d) 〈f , f 〉B ≥ 0.




Defintion:




(c) 〈f , g〉B = 〈g , f 〉∗B;

(d) 〈f , f 〉B ≥ 0.



Hilbert C ∗-modules

In applications one often is in the situation that one has aA0-module V0, where A0 is a dense involutive subalgebra of A,and that there is a pairing 〈., .〉0 from V0 × V0 → A0 satisfyingall the conditions of a Hilbert C ∗-module, when positivity isconsidered with respect to A. Then one can complete V0 toobtain a Hilbert C ∗-module V over A.The challenge in the construction of Hilbert C ∗-modules is tofind appropriate function spaces V0.

Hilbert C ∗-modules are also Banach modules for the HilbertC ∗-module norm, i.e. A‖A · g‖ ≤ ‖A‖A‖g‖.In some cases the structures of left and right HilbertC ∗-modules over two C ∗-algebras A and B are compatible witheach other.



In applications one often is in the situation that one has aA0-module V0, where A0 is a dense involutive subalgebra of A,and that there is a pairing 〈., .〉0 from V0 × V0 → A0 satisfyingall the conditions of a Hilbert C ∗-module, when positivity isconsidered with respect to A. Then one can complete V0 toobtain a Hilbert C ∗-module V over A.The challenge in the construction of Hilbert C ∗-modules is tofind appropriate function spaces V0.Hilbert C ∗-modules are also Banach modules for the HilbertC ∗-module norm, i.e. A‖A · g‖ ≤ ‖A‖A‖g‖.

In some cases the structures of left and right HilbertC ∗-modules over two C ∗-algebras A and B are compatible witheach other.



In applications one often is in the situation that one has aA0-module V0, where A0 is a dense involutive subalgebra of A,and that there is a pairing 〈., .〉0 from V0 × V0 → A0 satisfyingall the conditions of a Hilbert C ∗-module, when positivity isconsidered with respect to A. Then one can complete V0 toobtain a Hilbert C ∗-module V over A.The challenge in the construction of Hilbert C ∗-modules is tofind appropriate function spaces V0.Hilbert C ∗-modules are also Banach modules for the HilbertC ∗-module norm, i.e. A‖A · g‖ ≤ ‖A‖A‖g‖.In some cases the structures of left and right HilbertC ∗-modules over two C ∗-algebras A and B are compatible witheach other.



In applications one often is in the situation that one has aA0-module V0, where A0 is a dense involutive subalgebra of A,and that there is a pairing 〈., .〉0 from V0 × V0 → A0 satisfyingall the conditions of a Hilbert C ∗-module, when positivity isconsidered with respect to A. Then one can complete V0 toobtain a Hilbert C ∗-module V over A.The challenge in the construction of Hilbert C ∗-modules is tofind appropriate function spaces V0.Hilbert C ∗-modules are also Banach modules for the HilbertC ∗-module norm, i.e. A‖A · g‖ ≤ ‖A‖A‖g‖.In some cases the structures of left and right HilbertC ∗-modules over two C ∗-algebras A and B are compatible witheach other.


Maps on Hilbert C ∗-modules

Suppose V is a Hilbert A-module. Then a modulemapping T : V → V is adjointable, if there is a mappingT ∗ : V → V such that

A〈Tf , g〉 = A〈f ,T ∗g〉 for all f , g ∈ V .

L(V ) denotes the space of all adjointable mappings on V .

We define the norm of T by‖T‖ = supA‖Tg‖ : A‖g‖ ≤ 1.

L(V ) is a C ∗-algebra with respect to this norm.

rank one operators ΘAg ,hf = A〈f , g〉 · h are adjointableoperators.

The closed linear subspace of L(V ) spanned byΘAg ,h : f , g ∈ V is the algebra of compact A-moduleoperators.






We define the norm of T by‖T‖ = supA‖Tg‖ : A‖g‖ ≤ 1.L(V ) is a C ∗-algebra with respect to this norm.




























Properties of rank one Hilbert module operators

ΘAg ,hΘAg ′,h′ = ΘAk,h for k = A〈g , h′〉g ′

(ΘAg ,h)∗ = ΘAh,g

For T ∈ L(V ) we have T ΘAg ,h = ΘAg ,Th.

‖ΘAg ,h‖ ≤ A‖g‖A‖h‖.Suppose A is unital and A〈g , g〉 = I , then ‖ΘAg ,h‖ = A‖h‖.

We denote the unit sphere of AV byS(AV ) = g ∈ AV : A〈g , g〉 = I.













‖ΘAg ,h‖ ≤ A‖g‖A‖h‖.

Suppose A is unital and A〈g , g〉 = I , then ‖ΘAg ,h‖ = A‖h‖.

























For the sake of simplicity I restrict my discussion to the case offinitely generated Hilbert C ∗-modules.

Let A be a unital C ∗-algebra. A sequencegj : j = 1, ..., n in a (left) Hilbert A-module AV is calleda standard module frame if there are positive reals C ,Dsuch that

C A〈f , f 〉 ≤n∑

j=1

A〈f , gj〉A〈gj , f 〉 ≤ D A〈f , f 〉

for each f ∈ AV .

Let A be a unital C ∗-algebra. A sequencegj : j = 1, ..., n in a (left) Hilbert A-module AV is astandard module frame if the reconstruction formula

f =n∑

j=1

A〈f , gj〉 · gj for all f ∈ AV .





C A〈f , f 〉 ≤n∑

j=1

A〈f , gj〉A〈gj , f 〉 ≤ D A〈f , f 〉

for each f ∈ AV .Let A be a unital C ∗-algebra. A sequencegj : j = 1, ..., n in a (left) Hilbert A-module AV is astandard module frame if the reconstruction formula

f =n∑

j=1






C A〈f , f 〉 ≤n∑

j=1

A〈f , gj〉A〈gj , f 〉 ≤ D A〈f , f 〉

for each f ∈ AV .Let A be a unital C ∗-algebra. A sequencegj : j = 1, ..., n in a (left) Hilbert A-module AV is astandard module frame if the reconstruction formula

f =n∑

j=1



Frames for Hilbert C ∗-module

The existence of a standard module frame of finitecardinality is equivalent to AV being a projective module,i.e it can be embedded into An as a direct summand.

In terms of rank one Hilbert A-module operators thedefinition of a frame becomes:

C A〈f , f 〉 ≤n∑

j=1

A〈ΘAgj ,gj f , f 〉A ≤ D A〈f , f 〉.

One has the following: A sequence gj : j = 1, ..., n in a(left) Hilbert A-module AV is a frame if and only if∑n

j=1 AΘAgj ,gj converges in the strict topology to abounded invertible operator in L(AV ).





C A〈f , f 〉 ≤n∑

j=1








C A〈f , f 〉 ≤n∑

j=1






Suppose AV is a finitely generated Hilbert A-module.Then any set of generators gj : j = 1, ..., n is a standardmodule frame. The number of the shortest frame gives thenumber of factors of An into which AV is embeddable.

In other words, the positive module operatorS =

∑nj=1 ΘAgj ,gj is invertible and the upper and lower

frame bounds are given by ‖S‖2 and ‖S−1‖−2.

In particular, AV is singly generated if ΘAg ,g is invertible.

Is there a way to generate finitely generated HilbertC ∗-modules?


















Morita-Rieffel equivalence

Rieffel introduced in 1970’s the notion of strong Moritaequivalence for C ∗-algebras:

Definition:

Let A and B be C ∗-algebras. Then an A-B-equivalencebimodule AV B is an A-B-bimodule such that:

(a) AV B is a full left Hilbert A-module and a full right HilbertB-module;

(b) for all f , g ∈ AV B,A ∈ A and B ∈ B we have that〈A · f , g〉B = 〈f ,A∗ · g〉B and A〈f · B, g〉 = A〈f , g · B∗〉;

(c) for all f , g , h ∈ AV B, A〈f , g〉 · h = f · 〈g , h〉B.

The C ∗-algebras A and B are called Morita-Rieffel equivalent ifthere exists an A− B equivalence bimodule.




Definition:









Definition:









Definition:









Definition:







Morita-Rieffel equivalence - Consequences

In words, Condition (b) says that A acts by adjointableoperators on VB and that B acts by adjointable operatorson AV ,Condition (c) is an associativity condition between theA-inner product and the B-inner product.ΘBg ,hf = f · 〈g , h〉B is a rank one B-module operator.

Consequently, the invertiblity of ΘAg ,g is equivalent to the

invertiblity of ΘBg ,g . Note that this amounts to

ΘBg ,g f = f · 〈g , g〉B, i.e. f and g are “uncoupled”.

If A and B are Morita-Rieffel equivalent, then AVB is aprojective left A-module, and a projective rightB-module.There exist g1, ..., gn ∈ AVB such that

f =n∑

j=1

A〈f , gj〉 · gj =n∑

j=1

f · 〈gj , gj〉B

for all f ∈ AVB.






ΘBg ,g f = f · 〈g , g〉B, i.e. f and g are “uncoupled”.If A and B are Morita-Rieffel equivalent, then AVB is aprojective left A-module, and a projective rightB-module.

There exist g1, ..., gn ∈ AVB such that

f =n∑

j=1

A〈f , gj〉 · gj =n∑

j=1

f · 〈gj , gj〉B

for all f ∈ AVB.






ΘBg ,g f = f · 〈g , g〉B, i.e. f and g are “uncoupled”.If A and B are Morita-Rieffel equivalent, then AVB is aprojective left A-module, and a projective rightB-module.There exist g1, ..., gn ∈ AVB such that

f =n∑

j=1

A〈f , gj〉 · gj =n∑

j=1

f · 〈gj , gj〉B

for all f ∈ AVB.Luef, Franz Hilbert C∗-modules over noncommutative tori





ΘBg ,g f = f · 〈g , g〉B, i.e. f and g are “uncoupled”.If A and B are Morita-Rieffel equivalent, then AVB is aprojective left A-module, and a projective rightB-module.There exist g1, ..., gn ∈ AVB such that

f =n∑

j=1

A〈f , gj〉 · gj =n∑

j=1

f · 〈gj , gj〉B

for all f ∈ AVB.Luef, Franz Hilbert C∗-modules over noncommutative tori


The compact A-module operators are isomorphic to B.

Suppose A has a normalized trace trA. Then we canintroduce a trace trB on B by

trB(〈f , g〉B) = trA(A〈g , f 〉).

trB is not normalized and trB(I ) = dim(AV ), thedimension of the Hilbert A-module AV .


Morita equivalent spectrally invariant subalgebras

Recall that a unital subalgebra A of a unital C ∗-algebra B withcommon unit is called spectrally invariant, if for A ∈ A withA−1 ∈ B one actually has that A−1 ∈ A.

Theorem:

Let A and B be unital C ∗-algebras that are Morita equivalentvia an equivalence bimodule AVB.

Suppose we have dense∗-Banach (or Frechet) subalgebras A0 and B0 of A and Brespectively containing the identity elements.Furthermore weassume that A0 and B0 are spectrally invariant in A and Brespectively.Let V0 be a dense subspace of AVB which is closedunder the actions of A0 and B0, and such that the restrictionsof the inner products A〈., .〉 and 〈., .〉B have values in A0 andB0 respectively.Then V0 is a finitely generated projective leftA0-module.In addition we have that V0 is a finitely generatedprojective right B0-module.




Theorem:

Let A and B be unital C ∗-algebras that are Morita equivalentvia an equivalence bimodule AVB.Suppose we have dense∗-Banach (or Frechet) subalgebras A0 and B0 of A and Brespectively containing the identity elements.

Furthermore weassume that A0 and B0 are spectrally invariant in A and Brespectively.Let V0 be a dense subspace of AVB which is closedunder the actions of A0 and B0, and such that the restrictionsof the inner products A〈., .〉 and 〈., .〉B have values in A0 andB0 respectively.Then V0 is a finitely generated projective leftA0-module.In addition we have that V0 is a finitely generatedprojective right B0-module.




Theorem:

Let A and B be unital C ∗-algebras that are Morita equivalentvia an equivalence bimodule AVB.Suppose we have dense∗-Banach (or Frechet) subalgebras A0 and B0 of A and Brespectively containing the identity elements.Furthermore weassume that A0 and B0 are spectrally invariant in A and Brespectively.

Let V0 be a dense subspace of AVB which is closedunder the actions of A0 and B0, and such that the restrictionsof the inner products A〈., .〉 and 〈., .〉B have values in A0 andB0 respectively.Then V0 is a finitely generated projective leftA0-module.In addition we have that V0 is a finitely generatedprojective right B0-module.




Theorem:

Let A and B be unital C ∗-algebras that are Morita equivalentvia an equivalence bimodule AVB.Suppose we have dense∗-Banach (or Frechet) subalgebras A0 and B0 of A and Brespectively containing the identity elements.Furthermore weassume that A0 and B0 are spectrally invariant in A and Brespectively.Let V0 be a dense subspace of AVB which is closedunder the actions of A0 and B0, and such that the restrictionsof the inner products A〈., .〉 and 〈., .〉B have values in A0 andB0 respectively.

Then V0 is a finitely generated projective leftA0-module.In addition we have that V0 is a finitely generatedprojective right B0-module.




Theorem:

Let A and B be unital C ∗-algebras that are Morita equivalentvia an equivalence bimodule AVB.Suppose we have dense∗-Banach (or Frechet) subalgebras A0 and B0 of A and Brespectively containing the identity elements.Furthermore weassume that A0 and B0 are spectrally invariant in A and Brespectively.Let V0 be a dense subspace of AVB which is closedunder the actions of A0 and B0, and such that the restrictionsof the inner products A〈., .〉 and 〈., .〉B have values in A0 andB0 respectively.Then V0 is a finitely generated projective leftA0-module.

In addition we have that V0 is a finitely generatedprojective right B0-module.




Theorem:

Let A and B be unital C ∗-algebras that are Morita equivalentvia an equivalence bimodule AVB.Suppose we have dense∗-Banach (or Frechet) subalgebras A0 and B0 of A and Brespectively containing the identity elements.Furthermore weassume that A0 and B0 are spectrally invariant in A and Brespectively.Let V0 be a dense subspace of AVB which is closedunder the actions of A0 and B0, and such that the restrictionsof the inner products A〈., .〉 and 〈., .〉B have values in A0 andB0 respectively.Then V0 is a finitely generated projective leftA0-module.In addition we have that V0 is a finitely generatedprojective right B0-module.




Theorem:

Let A and B be unital C ∗-algebras that are Morita equivalentvia an equivalence bimodule AVB.Suppose we have dense∗-Banach (or Frechet) subalgebras A0 and B0 of A and Brespectively containing the identity elements.Furthermore weassume that A0 and B0 are spectrally invariant in A and Brespectively.Let V0 be a dense subspace of AVB which is closedunder the actions of A0 and B0, and such that the restrictionsof the inner products A〈., .〉 and 〈., .〉B have values in A0 andB0 respectively.Then V0 is a finitely generated projective leftA0-module.In addition we have that V0 is a finitely generatedprojective right B0-module.


Stone-von Neumann theorem

Rieffel demonstrated that the Stone-von Neumann theorem onthe uniqueness of the irreducible representations of theHeisenberg group is equivalent to the following fact:

Theorem:

The C ∗-algebra of compact operators on a separable Hilbertspace H is Morita-Rieffel equivalent to C.

We construct an equivalence bimodule using basic facts andnotions from time-frequency analysis.


Stone-von Neumann theorem

Rieffel demonstrated that the Stone-von Neumann theorem onthe uniqueness of the irreducible representations of theHeisenberg group is equivalent to the following fact:

Theorem:

The C ∗-algebra of compact operators on a separable Hilbertspace H is Morita-Rieffel equivalent to C.

We construct an equivalence bimodule using basic facts andnotions from time-frequency analysis.


Schrodinger representation

translation Tx f (t) = f (t − x) for x ∈ Rd , modulationMωf (t) = e2πit·ωf (t) for ω ∈ Rd

time-frequency shift π(x , ω)f (t) = MωTx f (t) for(x , ω) ∈ Rd × Rd

MωTx = e2πix ·ωTxMω

π(x + y , ω + η) = e−2πix ·ηπ(x , ω)π(y , η)π(x , ω)π(y , η) = e2πi(y ·ω−x ·η)π(y , η)π(x , ω)

csymp

((x , ω), (y , η)

)= c((x , ω), (y , η)

)c((y , η), (x , ω)

)c((x , ω), (y , η)

)= e2πiy ·ω

Short-Time Fourier Transform:Vg f (x , ω) = 〈f , π(x , ω)g〉.







csymp

((x , ω), (y , η)

)= c((x , ω), (y , η)

)c((y , η), (x , ω)

)c((x , ω), (y , η)

)= e2πiy ·ω








csymp

((x , ω), (y , η)

)= c((x , ω), (y , η)

)c((y , η), (x , ω)

)c((x , ω), (y , η)

)= e2πiy ·ω








csymp

((x , ω), (y , η)

)= c((x , ω), (y , η)

)c((y , η), (x , ω)

)c((x , ω), (y , η)

)= e2πiy ·ω



Integrated Schrodinger Representation

1 (L1(R2d), c) is an involutive Banach algebra:

(F ]G )(z) =

∫∫F (z ′)G (z − z ′)c(z , z − z ′)dz

F ∗(z) = c(z , z)F (z).

2 (x , ω) 7→ VωUx is a non-degenerate faithful involutiverepresentation πL1 of (L1(R2d), c),πL1(F ) =

∫∫F (x , ω)VωUxdxdω.

3 πL1(F )πL1(G ) = πL1(F ]G ) and πL1(F )∗ = πL1(F ∗).

4 PF =∫∫

F (x , ω)VωUxdxdω is a projection in (L1(R2d), c)if and only if F ]F = F ,F = F ∗.A natural choice for F isVgg .




(F ]G )(z) =

∫∫F (z ′)G (z − z ′)c(z , z − z ′)dz

F ∗(z) = c(z , z)F (z).




4 PF =∫∫





(F ]G )(z) =

∫∫F (z ′)G (z − z ′)c(z , z − z ′)dz

F ∗(z) = c(z , z)F (z).




4 PF =∫∫

F (x , ω)VωUxdxdω is a projection in (L1(R2d), c)if and only if F ]F = F ,F = F ∗.

A natural choice for F isVgg .




(F ]G )(z) =

∫∫F (z ′)G (z − z ′)c(z , z − z ′)dz

F ∗(z) = c(z , z)F (z).




4 PF =∫∫





(F ]G )(z) =

∫∫F (z ′)G (z − z ′)c(z , z − z ′)dz

F ∗(z) = c(z , z)F (z).




4 PF =∫∫



Feichtinger’s algebra

Definition:

Feichtinger’s algebra S0(Rd) = f ∈ L2(Rd) : ‖f ‖S0 =∫∫R2d |〈f , π(x , ω)g〉|dxdω <∞, also denoted by M1(Rd).

S0(Rd) is invariant under time-frequency shifts:‖π(u, η)g‖S0 ≤ ‖g‖S0 .

S0(Rd) is a Banach algebra w.r.t point-wise multiplication.

S0(R2d) is invariant under symplectic Fourier transformF (z) =

∫∫R2d F (z)e2πiΩ(z,z)dz .

If f , g ∈ S0(Rd), then Vg f ∈ S0(R2d).

Suppose f1, f2, g1, g2 ∈ S0(Rd).

(Vg1f1 · Vg2f2)(z) = (Vf2f1 · Vg2g1)(z)



Definition:








(Vg1f1 · Vg2f2)(z) = (Vf2f1 · Vg2g1)(z)



Definition:








(Vg1f1 · Vg2f2)(z) = (Vf2f1 · Vg2g1)(z)



Definition:








(Vg1f1 · Vg2f2)(z) = (Vf2f1 · Vg2g1)(z)



Definition:








(Vg1f1 · Vg2f2)(z) = (Vf2f1 · Vg2g1)(z)



Definition:








(Vg1f1 · Vg2f2)(z) = (Vf2f1 · Vg2g1)(z)



Definition:








(Vg1f1 · Vg2f2)(z) = (Vf2f1 · Vg2g1)(z)


Facts

Consider G(g ,R2d) and the associated analysis andsynthesis mapping

f 7→ (Vg f (z)), A 7→∫∫

A(z)π(z)gdz

For f , g in S0(Rd) and F in S0(R2d) these two mappingsare well-defined.

Consider the pseudodifferential operator

H =

∫∫R2d

A(z)π(z)dz

for A ∈ S0(R2d). Then H is trace-class.


Facts


f 7→ (Vg f (z)), A 7→∫∫

A(z)π(z)gdz



H =

∫∫R2d

A(z)π(z)dz



Facts


f 7→ (Vg f (z)), A 7→∫∫

A(z)π(z)gdz



H =

∫∫R2d

A(z)π(z)dz



Facts


f 7→ (Vg f (z)), A 7→∫∫

A(z)π(z)gdz



H =

∫∫R2d

A(z)π(z)dz



Different Perspective

Left action of K(L2(Rd)) on S0(Rd):For F ∈ S0(R2d) and g ∈ S0(Rd)

π(F ) · g =

∫∫F (z)π(z)gdz

Inner product with values in K(L2(Rd))

K〈f , g〉 =

∫∫Vg f (z)π(z)

for f , g ∈ S0(Rd).

K〈π(A) · f , g〉 = π(A)K〈f , g〉, i.e. the right hand sideamounts to A]Vg f .




π(F ) · g =

∫∫F (z)π(z)gdz


K〈f , g〉 =

∫∫Vg f (z)π(z)






π(F ) · g =

∫∫F (z)π(z)gdz


K〈f , g〉 =

∫∫Vg f (z)π(z)






π(F ) · g =

∫∫F (z)π(z)gdz


K〈f , g〉 =

∫∫Vg f (z)π(z)






π(F ) · g =

∫∫F (z)π(z)gdz


K〈f , g〉 =

∫∫Vg f (z)π(z)




Hilbert K-module

Theorem:

S0(Rd) becomes a full left Hilbert K-module KV whencompleted with respect to the norm K‖f ‖ = ‖K〈f , f 〉‖1/2 forf ∈ S0(Rd)

The proof relies on the properties of S0(Rd) and L1(R2d , c).

Rank-one KV -module operators are just the rank-one operatorson L2(Rd) Θg ,hf = 〈f , g〉 · h.


Hilbert K-module

Theorem:


The proof relies on the properties of S0(Rd) and L1(R2d , c).Rank-one KV -module operators are just the rank-one operatorson L2(Rd) Θg ,hf = 〈f , g〉 · h.


Hilbert K-module

Theorem:


The proof relies on the properties of S0(Rd) and L1(R2d , c).Rank-one KV -module operators are just the rank-one operatorson L2(Rd) Θg ,hf = 〈f , g〉 · h.


Equivalence bimodule between K and C

Since we need a right Hilbert C-module, we take the conjugateHilbert space of L2(Rd) with inner product〈f , g〉C = 〈g , f 〉L2(Rd ).

K〈f , g〉 · h = f · 〈g , h〉C

Equivalent to

〈Vg f ,Vkh〉L2(R2d ) = 〈f , k〉L2(Rd )〈h, g〉L2(Rd )

Theorem:

S0(Rd) is an equivalence bimodule between K and C.

Morita-Rieffel equivalent C ∗-algebras have equivalentcategories of representations.




K〈f , g〉 · h = f · 〈g , h〉CEquivalent to


Theorem:








Theorem:








Theorem:








Theorem:




Noncommutative torus

Let Θ = (θj ,k) be a real skew-symmetric matrix and letU1, ...,U2d be unitary operators. Then the noncommutativetorus AΘ is defined as the universal C ∗-algebra generated byUj ’s satisfying

UkUj = e2πiθj,k UjUk .

For the connection between noncommutative tori and Gaboranalysis it is more useful to realize AΘ as the group C ∗-algebraof a lattice Λ in R2d .












Twisted group algebras

Let Λ be a lattice in R2d and c a continuous 2-cocycle withvalues in T.Then the twisted group algebra `1(Λ, c) is `1(Λ) with twistedconvolution \ as multiplication and ∗ as involution.

Twisted convolution of a and b is defined by

a\b(λ) =∑µ∈Λ

a(µ)b(λ− µ)c(µ, λ− µ) for λ, µ ∈ Λ,

Twisted involution of a given by

a∗(λ) = c(λ, λ)a(−λ) for λ ∈ Λ.

C ∗(Λ, c) is the enveloping C ∗-algebra of `1(Λ, c).Pick a basis v1, ..., v2d for Λ. Then Uvj together with(θvj ,vk ) provides a description of the noncommutative torus.



Let Λ be a lattice in R2d and c a continuous 2-cocycle withvalues in T.Then the twisted group algebra `1(Λ, c) is `1(Λ) with twistedconvolution \ as multiplication and ∗ as involution.Twisted convolution of a and b is defined by

a\b(λ) =∑µ∈Λ



a∗(λ) = c(λ, λ)a(−λ) for λ ∈ Λ.





a\b(λ) =∑µ∈Λ



a∗(λ) = c(λ, λ)a(−λ) for λ ∈ Λ.

C ∗(Λ, c) is the enveloping C ∗-algebra of `1(Λ, c).

Pick a basis v1, ..., v2d for Λ. Then Uvj together with(θvj ,vk ) provides a description of the noncommutative torus.




a\b(λ) =∑µ∈Λ



a∗(λ) = c(λ, λ)a(−λ) for λ ∈ Λ.





a\b(λ) =∑µ∈Λ



a∗(λ) = c(λ, λ)a(−λ) for λ ∈ Λ.



Representations of noncommutative torus

The mapping of λ 7→ π(λ) is a projective representation of Λ,which gives a non-degenerate involutive representation of`1(Λ, c) by

πΛ(a) :=∑λ∈Λ

a(λ)π(λ) for a = (a(λ)) ∈ `1(Λ).

πΛ(a\b) = πΛ(a)πΛ(b)

πΛ(a∗) = πΛ(a)∗

This involutive representation of `1(Λ, c) is faithful,πΛ(a) = 0 implies a = 0.Denote the image of the map a 7→ πΛ(a) for a ∈ `1

s (Λ) byA1

s (Λ, c).





a(λ)π(λ) for a = (a(λ)) ∈ `1(Λ).

πΛ(a\b) = πΛ(a)πΛ(b)πΛ(a∗) = πΛ(a)∗


s (Λ) byA1

s (Λ, c).





a(λ)π(λ) for a = (a(λ)) ∈ `1(Λ).


This involutive representation of `1(Λ, c) is faithful,

πΛ(a) = 0 implies a = 0.Denote the image of the map a 7→ πΛ(a) for a ∈ `1

s (Λ) byA1

s (Λ, c).





a(λ)π(λ) for a = (a(λ)) ∈ `1(Λ).


This involutive representation of `1(Λ, c) is faithful,πΛ(a) = 0 implies a = 0.

Denote the image of the map a 7→ πΛ(a) for a ∈ `1s (Λ) by

A1s (Λ, c).





a(λ)π(λ) for a = (a(λ)) ∈ `1(Λ).



s (Λ) byA1

s (Λ, c).





a(λ)π(λ) for a = (a(λ)) ∈ `1(Λ).



s (Λ) byA1

s (Λ, c).


Spectrally invariant subalgebras of C ∗(Λ, c)


A1s (Λ, c) = A ∈ B(L2(R)) : A =

∑λ

a(λ)π(λ), ‖a‖`1s<∞

smooth noncommutative torus A∞(Λ, c) =⋂

s≥0A1s (Λ, c)

Theorem:

Let Λ be a lattice in R2d . Then A1s (Λ, c) and A∞(Λ, c) are

spectrally invariant subalgebras of the noncommutative torusC ∗(Λ, c).

The statement for A∞(Λ, c) was obtained by Connes in 1980and the one for A1

s (Λ, c) is due to Grochenig-Leinert andindependently by Rosenberg.




A1s (Λ, c) = A ∈ B(L2(R)) : A =

∑λ

a(λ)π(λ), ‖a‖`1s<∞


s≥0A1s (Λ, c)

Theorem:








A1s (Λ, c) = A ∈ B(L2(R)) : A =

∑λ

a(λ)π(λ), ‖a‖`1s<∞


s≥0A1s (Λ, c)

Theorem:








A1s (Λ, c) = A ∈ B(L2(R)) : A =

∑λ

a(λ)π(λ), ‖a‖`1s<∞


s≥0A1s (Λ, c)

Theorem:








A1s (Λ, c) = A ∈ B(L2(R)) : A =

∑λ

a(λ)π(λ), ‖a‖`1s<∞


s≥0A1s (Λ, c)

Theorem:






Equivalence bimodule between noncommutativetori

Quote from Rieffel’s seminal paper from 1988:We need suitable spaces of functions for the construction ofequivalence bimodules between noncommutative tori.

In the present context this means we need a space of functionson Rd which behaves well under both the Fourier transformand restriction to subgroups.As suggested by Weil, the appropriate space is the space S(Rd)of Schwartz functions.Feichtinger constructed a Banach algebra in 1980, S0(Rd),that turned out to be a very important substitute of theSchwartz space.Shortly, after he introduced the class of modulation spacesMp,q

s (Rd) and it turned out that S0(Rd) is the modulationspace M1(Rd).



Quote from Rieffel’s seminal paper from 1988:We need suitable spaces of functions for the construction ofequivalence bimodules between noncommutative tori.In the present context this means we need a space of functionson Rd which behaves well under both the Fourier transformand restriction to subgroups.

As suggested by Weil, the appropriate space is the space S(Rd)of Schwartz functions.Feichtinger constructed a Banach algebra in 1980, S0(Rd),that turned out to be a very important substitute of theSchwartz space.Shortly, after he introduced the class of modulation spacesMp,q




Quote from Rieffel’s seminal paper from 1988:We need suitable spaces of functions for the construction ofequivalence bimodules between noncommutative tori.In the present context this means we need a space of functionson Rd which behaves well under both the Fourier transformand restriction to subgroups.As suggested by Weil, the appropriate space is the space S(Rd)of Schwartz functions.

Feichtinger constructed a Banach algebra in 1980, S0(Rd),that turned out to be a very important substitute of theSchwartz space.Shortly, after he introduced the class of modulation spacesMp,q




Quote from Rieffel’s seminal paper from 1988:We need suitable spaces of functions for the construction ofequivalence bimodules between noncommutative tori.In the present context this means we need a space of functionson Rd which behaves well under both the Fourier transformand restriction to subgroups.As suggested by Weil, the appropriate space is the space S(Rd)of Schwartz functions.Feichtinger constructed a Banach algebra in 1980, S0(Rd),that turned out to be a very important substitute of theSchwartz space.Shortly, after he introduced the class of modulation spacesMp,q




Quote from Rieffel’s seminal paper from 1988:We need suitable spaces of functions for the construction ofequivalence bimodules between noncommutative tori.In the present context this means we need a space of functionson Rd which behaves well under both the Fourier transformand restriction to subgroups.As suggested by Weil, the appropriate space is the space S(Rd)of Schwartz functions.Feichtinger constructed a Banach algebra in 1980, S0(Rd),that turned out to be a very important substitute of theSchwartz space.Shortly, after he introduced the class of modulation spacesMp,q



Symplectic Fourier transform

Observe that the characters of Rd × Rd are of the formz 7→ χs(z) = e2πiΩ(z,z ′) for some z ′ ∈ Rd × Rd ,where the standard symplectic form Ω of z = (x , ω) andz ′ = (y , η) is defined by

Ω(z , z ′) = y · ω − x · η.Consequently, the dual group of Rd × Rd is Rd × Rd .In terms of the Euclidean inner product 〈., .〉 on R2d thesymplectic form Ω can be expressed as follows:pause

Ω(z , z ′) = 〈Jz , z ′〉 for J =

(0 Id−Id 0

).

Λ = z ∈ R2d : e2πiΩ(λ,z) = 1 for all λ ∈ Λ.The relation between Λ⊥ and the adjoint lattice Λ isΛ = JΛ⊥.




Ω(z , z ′) = y · ω − x · η.Consequently, the dual group of Rd × Rd is Rd × Rd .

In terms of the Euclidean inner product 〈., .〉 on R2d thesymplectic form Ω can be expressed as follows:pause

Ω(z , z ′) = 〈Jz , z ′〉 for J =

(0 Id−Id 0

).






Ω(z , z ′) = 〈Jz , z ′〉 for J =

(0 Id−Id 0

).






Ω(z , z ′) = 〈Jz , z ′〉 for J =

(0 Id−Id 0

).

Λ = z ∈ R2d : e2πiΩ(λ,z) = 1 for all λ ∈ Λ.The relation between Λ⊥ and the adjoint lattice Λ isΛ = JΛ⊥. Luef, Franz Hilbert C∗-modules over noncommutative tori




Ω(z , z ′) = 〈Jz , z ′〉 for J =

(0 Id−Id 0

).

Λ = z ∈ R2d : e2πiΩ(λ,z) = 1 for all λ ∈ Λ.The relation between Λ⊥ and the adjoint lattice Λ isΛ = JΛ⊥. Luef, Franz Hilbert C∗-modules over noncommutative tori

Feichtinger’s algebra - Properties

Definition:

Weighted versions of Feichtinger’s algebraM1

s (Rd) = f ∈ L2(Rd) : ‖f ‖M1s

=∫∫R2d |〈π(x , ω)g〉|(1 + |x |2 + |ω|2)s/2dxdω <∞

If f , g ∈ M1s (Rd), then Vg f ∈ M1

s (R2d).

M1(R2d) is invariant under the symplectic Fouriertransform: F (z) =


(Vg1f1 · Vg2f2)(z) = (Vf2f1 · Vg2g1)(z)



Definition:


s (Rd) = f ∈ L2(Rd) : ‖f ‖M1s

=∫∫R2d |〈π(x , ω)g〉|(1 + |x |2 + |ω|2)s/2dxdω <∞


s (R2d).



(Vg1f1 · Vg2f2)(z) = (Vf2f1 · Vg2g1)(z)



Definition:


s (Rd) = f ∈ L2(Rd) : ‖f ‖M1s

=∫∫R2d |〈π(x , ω)g〉|(1 + |x |2 + |ω|2)s/2dxdω <∞


s (R2d).



(Vg1f1 · Vg2f2)(z) = (Vf2f1 · Vg2g1)(z)



Definition:


s (Rd) = f ∈ L2(Rd) : ‖f ‖M1s

=∫∫R2d |〈π(x , ω)g〉|(1 + |x |2 + |ω|2)s/2dxdω <∞


s (R2d).



(Vg1f1 · Vg2f2)(z) = (Vf2f1 · Vg2g1)(z)


Janssen representation

Poisson summation formula for the symplectic Fouriertransform: ∑

λ∈Λ

F (λ) = vol(Λ)−1∑λ∈Λ

F (λ).

Fundamental Identity of Gabor analysis:

∑λ∈Λ

〈f , π(λ)g〉〈π(λ)h, k〉 = vol(Λ)−1∑λ∈Λ

〈f , π(λ)k〉〈π(λ)h, g〉.




λ∈Λ


F (λ).

Fundamental Identity of Gabor analysis:∑λ∈Λ

〈f , π(λ)g〉〈π(λ)h, k〉 = vol(Λ)−1∑λ∈Λ

〈f , π(λ)k〉〈π(λ)h, g〉.




λ∈Λ


F (λ).

Fundamental Identity of Gabor analysis:∑λ∈Λ

〈f , π(λ)g〉〈π(λ)h, k〉 = vol(Λ)−1∑λ∈Λ

〈f , π(λ)k〉〈π(λ)h, g〉.



Theorem:

For f , g , h ∈ M1s (Rd) or in S(Rd) we have that∑

λ∈Λ

〈f , π(λ)g〉〈π(λ)h = vol(Λ)−1∑λ∈Λ

〈h, π(λ)g〉π(λ)f ,


Rieffel’s Theorem

A deep result of Rieffel relates these different noncommutativetori.

Theorem:

C ∗(Λ, c) and C ∗(Λ, c) are Morita-Rieffel equivalent.


Rieffel’s Theorem

A deep result of Rieffel relates these different noncommutativetori.

Theorem:

C ∗(Λ, c) and C ∗(Λ, c) are Morita-Rieffel equivalent.


Gabor frames – traditional point of view

Let G(g ,Λ) = π(λ)g : λ ∈ Λ be a Gabor system.

analysis operator: Cg f = (〈f , π(λ)g〉)λ∈Λ

synthesis operator: Dga =∑

λ∈Λ a(λ)π(λ)f

frame operator: Sg ,Λf =∑

λ∈Λ〈f , π(λ)g〉π(λ)g

G(g ,Λ) is a frame for L2(Rd) if Sg ,Λ is invertible onL2(Rd).

A‖f ‖22 ≤

∑λ∈Λ

|〈f , π(λ)g〉|2 ≤ B‖f ‖22










A‖f ‖22 ≤

∑λ∈Λ

|〈f , π(λ)g〉|2 ≤ B‖f ‖22










A‖f ‖22 ≤

∑λ∈Λ

|〈f , π(λ)g〉|2 ≤ B‖f ‖22










A‖f ‖22 ≤

∑λ∈Λ

|〈f , π(λ)g〉|2 ≤ B‖f ‖22










A‖f ‖22 ≤

∑λ∈Λ

|〈f , π(λ)g〉|2 ≤ B‖f ‖22










A‖f ‖22 ≤

∑λ∈Λ

|〈f , π(λ)g〉|2 ≤ B‖f ‖22










A‖f ‖22 ≤

∑λ∈Λ

|〈f , π(λ)g〉|2 ≤ B‖f ‖22


Discrete reconstruction

f = S−1Sf =∑λ∈Λ

〈f , π(λ)S−1g〉π(λ)g

= SS−1f =∑λ∈Λ

〈f , π(λ)g〉π(λ)S−1g

= S−1/2SS−1/2f =∑λ∈Λ

〈f , π(λ)S−1/2g〉π(λ)S−1/2g .

canonical dual atom g := (SΛg ,g )−1g

canonical tight atom h0 := (SΛg ,g )−1/2g


Observation

Left action of A1s (Λ, c) on M1

s (Rd) by

Dag = πΛ(a)·g =[∑λ∈Λ

a(λ)π(λ)]g for a ∈ `1

s (Λ), g ∈ M1s (Rd)

Λ〈f , g〉 =∑λ∈Λ

〈f , π(λ)g〉π(λ)

For f , g ∈ M1s (Rd) define

Λ〈f , g〉 = πΛ(Vg f ) =∑

λ∈Λ〈f , π(λ)g〉π(λ)

Λ〈f , g〉 = Λ〈g , f 〉∗

Λ〈πΛ(a)f , g〉 = πΛ(a) Λ〈f , g〉Λ〈f , f 〉 is positive in C ∗(Λ, c), acutally in A1

s (Λ, c)


Observation


s (Rd) by



s (Λ), g ∈ M1s (Rd)

Λ〈f , g〉 =∑λ∈Λ



Λ〈f , g〉 = πΛ(Vg f ) =∑


Λ〈f , g〉 = Λ〈g , f 〉∗


s (Λ, c)


Observation


s (Rd) by



s (Λ), g ∈ M1s (Rd)

Λ〈f , g〉 =∑λ∈Λ



Λ〈f , g〉 = πΛ(Vg f ) =∑


Λ〈f , g〉 = Λ〈g , f 〉∗


s (Λ, c)


Observation


s (Rd) by



s (Λ), g ∈ M1s (Rd)

Λ〈f , g〉 =∑λ∈Λ



Λ〈f , g〉 = πΛ(Vg f ) =∑


Λ〈f , g〉 = Λ〈g , f 〉∗


s (Λ, c)


Observation


s (Rd) by



s (Λ), g ∈ M1s (Rd)

Λ〈f , g〉 =∑λ∈Λ



Λ〈f , g〉 = πΛ(Vg f ) =∑


Λ〈f , g〉 = Λ〈g , f 〉∗

Λ〈πΛ(a)f , g〉 = πΛ(a) Λ〈f , g〉

Λ〈f , f 〉 is positive in C ∗(Λ, c), acutally in A1s (Λ, c)


Observation


s (Rd) by



s (Λ), g ∈ M1s (Rd)

Λ〈f , g〉 =∑λ∈Λ



Λ〈f , g〉 = πΛ(Vg f ) =∑


Λ〈f , g〉 = Λ〈g , f 〉∗


s (Λ, c)


Observation


s (Rd) by



s (Λ), g ∈ M1s (Rd)

Λ〈f , g〉 =∑λ∈Λ



Λ〈f , g〉 = πΛ(Vg f ) =∑


Λ〈f , g〉 = Λ〈g , f 〉∗


s (Λ, c)


Observation


s (Rd) by



s (Λ), g ∈ M1s (Rd)

Λ〈f , g〉 =∑λ∈Λ



Λ〈f , g〉 = πΛ(Vg f ) =∑


Λ〈f , g〉 = Λ〈g , f 〉∗


s (Λ, c)


Hilbert C ∗(Λ, c)-module

Theorem:

M1s (Rd) becomes a full left Hilbert C ∗(Λ, c)-module ΛV w.r.t

to right action on M1s (Rd) and the inner product 〈., .〉Λ when

completed w.r.t. ‖f ‖Λ = ‖〈f , f 〉Λ‖1/2op .

There is an analogous result for the opposite C ∗-algebra ofC ∗(Λ, c), i.e. C ∗(Λ, c).

What is the appropriate right action of C ∗(Λ, c) on M1s (Rd)?



Theorem:




There is an analogous result for the opposite C ∗-algebra ofC ∗(Λ, c), i.e. C ∗(Λ, c).What is the appropriate right action of C ∗(Λ, c) on M1

s (Rd)?



Theorem:




There is an analogous result for the opposite C ∗-algebra ofC ∗(Λ, c), i.e. C ∗(Λ, c).What is the appropriate right action of C ∗(Λ, c) on M1

s (Rd)?



FIGA = vol(Λ)−1∑λ∈Λ

〈π(λ)∗f , k〉〈π(λ)h, g〉

=⟨vol(Λ)−1

∑λ∈Λ

〈π(λ)h, g〉π(λ)∗f , k⟩

g · πΛ(b) =∑λ∈Λ

π(λ)∗gb(λ) b ∈ `1s (Λ), g ∈ M1

s (R)

〈f , g〉Λ =∑

π(λ)∗〈g , π(λ)f 〉 f , g ∈ M1s (Rd)

Theorem:

M1s (Rd) becomes a full right Hilbert C ∗(Λ, c)-module VΛ

w.r.t to right action on M1s (Rd) and the inner product 〈., .〉Λ

when completed w.r.t. ‖f ‖Λ = ‖〈f , f 〉Λ‖1/2op .




〈π(λ)∗f , k〉〈π(λ)h, g〉

=⟨vol(Λ)−1

∑λ∈Λ

〈π(λ)h, g〉π(λ)∗f , k⟩


π(λ)∗gb(λ) b ∈ `1s (Λ), g ∈ M1

s (R)

〈f , g〉Λ =∑

π(λ)∗〈g , π(λ)f 〉 f , g ∈ M1s (Rd)

Theorem:







〈π(λ)∗f , k〉〈π(λ)h, g〉

=⟨vol(Λ)−1

∑λ∈Λ

〈π(λ)h, g〉π(λ)∗f , k⟩


π(λ)∗gb(λ) b ∈ `1s (Λ), g ∈ M1

s (R)

〈f , g〉Λ =∑

π(λ)∗〈g , π(λ)f 〉 f , g ∈ M1s (Rd)

Theorem:







〈π(λ)∗f , k〉〈π(λ)h, g〉

=⟨vol(Λ)−1

∑λ∈Λ

〈π(λ)h, g〉π(λ)∗f , k⟩


π(λ)∗gb(λ) b ∈ `1s (Λ), g ∈ M1

s (R)

〈f , g〉Λ =∑

π(λ)∗〈g , π(λ)f 〉 f , g ∈ M1s (Rd)

Theorem:





Adjointable C ∗(Λ, c)-module operators

By definition T is adjointable if there exists a T ? such that

Λ〈f ,Tg〉 = Λ〈f ,T ?g〉∑λ∈Λ

〈Tf , π(λ)g〉π(λ) =∑λ∈Λ

〈f , π(λ)T ?g〉π(λ)

T ? exists only if Tπ(λ) = π(λ)T for all λ ∈ Λ, such anoperator T is Λ-invariant.

Therefore, T ? is the adjoint T ∗ of a Λ-invariant operator.Most important example, ΘΛ

g ,hf = Λ〈f , g〉 · h and its adjoint is

ΘΛh,g .

ΘΛg ,h are known as Gabor frame-type operator






〈f , π(λ)T ?g〉π(λ)

T ? exists only if Tπ(λ) = π(λ)T for all λ ∈ Λ, such anoperator T is Λ-invariant.Therefore, T ? is the adjoint T ∗ of a Λ-invariant operator.

Most important example, ΘΛg ,hf = Λ〈f , g〉 · h and its adjoint is

ΘΛh,g .







〈f , π(λ)T ?g〉π(λ)

T ? exists only if Tπ(λ) = π(λ)T for all λ ∈ Λ, such anoperator T is Λ-invariant.Therefore, T ? is the adjoint T ∗ of a Λ-invariant operator.Most important example, ΘΛ


ΘΛh,g .







〈f , π(λ)T ?g〉π(λ)

T ? exists only if Tπ(λ) = π(λ)T for all λ ∈ Λ, such anoperator T is Λ-invariant.Therefore, T ? is the adjoint T ∗ of a Λ-invariant operator.Most important example, ΘΛ


ΘΛh,g .



Invertible adjointable C ∗(Λ, c)-module operators

If ΘΛg ,g is invertible, then G (g ,Λ) is a Gabor frame for L2(Rd).

Equivalently, ΛV is singly generated.

There are various criteria about the invertibility of the Gaborframe operator.

For g ∈ M1(Rd) the Gabor frame operator ΘΛg ,g is

invertible if and only if the analysis operator C Λg is

one-to-one from M1(Rd)′ to `∞(Λ).For g ∈ M1(Rd) the Gabor frame operator ΘΛ

g ,g is

invertible if and only if the synthesis operator DΛg is

one-to-one from `∞(Λ) to M1(Rd)′.For g ∈ M1(Rd) the Gabor frame operator has as Weylsymbol σg ,Λ =

∑λ∈Λ TλW (g , g), where W (g , g) is the

Wigner distribution.Then ΘΛ

g ,g is invertible if and only if σg ,Λ is invertible in

M∞,1(R2d) with respect to twisted convolution.




Equivalently, ΛV is singly generated.There are various criteria about the invertibility of the Gaborframe operator.



one-to-one from M1(Rd)′ to `∞(Λ).















g ,g is




Wigner distribution.

Then ΘΛg ,g is invertible if and only if σg ,Λ is invertible in









g ,g is






M∞,1(R2d) with respect to twisted convolution.Luef, Franz Hilbert C∗-modules over noncommutative tori







g ,g is






M∞,1(R2d) with respect to twisted convolution.Luef, Franz Hilbert C∗-modules over noncommutative tori

Morita equivalent C ∗-algebras in Gabor analysis

Therefore V0 is an equivalence bimodule between A0 and B0.If we are in the situation of the preceding theorem, then we callthe algebras A0 and B0 Morita-Rieffel equivalent.

The inner products Λ〈., .〉 and 〈., .〉Λ satisfy Rieffel’sassociativity condition

Λ〈f , g〉 · h = f 〈g , h〉Λ

rank-one operators on the Hilbert C ∗(Λ, c)-module:ΘΛ

f ,gh = Λ〈f , g〉 · h = SΛg ,hf are Gabor frame-type operators

and the associativity condition is the Janssen representation forthese Gabor frame-type operators.

Theorem:

M1s (Rd) is an equivalence bimodule between C ∗(Λ, c) and

C ∗(Λ, c). Moreover M1s (Rd) is an equivalence bimodule

between A1s (Λ, c) and A1

s (Λ, c).



Therefore V0 is an equivalence bimodule between A0 and B0.If we are in the situation of the preceding theorem, then we callthe algebras A0 and B0 Morita-Rieffel equivalent.The inner products Λ〈., .〉 and 〈., .〉Λ satisfy Rieffel’sassociativity condition

Λ〈f , g〉 · h = f 〈g , h〉Λ




Theorem:




s (Λ, c).




Λ〈f , g〉 · h = f 〈g , h〉Λ




Theorem:




s (Λ, c).




Λ〈f , g〉 · h = f 〈g , h〉Λ




Theorem:




s (Λ, c).




Λ〈f , g〉 · h = f 〈g , h〉Λ




Theorem:




s (Λ, c).Luef, Franz Hilbert C∗-modules over noncommutative tori



Λ〈f , g〉 · h = f 〈g , h〉Λ




Theorem:




s (Λ, c).Luef, Franz Hilbert C∗-modules over noncommutative tori

Projective modules over noncommutative tori

Theorem:

M1s (Rd) is a finitely generated projective right A1(Λ, c),i.e.

there exist g1, ..., gn in M1s (Rd)

f =n∑

i=1

Λ〈f , gi 〉 · gi =n∑

i=1

f 〈gi , gi 〉Λ .

Λ〈f , f 〉 =n∑

i=1

Λ〈gi , f 〉Λ〈f , gi 〉

for all f in M1s (Rd) (or S(Rd)).In particular

‖f ‖22 =

n∑i=1

∑λ∈Λ

|〈f , π(λ)gi 〉|2.

i.e. G(g1, ..., gn,Λ) is a multi-window Gabor frame forM1

s (Rd) (or S(Rd)).



Theorem:



f =n∑

i=1

Λ〈f , gi 〉 · gi =n∑

i=1


Λ〈f , f 〉 =n∑

i=1


for all f in M1s (Rd) (or S(Rd)).

In particular

‖f ‖22 =

n∑i=1

∑λ∈Λ

|〈f , π(λ)gi 〉|2.


s (Rd) (or S(Rd)).



Theorem:



f =n∑

i=1

Λ〈f , gi 〉 · gi =n∑

i=1


Λ〈f , f 〉 =n∑

i=1



‖f ‖22 =

n∑i=1

∑λ∈Λ

|〈f , π(λ)gi 〉|2.


s (Rd) (or S(Rd)).



Theorem:



f =n∑

i=1

Λ〈f , gi 〉 · gi =n∑

i=1


Λ〈f , f 〉 =n∑

i=1



‖f ‖22 =

n∑i=1

∑λ∈Λ

|〈f , π(λ)gi 〉|2.


s (Rd) (or S(Rd)).


Generalizations

All our results hold for Λ a lattice in G × G , where G is alocally compact abelian group.

There exist weighted variants of our statements:

v is submultiplicative, i.e. v(x + y , ω+ η) ≤ v(x , ω)v(y , η)for all (x , ω), (y , η) ∈ R2d .

v(x , ω) ≥ 1 and v(−x ,−ω) = v(x , ω) for all (x , ω) ∈ R2d .

(`1v (Λ), c) is a Banach algebra with continuous involution.

Grochenig: A1v (Λ, c) is spectrally invariant in C ∗(Λ, c) if

and only if v is a GRS-weight, i.e. lim v(nλ)1/n = 1 for allλ ∈ Λ.


Generalizations









Generalizations









Generalizations









Generalizations









Generalizations









Projections in a C ∗-algebra

Lemma:

• Let g be in AVB. Then Pg := A〈g , g〉 is a projection in Aif and only if g〈g , g〉B = g .

• Any element g in the unit sphere S(VB) of the HilbertB-module VB, i.e. of all g ∈ VB such that 〈g , g〉B = IB,gives a projection Pg in A.

In particular g0 = g〈g , g〉−1/2B is in S(VB), therefore Pg0 is

a (canonical) projection in A.



Lemma:







Lemma:






Projections in noncommutative tori

Recall p is called a projection if p = p∗ = p2.

Theorem:

Let G(g ,Λ) be a Gabor system on L2(Rd).

Then pg = Λ〈g , g〉 isa projection in C ∗(Λ, c) if and only if one of the followingcondition holds:

G(g ,Λ) is a tight Gabor frame for L2(Rd).

G(g ,Λ) is an orthogonal system.

〈g , g〉Λ = I .

〈g , π(λ)g〉 = vol(Λ)δλ,0 for all λ ∈ Λ.

In particular the canonical tight Gabor atom h0 := (SΛg ,g )−1/2g

yields a projection ph0 in C ∗(Λ, c).




Theorem:

Let G(g ,Λ) be a Gabor system on L2(Rd).Then pg = Λ〈g , g〉 isa projection in C ∗(Λ, c) if and only if one of the followingcondition holds:



〈g , g〉Λ = I .







Theorem:




〈g , g〉Λ = I .







Theorem:




〈g , g〉Λ = I .







Theorem:




〈g , g〉Λ = I .







Theorem:




〈g , g〉Λ = I .







Theorem:




〈g , g〉Λ = I .







Theorem:




〈g , g〉Λ = I .







Theorem:




〈g , g〉Λ = I .





Projections in noncommutative tori – continued

Lema:

Let g be in ΛVΛ . Then Pg := Λ〈g , g〉 is a projection inC ∗(Λ, c) if and only if g〈g , g〉Λ = g . If g ∈ M1

s (R) or S (R),then Pg gives a projection in A1

s (Λ, c) or A∞(Λ, c),respectively.

First we assume that g〈g , g〉Λ = g for some g in ΛVΛ . Thenwe have that

P2g = Λ〈g , g〉Λ〈g , g〉 = Λ

⟨Λ〈g , g〉g , g

⟩= Λ〈g〈g , g〉Λ , g〉 = Λ〈g , g〉 = Pg

and P∗g = Pg .

Now we suppose that Λ〈g , g〉 is a projection in C ∗(Λ, c). Thenan elementary computation yields the assertion:

Λ

⟨g〈g , g〉Λ − g , g〈g , g〉Λ − g

⟩= 0.



Lema:





P2g = Λ〈g , g〉Λ〈g , g〉 = Λ

⟨Λ〈g , g〉g , g

⟩= Λ〈g〈g , g〉Λ , g〉 = Λ〈g , g〉 = Pg

and P∗g = Pg .Now we suppose that Λ〈g , g〉 is a projection in C ∗(Λ, c). Thenan elementary computation yields the assertion:

Λ

⟨g〈g , g〉Λ − g , g〈g , g〉Λ − g

⟩= 0.



Lema:





P2g = Λ〈g , g〉Λ〈g , g〉 = Λ

⟨Λ〈g , g〉g , g

⟩= Λ〈g〈g , g〉Λ , g〉 = Λ〈g , g〉 = Pg


Λ

⟨g〈g , g〉Λ − g , g〈g , g〉Λ − g

⟩= 0.



Lema:





P2g = Λ〈g , g〉Λ〈g , g〉 = Λ

⟨Λ〈g , g〉g , g

⟩= Λ〈g〈g , g〉Λ , g〉 = Λ〈g , g〉 = Pg


Λ

⟨g〈g , g〉Λ − g , g〈g , g〉Λ − g

⟩= 0.


Consequences

The unit sphere of the Hilbert C ∗(Λ, c)-module VΛ isdefined by S(VΛ) = g ∈ VΛ : 〈g , g〉Λ = I, which is the setof all tight Gabor frames.g1(t) = (2)1/4e−πt

2a Gaussian, g2(t) = (π2 )1/2 1

cosh(πt) the

hyperbolic secant and g3(t) = e−π|t| the two-sided exponential.

Theorem:

Let Λ = αZ× βZ. Then pgi = Λ〈gi , gi 〉 is a projection inC ∗(Λ, c) if and only if αβ < 1.

The case of the Gaussian g1 is known as Boca’s projection. InManin’s work pg1 = Λ〈g1, g1〉 appear as quantum thetafunctions.


Consequences


2a Gaussian, g2(t) = (π2 )1/2 1

cosh(πt) the


Theorem:




Consequences


2a Gaussian, g2(t) = (π2 )1/2 1

cosh(πt) the


Theorem:




Consequences


2a Gaussian, g2(t) = (π2 )1/2 1

cosh(πt) the


Theorem:




Extensions

Feichtinger-Kaiblinger showed in particular that the set ofGabor frames G(g0,Λ) with g0 ∈ M1

s (Rd) is open, i.e. thereexists a ε > 0 such that for ‖g − g0‖M1

s< ε then G(g ,Λ) is

also a Gabor frame.Consequently, we have an open set of projections Λ〈g , g〉 closeto Λ〈g0, g0〉.

Actually, Feichtinger and Kaiblinger proved that one can alsovary the lattice.Therefore, a formulation in our setting amounts to incorporatethe fact that noncommutative tori are also give rise to acontinuous field of C ∗-algebras.


Extensions




also a Gabor frame.Consequently, we have an open set of projections Λ〈g , g〉 closeto Λ〈g0, g0〉.Actually, Feichtinger and Kaiblinger proved that one can alsovary the lattice.

Therefore, a formulation in our setting amounts to incorporatethe fact that noncommutative tori are also give rise to acontinuous field of C ∗-algebras.


Extensions




also a Gabor frame.Consequently, we have an open set of projections Λ〈g , g〉 closeto Λ〈g0, g0〉.Actually, Feichtinger and Kaiblinger proved that one can alsovary the lattice.Therefore, a formulation in our setting amounts to incorporatethe fact that noncommutative tori are also give rise to acontinuous field of C ∗-algebras.


Extensions






Extensions






Extensions

Rieffel demonstrated that viewed as quantum metric spaces oneis dealing with a continuous field of quantum metric spaces.Another problem fitting into this framework is theapproximation of continuous Gabor frames byfinite-dimensional Gabor frames.

In other words one is looking to control the changes ofprojective modules over “close” noncommutative tori.


Extensions

Rieffel demonstrated that viewed as quantum metric spaces oneis dealing with a continuous field of quantum metric spaces.Another problem fitting into this framework is theapproximation of continuous Gabor frames byfinite-dimensional Gabor frames.In other words one is looking to control the changes ofprojective modules over “close” noncommutative tori.


Extensions

Rieffel demonstrated that viewed as quantum metric spaces oneis dealing with a continuous field of quantum metric spaces.Another problem fitting into this framework is theapproximation of continuous Gabor frames byfinite-dimensional Gabor frames.In other words one is looking to control the changes ofprojective modules over “close” noncommutative tori.


Rotation algebras

Theorem:

L Let α be an irrational number. Then C ∗(αZ,R/Z) andC ∗(Z,R/αZ) are Morita-Rieffel equivalent.

C ∗(αZ,R/Z) is the C ∗-algebra for Z acting on thecircle by 2πα, and it is the completion of the ∗-algebraCc(T× Z) where the convolution product is given by

F ∗ G (t, n) =∑m∈Z

F (m, t)G (n −m, e−2πimθt)

and involution is given by

F ∗(n, t) = F (−n,−e−2πiθ)t

C ∗(Z,R/αZ) is the C ∗-algebra for Z acting on the circle by2π/α and it is isomorphic to C ∗(α−1Z,R/Z).


Rotation algebras

Theorem:



F ∗ G (t, n) =∑m∈Z



F ∗(n, t) = F (−n,−e−2πiθ)t



Rotation algebras

Theorem:



F ∗ G (t, n) =∑m∈Z



F ∗(n, t) = F (−n,−e−2πiθ)t



Morita-Rieffel equivalence for rotation algebras

Let A = C ∗(αZ,R/Z) and B = C ∗(Z,R/αZ). Then we defineon Cc(R) the structure of a right pre A Hilbert module and aleft pre B Hilbert module.For f , g ∈ Cc(R), F ∈ C ∗(αZ,R/Z) and G ∈ C ∗(Z,R/αZ) wedefine:

(f F )(t) =∑n∈Z

f (t − nα)F (nα, t − nα)

〈f , g〉A(mα, t) =∑n∈Z

f (r − n)g(r − n + mα)

(G f )(t) =∑n∈Z

G (n, t)f (t − n)

B〈f , g〉(m, t) =∑n∈Z

f (t − nα)g(t − nα−m).




(f F )(t) =∑n∈Z


〈f , g〉A(mα, t) =∑n∈Z

f (r − n)g(r − n + mα)

(G f )(t) =∑n∈Z

G (n, t)f (t − n)

B〈f , g〉(m, t) =∑n∈Z





(f F )(t) =∑n∈Z


〈f , g〉A(mα, t) =∑n∈Z

f (r − n)g(r − n + mα)

(G f )(t) =∑n∈Z

G (n, t)f (t − n)

B〈f , g〉(m, t) =∑n∈Z





(f F )(t) =∑n∈Z


〈f , g〉A(mα, t) =∑n∈Z

f (r − n)g(r − n + mα)

(G f )(t) =∑n∈Z

G (n, t)f (t − n)

B〈f , g〉(m, t) =∑n∈Z



Wiener amalgam spaces

In his work on Generalized Harmonic Analysis and TauberianTheorems Norbert Wiener introduced the space W (L∞, `1).Later H.-G. Feichtinger generalized Wiener’s space in his workon function spaces which allows one to measure global andlocal information W (B1,B2).

These spaces found applications in approximation theory,sampling theory, symbol classes of pseudo-differentialoperators, stochastic processes, etc.We restrict our discussion to the case of weighted analogs ofW (L∞, `1).Let v be a weight on Z such that:

• v(k) = v(−k),

• v(k + l) ≤ v(k)v(l),

• limn→∞ v(kn)1/n = 1.



In his work on Generalized Harmonic Analysis and TauberianTheorems Norbert Wiener introduced the space W (L∞, `1).Later H.-G. Feichtinger generalized Wiener’s space in his workon function spaces which allows one to measure global andlocal information W (B1,B2).These spaces found applications in approximation theory,sampling theory, symbol classes of pseudo-differentialoperators, stochastic processes, etc.

We restrict our discussion to the case of weighted analogs ofW (L∞, `1).Let v be a weight on Z such that:

• v(k) = v(−k),

• v(k + l) ≤ v(k)v(l),

• limn→∞ v(kn)1/n = 1.



In his work on Generalized Harmonic Analysis and TauberianTheorems Norbert Wiener introduced the space W (L∞, `1).Later H.-G. Feichtinger generalized Wiener’s space in his workon function spaces which allows one to measure global andlocal information W (B1,B2).These spaces found applications in approximation theory,sampling theory, symbol classes of pseudo-differentialoperators, stochastic processes, etc.We restrict our discussion to the case of weighted analogs ofW (L∞, `1).

Let v be a weight on Z such that:

• v(k) = v(−k),

• v(k + l) ≤ v(k)v(l),

• limn→∞ v(kn)1/n = 1.



In his work on Generalized Harmonic Analysis and TauberianTheorems Norbert Wiener introduced the space W (L∞, `1).Later H.-G. Feichtinger generalized Wiener’s space in his workon function spaces which allows one to measure global andlocal information W (B1,B2).These spaces found applications in approximation theory,sampling theory, symbol classes of pseudo-differentialoperators, stochastic processes, etc.We restrict our discussion to the case of weighted analogs ofW (L∞, `1).Let v be a weight on Z such that:

• v(k) = v(−k),

• v(k + l) ≤ v(k)v(l),

• limn→∞ v(kn)1/n = 1.



In his work on Generalized Harmonic Analysis and TauberianTheorems Norbert Wiener introduced the space W (L∞, `1).Later H.-G. Feichtinger generalized Wiener’s space in his workon function spaces which allows one to measure global andlocal information W (B1,B2).These spaces found applications in approximation theory,sampling theory, symbol classes of pseudo-differentialoperators, stochastic processes, etc.We restrict our discussion to the case of weighted analogs ofW (L∞, `1).Let v be a weight on Z such that:

• v(k) = v(−k),

• v(k + l) ≤ v(k)v(l),

• limn→∞ v(kn)1/n = 1.


Wiener amalgam spaces - continued

For α > 0 we set Iα = [0, α) and denote by χα thecharacteristic function of Iα.A function f belongs to W (L∞, `1

v ) if

‖f ‖W (L∞,`1v ) =

∑k∈Z‖f · Tkαχα‖L∞v(k) <∞.

The definition is independent of α and we have the followingembeddings:

S ⊂W (L∞, `1v ) ⊂W (L∞, `1) ⊂ L2.




v ) if

‖f ‖W (L∞,`1v ) =



S ⊂W (L∞, `1v ) ⊂W (L∞, `1) ⊂ L2.




v ) if

‖f ‖W (L∞,`1v ) =



S ⊂W (L∞, `1v ) ⊂W (L∞, `1) ⊂ L2.


Walnut representation of Gabor frame operator

Recall B = C ∗(Z,R/αZ).

Lemma:

Let g , h be in W (L∞, `1v ). Then 〈g , h〉B is in L∞(R).

Theorem:

If f , g , h are in W (L∞, `1v ), then

〈g , h〉B f =∑k,n∈Z

〈f ,TαkMlg〉TαkMlh =: Sg ,hf .

In time-frequency analysis the representation in the lastTheorem is called the Walnut representation of a Gaborframe operator.


Walnut representation of Gabor frame operator

Recall B = C ∗(Z,R/αZ).

Lemma:

Let g , h be in W (L∞, `1v ). Then 〈g , h〉B is in L∞(R).

Theorem:

If f , g , h are in W (L∞, `1v ), then

〈g , h〉B f =∑k,n∈Z

〈f ,TαkMlg〉TαkMlh =: Sg ,hf .

In time-frequency analysis the representation in the lastTheorem is called the Walnut representation of a Gaborframe operator.


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