Hilbert C ∗ -modules over noncommutative tori
Texas A & M
Luef, Franz
UC 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 if
there exists positive constants A, B such that for all f ∈ H
X
Akf k2H ≤
|hf , gj i|2 ≤ Bkf k2H .
(1)
j∈J
analysis operator: Cf = {hf , gj i : j ∈ J}
Luef, Franz
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 if
there exists positive constants A, B such that for all f ∈ H
X
(1)
Akf k2H ≤
|hf , gj i|2 ≤ Bkf k2H .
j∈J
analysis operator: Cf = {hf , gj i : j ∈ J}
P
synthesis operator: Dc = j∈J cj gj for c = (cj )j∈J
Luef, Franz
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 if
there exists positive constants A, B such that for all f ∈ H
X
(1)
Akf k2H ≤
|hf , gj i|2 ≤ Bkf k2H .
j∈J
analysis operator: Cf = {hf , gj i : j ∈ J}
P
synthesis operator: Dc = j∈J cj gj for c = (cj )j∈J
P
frame operator: Sf = DCf = j∈J hf , gj igj
Luef, Franz
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 if
there exists positive constants A, B such that for all f ∈ H
X
(1)
Akf k2H ≤
|hf , gj i|2 ≤ Bkf k2H .
j∈J
analysis operator: Cf = {hf , gj i : j ∈ J}
P
synthesis operator: Dc = j∈J cj gj for c = (cj )j∈J
P
frame operator: Sf = DCf = j∈J hf , gj igj
Luef, Franz
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 if
there exists positive constants A, B such that for all f ∈ H
X
(1)
Akf k2H ≤
|hf , gj i|2 ≤ Bkf k2H .
j∈J
analysis operator: Cf = {hf , gj i : j ∈ J}
P
synthesis operator: Dc = j∈J cj gj for c = (cj )j∈J
P
frame operator: Sf = DCf = j∈J hf , gj igj
Basic fact: {gj : j ∈ J} is a frame for H ⇔, then the frame
operator S is a positive invertible operator on H.
Luef, Franz
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 if
there exists positive constants A, B such that for all f ∈ H
X
(1)
Akf k2H ≤
|hf , gj i|2 ≤ Bkf k2H .
j∈J
analysis operator: Cf = {hf , gj i : j ∈ J}
P
synthesis operator: Dc = j∈J cj gj for c = (cj )j∈J
P
frame operator: Sf = DCf = j∈J hf , gj igj
Basic fact: {gj : j ∈ J} is a frame for H ⇔, then the frame
operator S is a positive invertible operator on H.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Frames for Hilbert and Banach spaces
Another useful reformulation of the notion of frames:
Let g ⊗ h be the rank one operator defined by
(g ⊗ h)f = hf , g ih. Then
X
A · IH ≤
gj ⊗ gj ≤ B · IH ,
j∈J
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 −1 gj : j ∈ J} is a
frame with frame bounds B −1 , A−1 f ∈ H and
X
f =
hf , S −1 gj igj .
(2)
j∈J
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Frames for Hilbert and Banach spaces
Another useful reformulation of the notion of frames:
Let g ⊗ h be the rank one operator defined by
(g ⊗ h)f = hf , g ih. Then
X
A · IH ≤
gj ⊗ gj ≤ B · IH ,
j∈J
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 −1 gj : j ∈ J} is a
frame with frame bounds B −1 , A−1 f ∈ H and
X
f =
hf , S −1 gj igj .
(2)
j∈J
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Banach frames
Gröchenig introduced Banach frames:
A sequence {gj : j ∈ J} of a Banach space B is called a
Banach frame if there exists an associated sequence
space Bd (J) and a continuous reconstruction operator
R : Bd (J) → B such that for all f ∈ B
R((hf , gj i)j∈J )) = f ,
C −1 kf kB ≤ k(hf , gj i)j∈J )kBd ≤ C kf kB
for some constant C ≥ 1.
Coorbit spaces provide a natural class of Banach frames
for a wide class of function spaces.
Banach-Gelfand triples
Casazza, Han and Larson introduced the notion of a
framing for a Banach space.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Banach frames
Gröchenig introduced Banach frames:
A sequence {gj : j ∈ J} of a Banach space B is called a
Banach frame if there exists an associated sequence
space Bd (J) and a continuous reconstruction operator
R : Bd (J) → B such that for all f ∈ B
R((hf , gj i)j∈J )) = f ,
C −1 kf kB ≤ k(hf , gj i)j∈J )kBd ≤ C kf kB
for some constant C ≥ 1.
Coorbit spaces provide a natural class of Banach frames
for a wide class of function spaces.
Banach-Gelfand triples
Casazza, Han and Larson introduced the notion of a
framing for a Banach space.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Frames for Hilbert C ∗ -modules
Theory of frames for Hilbert C ∗ -modules due to Frank
and Larson.
Relevance of Hilbert C ∗ -modules for wavelets was
demonstrated by Packer and Rieffel (projective
multiresolution analysis).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Frames for Hilbert C ∗ -modules
Theory of frames for Hilbert C ∗ -modules due to Frank
and Larson.
Relevance of Hilbert C ∗ -modules for wavelets was
demonstrated by Packer and Rieffel (projective
multiresolution analysis).
Construction of equivalence bimodules between
noncommutative tori is Gabor analysis.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Frames for Hilbert C ∗ -modules
Theory of frames for Hilbert C ∗ -modules due to Frank
and Larson.
Relevance of Hilbert C ∗ -modules for wavelets was
demonstrated by Packer and Rieffel (projective
multiresolution analysis).
Construction of equivalence bimodules between
noncommutative tori is Gabor analysis.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Left Hilbert C ∗ -modules
In the 1970’s Paschke and Rieffel independently generalized the
notion of Hilbert spaces to so-called Hilbert C ∗ -modules.
Defintion:
Let A be a unital C ∗ -algebra. Then a vector space V is a left
Hilbert A-module, i.e. (A, g ) 7→ A · g is a map from
X × A → A, with a pairing A h., .i such that for all f , g , h ∈ V :
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Left Hilbert C ∗ -modules
In the 1970’s Paschke and Rieffel independently generalized the
notion of Hilbert spaces to so-called Hilbert C ∗ -modules.
Defintion:
Let A be a unital C ∗ -algebra. Then a vector space V is a left
Hilbert A-module, i.e. (A, g ) 7→ A · g is a map from
X × A → A, with a pairing A h., .i such that for all f , g , h ∈ V :
(a)
A hλg
+ µh, ki = λ A hg , ki + µ A hh, ki for all λ, µ ∈ C;
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Left Hilbert C ∗ -modules
In the 1970’s Paschke and Rieffel independently generalized the
notion of Hilbert spaces to so-called Hilbert C ∗ -modules.
Defintion:
Let A be a unital C ∗ -algebra. Then a vector space V is a left
Hilbert A-module, i.e. (A, g ) 7→ A · g is a map from
X × A → A, with a pairing A h., .i such that for all f , g , h ∈ V :
(a)
A hλg
(b)
A hA
+ µh, ki = λ A hg , ki + µ A hh, ki for all λ, µ ∈ C;
· f , g i = A · A hf , g i for all A ∈ A;
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Left Hilbert C ∗ -modules
In the 1970’s Paschke and Rieffel independently generalized the
notion of Hilbert spaces to so-called Hilbert C ∗ -modules.
Defintion:
Let A be a unital C ∗ -algebra. Then a vector space V is a left
Hilbert A-module, i.e. (A, g ) 7→ A · g is a map from
X × A → A, with a pairing A h., .i such that for all f , g , h ∈ V :
(a)
A hλg
+ µh, ki = λ A hg , ki + µ A hh, ki for all λ, µ ∈ C;
(b)
A hA
(c)
A hf , g i
· f , g i = A · A hf , g i for all A ∈ A;
= A hg , f i∗ ;
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Left Hilbert C ∗ -modules
In the 1970’s Paschke and Rieffel independently generalized the
notion of Hilbert spaces to so-called Hilbert C ∗ -modules.
Defintion:
Let A be a unital C ∗ -algebra. Then a vector space V is a left
Hilbert A-module, i.e. (A, g ) 7→ A · g is a map from
X × A → A, with a pairing A h., .i such that for all f , g , h ∈ V :
(a)
A hλg
+ µh, ki = λ A hg , ki + µ A hh, ki for all λ, µ ∈ C;
(b)
A hA
(c)
A hf , g i
= A hg , f i∗ ;
(d)
A hf , f i
≥ 0.
· f , g i = A · A hf , g i for all A ∈ A;
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Left Hilbert C ∗ -modules
In the 1970’s Paschke and Rieffel independently generalized the
notion of Hilbert spaces to so-called Hilbert C ∗ -modules.
Defintion:
Let A be a unital C ∗ -algebra. Then a vector space V is a left
Hilbert A-module, i.e. (A, g ) 7→ A · g is a map from
X × A → A, with a pairing A h., .i such that for all f , g , h ∈ V :
(a)
A hλg
+ µh, ki = λ A hg , ki + µ A hh, ki for all λ, µ ∈ C;
(b)
A hA
(c)
A hf , g i
= A hg , f i∗ ;
(d)
A hf , f i
≥ 0.
· f , g i = A · A hf , g i for all A ∈ A;
(e) V is complete with respect to the norm
1/2 . If the ideal
A kf k := kA hf , f ik
span{A hf , g i : f , g ∈ V } is dense in A, then V is called
full.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Left Hilbert C ∗ -modules
In the 1970’s Paschke and Rieffel independently generalized the
notion of Hilbert spaces to so-called Hilbert C ∗ -modules.
Defintion:
Let A be a unital C ∗ -algebra. Then a vector space V is a left
Hilbert A-module, i.e. (A, g ) 7→ A · g is a map from
X × A → A, with a pairing A h., .i such that for all f , g , h ∈ V :
(a)
A hλg
+ µh, ki = λ A hg , ki + µ A hh, ki for all λ, µ ∈ C;
(b)
A hA
(c)
A hf , g i
= A hg , f i∗ ;
(d)
A hf , f i
≥ 0.
· f , g i = A · A hf , g i for all A ∈ A;
(e) V is complete with respect to the norm
1/2 . If the ideal
A kf k := kA hf , f ik
span{A hf , g i : f , g ∈ V } is dense in A, then V is called
full.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Right Hilbert C ∗ -modules
Defintion:
Let B be a unital C ∗ -algebra. Then a vector space V is a right
Hilbert B-module, i.e. (g , B) 7→ g · B is a map from
B × V → B, with a pairing h., .iB which is linear in the second
variable, such that for all f , g , h ∈ V :
(a) hf , λg + µhiB = λhf , g iB + µhf , hiB for all λ, µ ∈ C;
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Right Hilbert C ∗ -modules
Defintion:
Let B be a unital C ∗ -algebra. Then a vector space V is a right
Hilbert B-module, i.e. (g , B) 7→ g · B is a map from
B × V → B, with a pairing h., .iB which is linear in the second
variable, such that for all f , g , h ∈ V :
(a) hf , λg + µhiB = λhf , g iB + µhf , hiB for all λ, µ ∈ C;
(b) h·f , g · Bi = hf , g iB B for all B ∈ B;
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Right Hilbert C ∗ -modules
Defintion:
Let B be a unital C ∗ -algebra. Then a vector space V is a right
Hilbert B-module, i.e. (g , B) 7→ g · B is a map from
B × V → B, with a pairing h., .iB which is linear in the second
variable, such that for all f , g , h ∈ V :
(a) hf , λg + µhiB = λhf , g iB + µhf , hiB for all λ, µ ∈ C;
(b) h·f , g · Bi = hf , g iB B for all B ∈ B;
(c) hf , g iB = hg , f i∗B ;
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Right Hilbert C ∗ -modules
Defintion:
Let B be a unital C ∗ -algebra. Then a vector space V is a right
Hilbert B-module, i.e. (g , B) 7→ g · B is a map from
B × V → B, with a pairing h., .iB which is linear in the second
variable, such that for all f , g , h ∈ V :
(a) hf , λg + µhiB = λhf , g iB + µhf , hiB for all λ, µ ∈ C;
(b) h·f , g · Bi = hf , g iB B for all B ∈ B;
(c) hf , g iB = hg , f i∗B ;
(d) hf , f iB ≥ 0.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Right Hilbert C ∗ -modules
Defintion:
Let B be a unital C ∗ -algebra. Then a vector space V is a right
Hilbert B-module, i.e. (g , B) 7→ g · B is a map from
B × V → B, with a pairing h., .iB which is linear in the second
variable, such that for all f , g , h ∈ V :
(a) hf , λg + µhiB = λhf , g iB + µhf , hiB for all λ, µ ∈ C;
(b) h·f , g · Bi = hf , g iB B for all B ∈ B;
(c) hf , g iB = hg , f i∗B ;
(d) hf , f iB ≥ 0.
(e) V is complete with respect to the norm
kf kB := khf , f iB k1/2 . If the ideal
span{hf , g iB : f , g ∈ V } is dense in B, then V is called
full.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Right Hilbert C ∗ -modules
Defintion:
Let B be a unital C ∗ -algebra. Then a vector space V is a right
Hilbert B-module, i.e. (g , B) 7→ g · B is a map from
B × V → B, with a pairing h., .iB which is linear in the second
variable, such that for all f , g , h ∈ V :
(a) hf , λg + µhiB = λhf , g iB + µhf , hiB for all λ, µ ∈ C;
(b) h·f , g · Bi = hf , g iB B for all B ∈ B;
(c) hf , g iB = hg , f i∗B ;
(d) hf , f iB ≥ 0.
(e) V is complete with respect to the norm
kf kB := khf , f iB k1/2 . If the ideal
span{hf , g iB : f , g ∈ V } is dense in B, then V is called
full.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Right Hilbert C ∗ -modules
Defintion:
Let B be a unital C ∗ -algebra. Then a vector space V is a right
Hilbert B-module, i.e. (g , B) 7→ g · B is a map from
B × V → B, with a pairing h., .iB which is linear in the second
variable, such that for all f , g , h ∈ V :
(a) hf , λg + µhiB = λhf , g iB + µhf , hiB for all λ, µ ∈ C;
(b) h·f , g · Bi = hf , g iB B for all B ∈ B;
(c) hf , g iB = hg , f i∗B ;
(d) hf , f iB ≥ 0.
(e) V is complete with respect to the norm
kf kB := khf , f iB k1/2 . If the ideal
span{hf , g iB : f , g ∈ V } is dense in B, then V is called
full.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Hilbert C ∗ -modules
In applications one often is in the situation that one has a
A0 -module V0 , where A0 is a dense involutive subalgebra of A,
and that there is a pairing h., .i0 from V0 × V0 → A0 satisfying
all the conditions of a Hilbert C ∗ -module, when positivity is
considered with respect to A. Then one can complete V0 to
obtain a Hilbert C ∗ -module V over A.
The challenge in the construction of Hilbert C ∗ -modules is to
find appropriate function spaces V0 .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Hilbert C ∗ -modules
In applications one often is in the situation that one has a
A0 -module V0 , where A0 is a dense involutive subalgebra of A,
and that there is a pairing h., .i0 from V0 × V0 → A0 satisfying
all the conditions of a Hilbert C ∗ -module, when positivity is
considered with respect to A. Then one can complete V0 to
obtain a Hilbert C ∗ -module V over A.
The challenge in the construction of Hilbert C ∗ -modules is to
find appropriate function spaces V0 .
Hilbert C ∗ -modules are also Banach modules for the Hilbert
C ∗ -module norm, i.e. A kA · g k ≤ kAk A kg k.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Hilbert C ∗ -modules
In applications one often is in the situation that one has a
A0 -module V0 , where A0 is a dense involutive subalgebra of A,
and that there is a pairing h., .i0 from V0 × V0 → A0 satisfying
all the conditions of a Hilbert C ∗ -module, when positivity is
considered with respect to A. Then one can complete V0 to
obtain a Hilbert C ∗ -module V over A.
The challenge in the construction of Hilbert C ∗ -modules is to
find appropriate function spaces V0 .
Hilbert C ∗ -modules are also Banach modules for the Hilbert
C ∗ -module norm, i.e. A kA · g k ≤ kAk A kg k.
In some cases the structures of left and right Hilbert
C ∗ -modules over two C ∗ -algebras A and B are compatible with
each other.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Hilbert C ∗ -modules
In applications one often is in the situation that one has a
A0 -module V0 , where A0 is a dense involutive subalgebra of A,
and that there is a pairing h., .i0 from V0 × V0 → A0 satisfying
all the conditions of a Hilbert C ∗ -module, when positivity is
considered with respect to A. Then one can complete V0 to
obtain a Hilbert C ∗ -module V over A.
The challenge in the construction of Hilbert C ∗ -modules is to
find appropriate function spaces V0 .
Hilbert C ∗ -modules are also Banach modules for the Hilbert
C ∗ -module norm, i.e. A kA · g k ≤ kAk A kg k.
In some cases the structures of left and right Hilbert
C ∗ -modules over two C ∗ -algebras A and B are compatible with
each other.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Maps on Hilbert C ∗ -modules
Suppose V is a Hilbert A-module. Then a module
mapping T : V → V is adjointable, if there is a mapping
T ∗ : V → V such that
A hTf , g i
= A hf , T ∗ g i for all f , g ∈ V .
L(V ) denotes the space of all adjointable mappings on V .
We define the norm of T by
kT k = sup{A kTg k : A kg k ≤ 1}.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Maps on Hilbert C ∗ -modules
Suppose V is a Hilbert A-module. Then a module
mapping T : V → V is adjointable, if there is a mapping
T ∗ : V → V such that
A hTf , g i
= A hf , T ∗ g i for all f , g ∈ V .
L(V ) denotes the space of all adjointable mappings on V .
We define the norm of T by
kT k = sup{A kTg k : A kg k ≤ 1}.
L(V ) is a C ∗ -algebra with respect to this norm.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Maps on Hilbert C ∗ -modules
Suppose V is a Hilbert A-module. Then a module
mapping T : V → V is adjointable, if there is a mapping
T ∗ : V → V such that
A hTf , g i
= A hf , T ∗ g i for all f , g ∈ V .
L(V ) denotes the space of all adjointable mappings on V .
We define the norm of T by
kT k = sup{A kTg k : A kg k ≤ 1}.
L(V ) is a C ∗ -algebra with respect to this norm.
rank one operators ΘA
g ,h f = A hf , g i · h are adjointable
operators.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Maps on Hilbert C ∗ -modules
Suppose V is a Hilbert A-module. Then a module
mapping T : V → V is adjointable, if there is a mapping
T ∗ : V → V such that
A hTf , g i
= A hf , T ∗ g i for all f , g ∈ V .
L(V ) denotes the space of all adjointable mappings on V .
We define the norm of T by
kT k = sup{A kTg k : A kg k ≤ 1}.
L(V ) is a C ∗ -algebra with respect to this norm.
rank one operators ΘA
g ,h f = A hf , g i · h are adjointable
operators.
The closed linear subspace of L(V ) spanned by
{ΘA
g ,h : f , g ∈ V } is the algebra of compact A-module
operators.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Maps on Hilbert C ∗ -modules
Suppose V is a Hilbert A-module. Then a module
mapping T : V → V is adjointable, if there is a mapping
T ∗ : V → V such that
A hTf , g i
= A hf , T ∗ g i for all f , g ∈ V .
L(V ) denotes the space of all adjointable mappings on V .
We define the norm of T by
kT k = sup{A kTg k : A kg k ≤ 1}.
L(V ) is a C ∗ -algebra with respect to this norm.
rank one operators ΘA
g ,h f = A hf , g i · h are adjointable
operators.
The closed linear subspace of L(V ) spanned by
{ΘA
g ,h : f , g ∈ V } is the algebra of compact A-module
operators.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Properties of rank one Hilbert module operators
A
A
0 0
ΘA
g ,h Θg 0 ,h0 = Θk,h for k = A hg , h ig
∗
A
(ΘA
g ,h ) = Θh,g
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Properties of rank one Hilbert module operators
A
A
0 0
ΘA
g ,h Θg 0 ,h0 = Θk,h for k = A hg , h ig
∗
A
(ΘA
g ,h ) = Θh,g
A
For T ∈ L(V ) we have T ΘA
g ,h = Θg ,Th .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Properties of rank one Hilbert module operators
A
A
0 0
ΘA
g ,h Θg 0 ,h0 = Θk,h for k = A hg , h ig
∗
A
(ΘA
g ,h ) = Θh,g
A
For T ∈ L(V ) we have T ΘA
g ,h = Θg ,Th .
kΘA
g ,h k ≤ A kg kA khk.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Properties of rank one Hilbert module operators
A
A
0 0
ΘA
g ,h Θg 0 ,h0 = Θk,h for k = A hg , h ig
∗
A
(ΘA
g ,h ) = Θh,g
A
For T ∈ L(V ) we have T ΘA
g ,h = Θg ,Th .
kΘA
g ,h k ≤ A kg kA khk.
Suppose A is unital and A hg , g i = I , then kΘA
g ,h k = A khk.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Properties of rank one Hilbert module operators
A
A
0 0
ΘA
g ,h Θg 0 ,h0 = Θk,h for k = A hg , h ig
∗
A
(ΘA
g ,h ) = Θh,g
A
For T ∈ L(V ) we have T ΘA
g ,h = Θg ,Th .
kΘA
g ,h k ≤ A kg kA khk.
Suppose A is unital and A hg , g i = I , then kΘA
g ,h k = A khk.
We denote the unit sphere of A V by
S(A V ) = {g ∈ A V : A hg , g i = I }.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Properties of rank one Hilbert module operators
A
A
0 0
ΘA
g ,h Θg 0 ,h0 = Θk,h for k = A hg , h ig
∗
A
(ΘA
g ,h ) = Θh,g
A
For T ∈ L(V ) we have T ΘA
g ,h = Θg ,Th .
kΘA
g ,h k ≤ A kg kA khk.
Suppose A is unital and A hg , g i = I , then kΘA
g ,h k = A khk.
We denote the unit sphere of A V by
S(A V ) = {g ∈ A V : A hg , g i = I }.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Frames for Hilbert C ∗ -modules
For the sake of simplicity I restrict my discussion to the case of
finitely generated Hilbert C ∗ -modules.
Let A be a unital C ∗ -algebra. A sequence
{gj : j = 1, ..., n} in a (left) Hilbert A-module A V is called
a standard module frame if there are positive reals C , D
such that
n
X
C A hf , f i ≤
A hf , gj iA hgj , f i ≤ D A hf , f i
j=1
for each f ∈ A V .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Frames for Hilbert C ∗ -modules
For the sake of simplicity I restrict my discussion to the case of
finitely generated Hilbert C ∗ -modules.
Let A be a unital C ∗ -algebra. A sequence
{gj : j = 1, ..., n} in a (left) Hilbert A-module A V is called
a standard module frame if there are positive reals C , D
such that
n
X
C A hf , f i ≤
A hf , gj iA hgj , f i ≤ D A hf , f i
j=1
for each f ∈ A V .
Let A be a unital C ∗ -algebra. A sequence
{gj : j = 1, ..., n} in a (left) Hilbert A-module A V is a
standard module frame if the reconstruction formula
n
X
f =
A hf , gj i · gj for all f ∈ A V .
j=1
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Frames for Hilbert C ∗ -modules
For the sake of simplicity I restrict my discussion to the case of
finitely generated Hilbert C ∗ -modules.
Let A be a unital C ∗ -algebra. A sequence
{gj : j = 1, ..., n} in a (left) Hilbert A-module A V is called
a standard module frame if there are positive reals C , D
such that
n
X
C A hf , f i ≤
A hf , gj iA hgj , f i ≤ D A hf , f i
j=1
for each f ∈ A V .
Let A be a unital C ∗ -algebra. A sequence
{gj : j = 1, ..., n} in a (left) Hilbert A-module A V is a
standard module frame if the reconstruction formula
n
X
f =
A hf , gj i · gj for all f ∈ A V .
j=1
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Frames for Hilbert C ∗ -module
The existence of a standard module frame of finite
cardinality is equivalent to A V 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 the
definition of a frame becomes:
C A hf , f i ≤
n
X
A
A hΘgj ,gj f , f iA
≤ D A hf , f i.
j=1
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Frames for Hilbert C ∗ -module
The existence of a standard module frame of finite
cardinality is equivalent to A V 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 the
definition of a frame becomes:
C A hf , f i ≤
n
X
A
A hΘgj ,gj f , f iA
≤ D A hf , f i.
j=1
One has the following: A sequence {gj : j = 1, ..., n} in a
(left)
A-module A V is a frame if and only if
Pn Hilbert
A
Θ
A
gj ,gj converges in the strict topology to a
j=1
bounded invertible operator in L(A V ).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Frames for Hilbert C ∗ -module
The existence of a standard module frame of finite
cardinality is equivalent to A V 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 the
definition of a frame becomes:
C A hf , f i ≤
n
X
A
A hΘgj ,gj f , f iA
≤ D A hf , f i.
j=1
One has the following: A sequence {gj : j = 1, ..., n} in a
(left)
A-module A V is a frame if and only if
Pn Hilbert
A
Θ
A
gj ,gj converges in the strict topology to a
j=1
bounded invertible operator in L(A V ).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Frames for Hilbert C ∗ -module
Suppose A V is a finitely generated Hilbert A-module.
Then any set of generators {gj : j = 1, ..., n} is a standard
module frame. The number of the shortest frame gives the
number of factors of An into which A V is embeddable.
In other
P words, the positive module operator
S = nj=1 ΘA
gj ,gj is invertible and the upper and lower
frame bounds are given by kSk2 and kS −1 k−2 .
In particular, A V is singly generated if ΘA
g ,g is invertible.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Frames for Hilbert C ∗ -module
Suppose A V is a finitely generated Hilbert A-module.
Then any set of generators {gj : j = 1, ..., n} is a standard
module frame. The number of the shortest frame gives the
number of factors of An into which A V is embeddable.
In other
P words, the positive module operator
S = nj=1 ΘA
gj ,gj is invertible and the upper and lower
frame bounds are given by kSk2 and kS −1 k−2 .
In particular, A V is singly generated if ΘA
g ,g is invertible.
Is there a way to generate finitely generated Hilbert
C ∗ -modules?
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Frames for Hilbert C ∗ -module
Suppose A V is a finitely generated Hilbert A-module.
Then any set of generators {gj : j = 1, ..., n} is a standard
module frame. The number of the shortest frame gives the
number of factors of An into which A V is embeddable.
In other
P words, the positive module operator
S = nj=1 ΘA
gj ,gj is invertible and the upper and lower
frame bounds are given by kSk2 and kS −1 k−2 .
In particular, A V is singly generated if ΘA
g ,g is invertible.
Is there a way to generate finitely generated Hilbert
C ∗ -modules?
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Morita-Rieffel equivalence
Rieffel introduced in 1970’s the notion of strong Morita
equivalence for C ∗ -algebras:
Definition:
Let A and B be C ∗ -algebras. Then an A-B-equivalence
bimodule A V B is an A-B-bimodule such that:
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Morita-Rieffel equivalence
Rieffel introduced in 1970’s the notion of strong Morita
equivalence for C ∗ -algebras:
Definition:
Let A and B be C ∗ -algebras. Then an A-B-equivalence
bimodule A V B is an A-B-bimodule such that:
(a)
is a full left Hilbert A-module and a full right Hilbert
B-module;
AV B
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Morita-Rieffel equivalence
Rieffel introduced in 1970’s the notion of strong Morita
equivalence for C ∗ -algebras:
Definition:
Let A and B be C ∗ -algebras. Then an A-B-equivalence
bimodule A V B is an A-B-bimodule such that:
(a)
is a full left Hilbert A-module and a full right Hilbert
B-module;
AV B
(b) for all f , g ∈ A V B , A ∈ A and B ∈ B we have that
hA · f , g iB = hf , A∗ · g iB and A hf · B, g i = A hf , g · B ∗ i;
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Morita-Rieffel equivalence
Rieffel introduced in 1970’s the notion of strong Morita
equivalence for C ∗ -algebras:
Definition:
Let A and B be C ∗ -algebras. Then an A-B-equivalence
bimodule A V B is an A-B-bimodule such that:
(a)
is a full left Hilbert A-module and a full right Hilbert
B-module;
AV B
(b) for all f , g ∈ A V B , A ∈ A and B ∈ B we have that
hA · f , g iB = hf , A∗ · g iB and A hf · B, g i = A hf , g · B ∗ i;
(c) for all f , g , h ∈ A V B , A hf , g i · h = f · hg , hiB .
The C ∗ -algebras A and B are called Morita-Rieffel equivalent if
there exists an A − B equivalence bimodule.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Morita-Rieffel equivalence
Rieffel introduced in 1970’s the notion of strong Morita
equivalence for C ∗ -algebras:
Definition:
Let A and B be C ∗ -algebras. Then an A-B-equivalence
bimodule A V B is an A-B-bimodule such that:
(a)
is a full left Hilbert A-module and a full right Hilbert
B-module;
AV B
(b) for all f , g ∈ A V B , A ∈ A and B ∈ B we have that
hA · f , g iB = hf , A∗ · g iB and A hf · B, g i = A hf , g · B ∗ i;
(c) for all f , g , h ∈ A V B , A hf , g i · h = f · hg , hiB .
The C ∗ -algebras A and B are called Morita-Rieffel equivalent if
there exists an A − B equivalence bimodule.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Morita-Rieffel equivalence - Consequences
In words, Condition (b) says that A acts by adjointable
operators on VB and that B acts by adjointable operators
on A V ,
Condition (c) is an associativity condition between the
A-inner product and the B-inner product.
ΘB
g ,h f = f · hg , hiB is a rank one B-module operator.
Consequently, the invertiblity of ΘA
g ,g is equivalent to the
B
invertiblity of Θg ,g . Note that this amounts to
ΘB
g ,g f = f · hg , g iB , i.e. f and g are “uncoupled”.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Morita-Rieffel equivalence - Consequences
In words, Condition (b) says that A acts by adjointable
operators on VB and that B acts by adjointable operators
on A V ,
Condition (c) is an associativity condition between the
A-inner product and the B-inner product.
ΘB
g ,h f = f · hg , hiB is a rank one B-module operator.
Consequently, the invertiblity of ΘA
g ,g is equivalent to the
B
invertiblity of Θg ,g . Note that this amounts to
ΘB
g ,g f = f · hg , g iB , i.e. f and g are “uncoupled”.
If A and B are Morita-Rieffel equivalent, then A VB is a
projective left A-module, and a projective right
B-module.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Morita-Rieffel equivalence - Consequences
In words, Condition (b) says that A acts by adjointable
operators on VB and that B acts by adjointable operators
on A V ,
Condition (c) is an associativity condition between the
A-inner product and the B-inner product.
ΘB
g ,h f = f · hg , hiB is a rank one B-module operator.
Consequently, the invertiblity of ΘA
g ,g is equivalent to the
B
invertiblity of Θg ,g . Note that this amounts to
ΘB
g ,g f = f · hg , g iB , i.e. f and g are “uncoupled”.
If A and B are Morita-Rieffel equivalent, then A VB is a
projective left A-module, and a projective right
B-module.There exist g1 , ..., gn ∈ A VB such that
n
n
X
X
f =
hf
,
g
i
·
g
=
f · hgj , gj iB
A
j
j
j=1
Luef, Franz
j=1
Hilbert C ∗ -modules over noncommutative tori
Morita-Rieffel equivalence - Consequences
In words, Condition (b) says that A acts by adjointable
operators on VB and that B acts by adjointable operators
on A V ,
Condition (c) is an associativity condition between the
A-inner product and the B-inner product.
ΘB
g ,h f = f · hg , hiB is a rank one B-module operator.
Consequently, the invertiblity of ΘA
g ,g is equivalent to the
B
invertiblity of Θg ,g . Note that this amounts to
ΘB
g ,g f = f · hg , g iB , i.e. f and g are “uncoupled”.
If A and B are Morita-Rieffel equivalent, then A VB is a
projective left A-module, and a projective right
B-module.There exist g1 , ..., gn ∈ A VB such that
n
n
X
X
f =
hf
,
g
i
·
g
=
f · hgj , gj iB
A
j
j
j=1
Luef, Franz
j=1
Hilbert C ∗ -modules over noncommutative tori
Morita-Rieffel equivalence - Consequences
The compact A-module operators are isomorphic to B.
Suppose A has a normalized trace trA . Then we can
introduce a trace trB on B by
trB (hf , g iB ) = trA (A hg , f i).
trB is not normalized and trB (I ) = dim(A V ), the
dimension of the Hilbert A-module A V .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Morita equivalent spectrally invariant subalgebras
Recall that a unital subalgebra A of a unital C ∗ -algebra B with
common unit is called spectrally invariant, if for A ∈ A with
A−1 ∈ B one actually has that A−1 ∈ A.
Theorem:
Let A and B be unital C ∗ -algebras that are Morita equivalent
via an equivalence bimodule A VB .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Morita equivalent spectrally invariant subalgebras
Recall that a unital subalgebra A of a unital C ∗ -algebra B with
common unit is called spectrally invariant, if for A ∈ A with
A−1 ∈ B one actually has that A−1 ∈ A.
Theorem:
Let A and B be unital C ∗ -algebras that are Morita equivalent
via an equivalence bimodule A VB .Suppose we have dense
∗-Banach (or Frechet) subalgebras A0 and B0 of A and B
respectively containing the identity elements.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Morita equivalent spectrally invariant subalgebras
Recall that a unital subalgebra A of a unital C ∗ -algebra B with
common unit is called spectrally invariant, if for A ∈ A with
A−1 ∈ B one actually has that A−1 ∈ A.
Theorem:
Let A and B be unital C ∗ -algebras that are Morita equivalent
via an equivalence bimodule A VB .Suppose we have dense
∗-Banach (or Frechet) subalgebras A0 and B0 of A and B
respectively containing the identity elements.Furthermore we
assume that A0 and B0 are spectrally invariant in A and B
respectively.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Morita equivalent spectrally invariant subalgebras
Recall that a unital subalgebra A of a unital C ∗ -algebra B with
common unit is called spectrally invariant, if for A ∈ A with
A−1 ∈ B one actually has that A−1 ∈ A.
Theorem:
Let A and B be unital C ∗ -algebras that are Morita equivalent
via an equivalence bimodule A VB .Suppose we have dense
∗-Banach (or Frechet) subalgebras A0 and B0 of A and B
respectively containing the identity elements.Furthermore we
assume that A0 and B0 are spectrally invariant in A and B
respectively.Let V0 be a dense subspace of A VB which is closed
under the actions of A0 and B0 , and such that the restrictions
of the inner products A h., .i and h., .iB have values in A0 and
B0 respectively.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Morita equivalent spectrally invariant subalgebras
Recall that a unital subalgebra A of a unital C ∗ -algebra B with
common unit is called spectrally invariant, if for A ∈ A with
A−1 ∈ B one actually has that A−1 ∈ A.
Theorem:
Let A and B be unital C ∗ -algebras that are Morita equivalent
via an equivalence bimodule A VB .Suppose we have dense
∗-Banach (or Frechet) subalgebras A0 and B0 of A and B
respectively containing the identity elements.Furthermore we
assume that A0 and B0 are spectrally invariant in A and B
respectively.Let V0 be a dense subspace of A VB which is closed
under the actions of A0 and B0 , and such that the restrictions
of the inner products A h., .i and h., .iB have values in A0 and
B0 respectively.Then V0 is a finitely generated projective left
A0 -module.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Morita equivalent spectrally invariant subalgebras
Recall that a unital subalgebra A of a unital C ∗ -algebra B with
common unit is called spectrally invariant, if for A ∈ A with
A−1 ∈ B one actually has that A−1 ∈ A.
Theorem:
Let A and B be unital C ∗ -algebras that are Morita equivalent
via an equivalence bimodule A VB .Suppose we have dense
∗-Banach (or Frechet) subalgebras A0 and B0 of A and B
respectively containing the identity elements.Furthermore we
assume that A0 and B0 are spectrally invariant in A and B
respectively.Let V0 be a dense subspace of A VB which is closed
under the actions of A0 and B0 , and such that the restrictions
of the inner products A h., .i and h., .iB have values in A0 and
B0 respectively.Then V0 is a finitely generated projective left
A0 -module.In addition we have that V0 is a finitely generated
projective right B0 -module.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Morita equivalent spectrally invariant subalgebras
Recall that a unital subalgebra A of a unital C ∗ -algebra B with
common unit is called spectrally invariant, if for A ∈ A with
A−1 ∈ B one actually has that A−1 ∈ A.
Theorem:
Let A and B be unital C ∗ -algebras that are Morita equivalent
via an equivalence bimodule A VB .Suppose we have dense
∗-Banach (or Frechet) subalgebras A0 and B0 of A and B
respectively containing the identity elements.Furthermore we
assume that A0 and B0 are spectrally invariant in A and B
respectively.Let V0 be a dense subspace of A VB which is closed
under the actions of A0 and B0 , and such that the restrictions
of the inner products A h., .i and h., .iB have values in A0 and
B0 respectively.Then V0 is a finitely generated projective left
A0 -module.In addition we have that V0 is a finitely generated
projective right B0 -module.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Stone-von Neumann theorem
Rieffel demonstrated that the Stone-von Neumann theorem on
the uniqueness of the irreducible representations of the
Heisenberg group is equivalent to the following fact:
Theorem:
The C ∗ -algebra of compact operators on a separable Hilbert
space H is Morita-Rieffel equivalent to C.
We construct an equivalence bimodule using basic facts and
notions from time-frequency analysis.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Stone-von Neumann theorem
Rieffel demonstrated that the Stone-von Neumann theorem on
the uniqueness of the irreducible representations of the
Heisenberg group is equivalent to the following fact:
Theorem:
The C ∗ -algebra of compact operators on a separable Hilbert
space H is Morita-Rieffel equivalent to C.
We construct an equivalence bimodule using basic facts and
notions from time-frequency analysis.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Schrödinger representation
translation Tx f (t) = f (t − x)
for x ∈ Rd , modulation
bd
Mω f (t) = e 2πit·ω f (t) for ω ∈ R
time-frequency shift π(x, ω)f (t) = Mω Tx f (t) for
bd
(x, ω) ∈ Rd × R
Mω Tx = e 2πix·ω Tx Mω
π(x + y , ω + η) = e −2πix·η π(x, ω)π(y , η)
π(x, ω)π(y , η) = e 2πi(y ·ω−x·η) π(y , η)π(x, ω)
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Schrödinger representation
translation Tx f (t) = f (t − x)
for x ∈ Rd , modulation
bd
Mω f (t) = e 2πit·ω f (t) for ω ∈ R
time-frequency shift π(x, ω)f (t) = Mω Tx f (t) for
bd
(x, ω) ∈ Rd × R
Mω Tx = e 2πix·ω Tx Mω
π(x + y , ω + η) = e −2πix·η π(x, ω)π(y , η)
π(x, ω)π(y , η) = e 2πi(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) = e 2πiy ·ω
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Schrödinger representation
translation Tx f (t) = f (t − x)
for x ∈ Rd , modulation
bd
Mω f (t) = e 2πit·ω f (t) for ω ∈ R
time-frequency shift π(x, ω)f (t) = Mω Tx f (t) for
bd
(x, ω) ∈ Rd × R
Mω Tx = e 2πix·ω Tx Mω
π(x + y , ω + η) = e −2πix·η π(x, ω)π(y , η)
π(x, ω)π(y , η) = e 2πi(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) = e 2πiy ·ω
Short-Time Fourier Transform:
Vg f (x, ω) = hf , π(x, ω)g i.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Schrödinger representation
translation Tx f (t) = f (t − x)
for x ∈ Rd , modulation
bd
Mω f (t) = e 2πit·ω f (t) for ω ∈ R
time-frequency shift π(x, ω)f (t) = Mω Tx f (t) for
bd
(x, ω) ∈ Rd × R
Mω Tx = e 2πix·ω Tx Mω
π(x + y , ω + η) = e −2πix·η π(x, ω)π(y , η)
π(x, ω)π(y , η) = e 2πi(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) = e 2πiy ·ω
Short-Time Fourier Transform:
Vg f (x, ω) = hf , π(x, ω)g i.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Integrated Schrödinger Representation
1
(L1 (R2d ), c) is an involutive Banach algebra:
ZZ
(F ]G )(z) =
F (z 0 )G (z − z 0 )c(z, z − z 0 )dz
F ∗ (z) = c(z, z)F (z).
2
(x, ω) 7→ Vω Ux is a non-degenerate faithful involutive
1
2d
representation
RR πL1 of (L (R ), c),
πL1 (F ) =
F (x, ω)Vω Ux dxdω.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Integrated Schrödinger Representation
1
(L1 (R2d ), c) is an involutive Banach algebra:
ZZ
(F ]G )(z) =
F (z 0 )G (z − z 0 )c(z, z − z 0 )dz
F ∗ (z) = c(z, z)F (z).
2
3
(x, ω) 7→ Vω Ux is a non-degenerate faithful involutive
1
2d
representation
RR πL1 of (L (R ), c),
πL1 (F ) =
F (x, ω)Vω Ux dxdω.
πL1 (F )πL1 (G ) = πL1 (F ]G ) and πL1 (F )∗ = πL1 (F ∗ ).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Integrated Schrödinger Representation
1
(L1 (R2d ), c) is an involutive Banach algebra:
ZZ
(F ]G )(z) =
F (z 0 )G (z − z 0 )c(z, z − z 0 )dz
F ∗ (z) = c(z, z)F (z).
2
3
4
(x, ω) 7→ Vω Ux is a non-degenerate faithful involutive
1
2d
representation
RR πL1 of (L (R ), c),
πL1 (F ) =
F (x, ω)Vω Ux dxdω.
πL1 (F )πL1 (G ) = πL1 (F ]G ) and πL1 (F )∗ = πL1 (F ∗ ).
RR
PF =
F (x, ω)Vω Ux dxdω is a projection in (L1 (R2d ), c)
if and only if F ]F = F , F = F ∗ .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Integrated Schrödinger Representation
1
(L1 (R2d ), c) is an involutive Banach algebra:
ZZ
(F ]G )(z) =
F (z 0 )G (z − z 0 )c(z, z − z 0 )dz
F ∗ (z) = c(z, z)F (z).
2
3
4
(x, ω) 7→ Vω Ux is a non-degenerate faithful involutive
1
2d
representation
RR πL1 of (L (R ), c),
πL1 (F ) =
F (x, ω)Vω Ux dxdω.
πL1 (F )πL1 (G ) = πL1 (F ]G ) and πL1 (F )∗ = πL1 (F ∗ ).
RR
PF =
F (x, ω)Vω Ux dxdω is a projection in (L1 (R2d ), c)
if and only if F ]F = F , F = F ∗ .A natural choice for F is
Vg g .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Integrated Schrödinger Representation
1
(L1 (R2d ), c) is an involutive Banach algebra:
ZZ
(F ]G )(z) =
F (z 0 )G (z − z 0 )c(z, z − z 0 )dz
F ∗ (z) = c(z, z)F (z).
2
3
4
(x, ω) 7→ Vω Ux is a non-degenerate faithful involutive
1
2d
representation
RR πL1 of (L (R ), c),
πL1 (F ) =
F (x, ω)Vω Ux dxdω.
πL1 (F )πL1 (G ) = πL1 (F ]G ) and πL1 (F )∗ = πL1 (F ∗ ).
RR
PF =
F (x, ω)Vω Ux dxdω is a projection in (L1 (R2d ), c)
if and only if F ]F = F , F = F ∗ .A natural choice for F is
Vg g .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Feichtinger’s algebra
Definition:
Feichtinger’s
algebra S0 (Rd ) = {f ∈ L2 (Rd ) : kf kS0 =
RR
1
d
R2d |hf , π(x, ω)g i|dxdω < ∞}, also denoted by M (R ).
S0 (Rd ) is invariant under time-frequency shifts:
kπ(u, η)g kS0 ≤ kg kS0 .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Feichtinger’s algebra
Definition:
Feichtinger’s
algebra S0 (Rd ) = {f ∈ L2 (Rd ) : kf kS0 =
RR
1
d
R2d |hf , π(x, ω)g i|dxdω < ∞}, also denoted by M (R ).
S0 (Rd ) is invariant under time-frequency shifts:
kπ(u, η)g kS0 ≤ kg kS0 .
S0 (Rd ) is a Banach algebra w.r.t point-wise multiplication.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Feichtinger’s algebra
Definition:
Feichtinger’s
algebra S0 (Rd ) = {f ∈ L2 (Rd ) : kf kS0 =
RR
1
d
R2d |hf , π(x, ω)g i|dxdω < ∞}, also denoted by M (R ).
S0 (Rd ) is invariant under time-frequency shifts:
kπ(u, η)g kS0 ≤ kg kS0 .
S0 (Rd ) is a Banach algebra w.r.t point-wise multiplication.
S0 (R2d )RR
is invariant under symplectic Fourier transform
Fb (b
z ) = R2d F (z)e 2πiΩ(bz ,z) dz.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Feichtinger’s algebra
Definition:
Feichtinger’s
algebra S0 (Rd ) = {f ∈ L2 (Rd ) : kf kS0 =
RR
1
d
R2d |hf , π(x, ω)g i|dxdω < ∞}, also denoted by M (R ).
S0 (Rd ) is invariant under time-frequency shifts:
kπ(u, η)g kS0 ≤ kg kS0 .
S0 (Rd ) is a Banach algebra w.r.t point-wise multiplication.
S0 (R2d )RR
is invariant under symplectic Fourier transform
Fb (b
z ) = R2d F (z)e 2πiΩ(bz ,z) dz.
If f , g ∈ S0 (Rd ), then Vg f ∈ S0 (R2d ).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Feichtinger’s algebra
Definition:
Feichtinger’s
algebra S0 (Rd ) = {f ∈ L2 (Rd ) : kf kS0 =
RR
1
d
R2d |hf , π(x, ω)g i|dxdω < ∞}, also denoted by M (R ).
S0 (Rd ) is invariant under time-frequency shifts:
kπ(u, η)g kS0 ≤ kg kS0 .
S0 (Rd ) is a Banach algebra w.r.t point-wise multiplication.
S0 (R2d )RR
is invariant under symplectic Fourier transform
Fb (b
z ) = R2d F (z)e 2πiΩ(bz ,z) dz.
If f , g ∈ S0 (Rd ), then Vg f ∈ S0 (R2d ).
Suppose f1 , f2 , g1 , g2 ∈ S0 (Rd ).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Feichtinger’s algebra
Definition:
Feichtinger’s
algebra S0 (Rd ) = {f ∈ L2 (Rd ) : kf kS0 =
RR
1
d
R2d |hf , π(x, ω)g i|dxdω < ∞}, also denoted by M (R ).
S0 (Rd ) is invariant under time-frequency shifts:
kπ(u, η)g kS0 ≤ kg kS0 .
S0 (Rd ) is a Banach algebra w.r.t point-wise multiplication.
S0 (R2d )RR
is invariant under symplectic Fourier transform
Fb (b
z ) = R2d F (z)e 2πiΩ(bz ,z) dz.
If f , g ∈ S0 (Rd ), then Vg f ∈ S0 (R2d ).
Suppose f1 , f2 , g1 , g2 ∈ S0 (Rd ).
(Vg1\
f1 · Vg2 f2 )(b
z ) = (Vf2 f1 · Vg2 g1 )(b
z)
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Feichtinger’s algebra
Definition:
Feichtinger’s
algebra S0 (Rd ) = {f ∈ L2 (Rd ) : kf kS0 =
RR
1
d
R2d |hf , π(x, ω)g i|dxdω < ∞}, also denoted by M (R ).
S0 (Rd ) is invariant under time-frequency shifts:
kπ(u, η)g kS0 ≤ kg kS0 .
S0 (Rd ) is a Banach algebra w.r.t point-wise multiplication.
S0 (R2d )RR
is invariant under symplectic Fourier transform
Fb (b
z ) = R2d F (z)e 2πiΩ(bz ,z) dz.
If f , g ∈ S0 (Rd ), then Vg f ∈ S0 (R2d ).
Suppose f1 , f2 , g1 , g2 ∈ S0 (Rd ).
(Vg1\
f1 · Vg2 f2 )(b
z ) = (Vf2 f1 · Vg2 g1 )(b
z)
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Facts
Consider G(g , R2d ) and the associated analysis and
synthesis mapping
ZZ
f 7→ (Vg f (z)), A 7→
A(z)π(z)gdz
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Facts
Consider G(g , R2d ) and the associated analysis and
synthesis mapping
ZZ
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 mappings
are well-defined.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Facts
Consider G(g , R2d ) and the associated analysis and
synthesis mapping
ZZ
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 mappings
are well-defined.
Consider the pseudodifferential operator
ZZ
H=
A(z)π(z)dz
R2d
for A ∈ S0 (R2d ). Then H is trace-class.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Facts
Consider G(g , R2d ) and the associated analysis and
synthesis mapping
ZZ
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 mappings
are well-defined.
Consider the pseudodifferential operator
ZZ
H=
A(z)π(z)dz
R2d
for A ∈ S0 (R2d ). Then H is trace-class.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Different Perspective
Left action of K(L2 (Rd )) on S0 (Rd ):
For F ∈ S0 (R2d ) and g ∈ S0 (Rd )
ZZ
π(F ) · g =
F (z)π(z)gdz
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Different Perspective
Left action of K(L2 (Rd )) on S0 (Rd ):
For F ∈ S0 (R2d ) and g ∈ S0 (Rd )
ZZ
π(F ) · g =
F (z)π(z)gdz
Inner product with values in K(L2 (Rd ))
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Different Perspective
Left action of K(L2 (Rd )) on S0 (Rd ):
For F ∈ S0 (R2d ) and g ∈ S0 (Rd )
ZZ
π(F ) · g =
F (z)π(z)gdz
Inner product with values in K(L2 (Rd ))
ZZ
hf
,
g
i
=
Vg f (z)π(z)
K
for f , g ∈ S0 (Rd ).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Different Perspective
Left action of K(L2 (Rd )) on S0 (Rd ):
For F ∈ S0 (R2d ) and g ∈ S0 (Rd )
ZZ
π(F ) · g =
F (z)π(z)gdz
Inner product with values in K(L2 (Rd ))
ZZ
hf
,
g
i
=
Vg f (z)π(z)
K
for f , g ∈ S0 (Rd ).
K hπ(A)
· f , g i = π(A)K hf , g i, i.e. the right hand side
amounts to A]Vg f .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Different Perspective
Left action of K(L2 (Rd )) on S0 (Rd ):
For F ∈ S0 (R2d ) and g ∈ S0 (Rd )
ZZ
π(F ) · g =
F (z)π(z)gdz
Inner product with values in K(L2 (Rd ))
ZZ
hf
,
g
i
=
Vg f (z)π(z)
K
for f , g ∈ S0 (Rd ).
K hπ(A)
· f , g i = π(A)K hf , g i, i.e. the right hand side
amounts to A]Vg f .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Hilbert K-module
Theorem:
S0 (Rd ) becomes a full left Hilbert K-module K V when
completed with respect to the norm K kf k = kK hf , f ik1/2 for
f ∈ S0 (Rd )
The proof relies on the properties of S0 (Rd ) and L1 (R2d , c).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Hilbert K-module
Theorem:
S0 (Rd ) becomes a full left Hilbert K-module K V when
completed with respect to the norm K kf k = kK hf , f ik1/2 for
f ∈ S0 (Rd )
The proof relies on the properties of S0 (Rd ) and L1 (R2d , c).
Rank-one K V -module operators are just the rank-one operators
on L2 (Rd ) Θg ,h f = hf , g i · h.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Hilbert K-module
Theorem:
S0 (Rd ) becomes a full left Hilbert K-module K V when
completed with respect to the norm K kf k = kK hf , f ik1/2 for
f ∈ S0 (Rd )
The proof relies on the properties of S0 (Rd ) and L1 (R2d , c).
Rank-one K V -module operators are just the rank-one operators
on L2 (Rd ) Θg ,h f = hf , g i · h.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Equivalence bimodule between K and C
Since we need a right Hilbert C-module, we take the conjugate
Hilbert space of L2 (Rd ) with inner product
hf , g iC = hg , f iL2 (Rd ) .
K hf , g i
Luef, Franz
· h = f · hg , hiC
Hilbert C ∗ -modules over noncommutative tori
Equivalence bimodule between K and C
Since we need a right Hilbert C-module, we take the conjugate
Hilbert space of L2 (Rd ) with inner product
hf , g iC = hg , f iL2 (Rd ) .
K hf , g i
· h = f · hg , hiC
Equivalent to
hVg f , Vk hiL2 (R2d ) = hf , kiL2 (Rd ) hh, g iL2 (Rd )
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Equivalence bimodule between K and C
Since we need a right Hilbert C-module, we take the conjugate
Hilbert space of L2 (Rd ) with inner product
hf , g iC = hg , f iL2 (Rd ) .
K hf , g i
· h = f · hg , hiC
Equivalent to
hVg f , Vk hiL2 (R2d ) = hf , kiL2 (Rd ) hh, g iL2 (Rd )
Theorem:
S0 (Rd ) is an equivalence bimodule between K and C.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Equivalence bimodule between K and C
Since we need a right Hilbert C-module, we take the conjugate
Hilbert space of L2 (Rd ) with inner product
hf , g iC = hg , f iL2 (Rd ) .
K hf , g i
· h = f · hg , hiC
Equivalent to
hVg f , Vk hiL2 (R2d ) = hf , kiL2 (Rd ) hh, g iL2 (Rd )
Theorem:
S0 (Rd ) is an equivalence bimodule between K and C.
Morita-Rieffel equivalent C ∗ -algebras have equivalent
categories of representations.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Equivalence bimodule between K and C
Since we need a right Hilbert C-module, we take the conjugate
Hilbert space of L2 (Rd ) with inner product
hf , g iC = hg , f iL2 (Rd ) .
K hf , g i
· h = f · hg , hiC
Equivalent to
hVg f , Vk hiL2 (R2d ) = hf , kiL2 (Rd ) hh, g iL2 (Rd )
Theorem:
S0 (Rd ) is an equivalence bimodule between K and C.
Morita-Rieffel equivalent C ∗ -algebras have equivalent
categories of representations.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Noncommutative torus
Let Θ = (θj,k ) be a real skew-symmetric matrix and let
U1 , ..., U2d be unitary operators. Then the noncommutative
torus AΘ is defined as the universal C ∗ -algebra generated by
Uj ’s satisfying
Uk Uj = e 2πiθj,k Uj Uk .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Noncommutative torus
Let Θ = (θj,k ) be a real skew-symmetric matrix and let
U1 , ..., U2d be unitary operators. Then the noncommutative
torus AΘ is defined as the universal C ∗ -algebra generated by
Uj ’s satisfying
Uk Uj = e 2πiθj,k Uj Uk .
For the connection between noncommutative tori and Gabor
analysis it is more useful to realize AΘ as the group C ∗ -algebra
of a lattice Λ in R2d .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Noncommutative torus
Let Θ = (θj,k ) be a real skew-symmetric matrix and let
U1 , ..., U2d be unitary operators. Then the noncommutative
torus AΘ is defined as the universal C ∗ -algebra generated by
Uj ’s satisfying
Uk Uj = e 2πiθj,k Uj Uk .
For the connection between noncommutative tori and Gabor
analysis it is more useful to realize AΘ as the group C ∗ -algebra
of a lattice Λ in R2d .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Twisted group algebras
Let Λ be a lattice in R2d and c a continuous 2-cocycle with
values in T.
Then the twisted group algebra `1 (Λ, c) is `1 (Λ) with twisted
convolution \ as multiplication and ∗ as involution.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Twisted group algebras
Let Λ be a lattice in R2d and c a continuous 2-cocycle with
values in T.
Then the twisted group algebra `1 (Λ, c) is `1 (Λ) with twisted
convolution \ as multiplication and ∗ as involution.
Twisted convolution of a and b is defined by
X
a\b(λ) =
a(µ)b(λ − µ)c(µ, λ − µ) for λ, µ ∈ Λ,
µ∈Λ
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Twisted group algebras
Let Λ be a lattice in R2d and c a continuous 2-cocycle with
values in T.
Then the twisted group algebra `1 (Λ, c) is `1 (Λ) with twisted
convolution \ as multiplication and ∗ as involution.
Twisted convolution of a and b is defined by
X
a\b(λ) =
a(µ)b(λ − µ)c(µ, λ − µ) for λ, µ ∈ 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).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Twisted group algebras
Let Λ be a lattice in R2d and c a continuous 2-cocycle with
values in T.
Then the twisted group algebra `1 (Λ, c) is `1 (Λ) with twisted
convolution \ as multiplication and ∗ as involution.
Twisted convolution of a and b is defined by
X
a\b(λ) =
a(µ)b(λ − µ)c(µ, λ − µ) for λ, µ ∈ 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.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Twisted group algebras
Let Λ be a lattice in R2d and c a continuous 2-cocycle with
values in T.
Then the twisted group algebra `1 (Λ, c) is `1 (Λ) with twisted
convolution \ as multiplication and ∗ as involution.
Twisted convolution of a and b is defined by
X
a\b(λ) =
a(µ)b(λ − µ)c(µ, λ − µ) for λ, µ ∈ 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.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
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
X
πΛ (a) :=
a(λ)π(λ) for a = (a(λ)) ∈ `1 (Λ).
λ∈Λ
πΛ (a\b) = πΛ (a)πΛ (b)
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
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
X
πΛ (a) :=
a(λ)π(λ) for a = (a(λ)) ∈ `1 (Λ).
λ∈Λ
πΛ (a\b) = πΛ (a)πΛ (b)
πΛ (a∗ ) = πΛ (a)∗
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
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
X
πΛ (a) :=
a(λ)π(λ) for a = (a(λ)) ∈ `1 (Λ).
λ∈Λ
πΛ (a\b) = πΛ (a)πΛ (b)
πΛ (a∗ ) = πΛ (a)∗
This involutive representation of `1 (Λ, c) is faithful,
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
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
X
πΛ (a) :=
a(λ)π(λ) for a = (a(λ)) ∈ `1 (Λ).
λ∈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.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
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
X
πΛ (a) :=
a(λ)π(λ) for a = (a(λ)) ∈ `1 (Λ).
λ∈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 ∈ `1s (Λ) by
A1s (Λ, c).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
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
X
πΛ (a) :=
a(λ)π(λ) for a = (a(λ)) ∈ `1 (Λ).
λ∈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 ∈ `1s (Λ) by
A1s (Λ, c).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Spectrally invariant subalgebras of C ∗ (Λ, c)
Recall that a unital subalgebra A of a unital C ∗ -algebra B with
common unit is called spectrally invariant, if for A ∈ A with
A−1 ∈ B one actually has that A−1 ∈ A.
X
A1s (Λ, c) = {A ∈ B(L2 (R)) : A =
a(λ)π(λ), kak`1s < ∞}
λ
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Spectrally invariant subalgebras of C ∗ (Λ, c)
Recall that a unital subalgebra A of a unital C ∗ -algebra B with
common unit is called spectrally invariant, if for A ∈ A with
A−1 ∈ B one actually has that A−1 ∈ A.
X
A1s (Λ, c) = {A ∈ B(L2 (R)) : A =
a(λ)π(λ), kak`1s < ∞}
λ
smooth noncommutative torus
Luef, Franz
A∞ (Λ, c)
=
T
1
s≥0 As (Λ, c)
Hilbert C ∗ -modules over noncommutative tori
Spectrally invariant subalgebras of C ∗ (Λ, c)
Recall that a unital subalgebra A of a unital C ∗ -algebra B with
common unit is called spectrally invariant, if for A ∈ A with
A−1 ∈ B one actually has that A−1 ∈ A.
X
A1s (Λ, c) = {A ∈ B(L2 (R)) : A =
a(λ)π(λ), kak`1s < ∞}
λ
smooth noncommutative torus
A∞ (Λ, c)
=
T
1
s≥0 As (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 torus
C ∗ (Λ, c).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Spectrally invariant subalgebras of C ∗ (Λ, c)
Recall that a unital subalgebra A of a unital C ∗ -algebra B with
common unit is called spectrally invariant, if for A ∈ A with
A−1 ∈ B one actually has that A−1 ∈ A.
X
A1s (Λ, c) = {A ∈ B(L2 (R)) : A =
a(λ)π(λ), kak`1s < ∞}
λ
smooth noncommutative torus
A∞ (Λ, c)
=
T
1
s≥0 As (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 torus
C ∗ (Λ, c).
The statement for A∞ (Λ, c) was obtained by Connes in 1980
and the one for A1s (Λ, c) is due to Gröchenig-Leinert and
independently by Rosenberg.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Spectrally invariant subalgebras of C ∗ (Λ, c)
Recall that a unital subalgebra A of a unital C ∗ -algebra B with
common unit is called spectrally invariant, if for A ∈ A with
A−1 ∈ B one actually has that A−1 ∈ A.
X
A1s (Λ, c) = {A ∈ B(L2 (R)) : A =
a(λ)π(λ), kak`1s < ∞}
λ
smooth noncommutative torus
A∞ (Λ, c)
=
T
1
s≥0 As (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 torus
C ∗ (Λ, c).
The statement for A∞ (Λ, c) was obtained by Connes in 1980
and the one for A1s (Λ, c) is due to Gröchenig-Leinert and
independently by Rosenberg.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Equivalence bimodule between noncommutative
tori
Quote from Rieffel’s seminal paper from 1988:
We need suitable spaces of functions for the construction of
equivalence bimodules between noncommutative tori.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Equivalence bimodule between noncommutative
tori
Quote from Rieffel’s seminal paper from 1988:
We need suitable spaces of functions for the construction of
equivalence bimodules between noncommutative tori.
In the present context this means we need a space of functions
on Rd which behaves well under both the Fourier transform
and restriction to subgroups.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Equivalence bimodule between noncommutative
tori
Quote from Rieffel’s seminal paper from 1988:
We need suitable spaces of functions for the construction of
equivalence bimodules between noncommutative tori.
In the present context this means we need a space of functions
on Rd which behaves well under both the Fourier transform
and restriction to subgroups.
As suggested by Weil, the appropriate space is the space S(Rd )
of Schwartz functions.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Equivalence bimodule between noncommutative
tori
Quote from Rieffel’s seminal paper from 1988:
We need suitable spaces of functions for the construction of
equivalence bimodules between noncommutative tori.
In the present context this means we need a space of functions
on Rd which behaves well under both the Fourier transform
and 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 the
Schwartz space.
Shortly, after he introduced the class of modulation spaces
Msp,q (Rd ) and it turned out that S0 (Rd ) is the modulation
space M 1 (Rd ).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Equivalence bimodule between noncommutative
tori
Quote from Rieffel’s seminal paper from 1988:
We need suitable spaces of functions for the construction of
equivalence bimodules between noncommutative tori.
In the present context this means we need a space of functions
on Rd which behaves well under both the Fourier transform
and 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 the
Schwartz space.
Shortly, after he introduced the class of modulation spaces
Msp,q (Rd ) and it turned out that S0 (Rd ) is the modulation
space M 1 (Rd ).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Symplectic Fourier transform
b d are of the form
Observe that the characters of Rd × R
0)
2πiΩ(z,z
0
bd ,
z 7→ χs (z) = e
for some z ∈ Rd × R
where the standard symplectic form Ω of z = (x, ω) and
z 0 = (y , η) is defined by
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Symplectic Fourier transform
b d are of the form
Observe that the characters of Rd × R
0)
2πiΩ(z,z
0
bd ,
z 7→ χs (z) = e
for some z ∈ Rd × R
where the standard symplectic form Ω of z = (x, ω) and
z 0 = (y , η) is defined by
Ω(z, z 0 ) = y · ω − x · η.
b d is R
b d × Rd .
Consequently, the dual group of Rd × R
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Symplectic Fourier transform
b d are of the form
Observe that the characters of Rd × R
0)
2πiΩ(z,z
0
bd ,
z 7→ χs (z) = e
for some z ∈ Rd × R
where the standard symplectic form Ω of z = (x, ω) and
z 0 = (y , η) is defined by
Ω(z, z 0 ) = y · ω − x · η.
b d is R
b d × Rd .
Consequently, the dual group of Rd × R
In terms of the Euclidean inner product h., .i on R2d the
symplectic form Ω can be expressed as follows:
pause
0 Id
Ω(z, z 0 ) = hJz, z 0 i for J =
.
−Id 0
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Symplectic Fourier transform
b d are of the form
Observe that the characters of Rd × R
0)
2πiΩ(z,z
0
bd ,
z 7→ χs (z) = e
for some z ∈ Rd × R
where the standard symplectic form Ω of z = (x, ω) and
z 0 = (y , η) is defined by
Ω(z, z 0 ) = y · ω − x · η.
b d is R
b d × Rd .
Consequently, the dual group of Rd × R
In terms of the Euclidean inner product h., .i on R2d the
symplectic form Ω can be expressed as follows:
pause
0 Id
Ω(z, z 0 ) = hJz, z 0 i for J =
.
−Id 0
Λ◦ = {z ∈ R2d : e 2πiΩ(λ,z) = 1 for all λ ∈ Λ}.
The relation between Λ⊥ and the adjoint lattice Λ◦ is
Λ◦ = JΛ⊥ .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Symplectic Fourier transform
b d are of the form
Observe that the characters of Rd × R
0)
2πiΩ(z,z
0
bd ,
z 7→ χs (z) = e
for some z ∈ Rd × R
where the standard symplectic form Ω of z = (x, ω) and
z 0 = (y , η) is defined by
Ω(z, z 0 ) = y · ω − x · η.
b d is R
b d × Rd .
Consequently, the dual group of Rd × R
In terms of the Euclidean inner product h., .i on R2d the
symplectic form Ω can be expressed as follows:
pause
0 Id
Ω(z, z 0 ) = hJz, z 0 i for J =
.
−Id 0
Λ◦ = {z ∈ R2d : e 2πiΩ(λ,z) = 1 for all λ ∈ Λ}.
The relation between Λ⊥ and the adjoint lattice Λ◦ is
Λ◦ = JΛ⊥ .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Feichtinger’s algebra - Properties
Definition:
Weighted versions of Feichtinger’s algebra
M 1 (Rd ) = {f ∈ L2 (Rd ) : kf kMs1 =
RRs
2
2 s/2 dxdω < ∞}
R2d |hπ(x, ω)g i|(1 + |x| + |ω| )
If f , g ∈ Ms1 (Rd ), then Vg f ∈ Ms1 (R2d ).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Feichtinger’s algebra - Properties
Definition:
Weighted versions of Feichtinger’s algebra
M 1 (Rd ) = {f ∈ L2 (Rd ) : kf kMs1 =
RRs
2
2 s/2 dxdω < ∞}
R2d |hπ(x, ω)g i|(1 + |x| + |ω| )
If f , g ∈ Ms1 (Rd ), then Vg f ∈ Ms1 (R2d ).
M 1 (R2d ) is invariant
symplectic Fourier
RRunder the2πiΩ(b
z ,z) dz.
b
transform: F (b
z ) = R2d F (z)e
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Feichtinger’s algebra - Properties
Definition:
Weighted versions of Feichtinger’s algebra
M 1 (Rd ) = {f ∈ L2 (Rd ) : kf kMs1 =
RRs
2
2 s/2 dxdω < ∞}
R2d |hπ(x, ω)g i|(1 + |x| + |ω| )
If f , g ∈ Ms1 (Rd ), then Vg f ∈ Ms1 (R2d ).
M 1 (R2d ) is invariant
symplectic Fourier
RRunder the2πiΩ(b
z ,z) dz.
b
transform: F (b
z ) = R2d F (z)e
(Vg1\
f1 · Vg2 f2 )(z) = (Vf2 f1 · Vg2 g1 )(z)
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Feichtinger’s algebra - Properties
Definition:
Weighted versions of Feichtinger’s algebra
M 1 (Rd ) = {f ∈ L2 (Rd ) : kf kMs1 =
RRs
2
2 s/2 dxdω < ∞}
R2d |hπ(x, ω)g i|(1 + |x| + |ω| )
If f , g ∈ Ms1 (Rd ), then Vg f ∈ Ms1 (R2d ).
M 1 (R2d ) is invariant
symplectic Fourier
RRunder the2πiΩ(b
z ,z) dz.
b
transform: F (b
z ) = R2d F (z)e
(Vg1\
f1 · Vg2 f2 )(z) = (Vf2 f1 · Vg2 g1 )(z)
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Janssen representation
Poisson summation formula for the symplectic Fourier
transform:
X
X
F (λ) = vol(Λ)−1
Fb (λ◦ ).
λ∈Λ
λ◦ ∈Λ◦
Fundamental Identity of Gabor analysis:
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Janssen representation
Poisson summation formula for the symplectic Fourier
transform:
X
X
F (λ) = vol(Λ)−1
Fb (λ◦ ).
λ∈Λ
λ◦ ∈Λ◦
Fundamental Identity of Gabor analysis:
X
X
hf , π(λ◦ )kihπ(λ◦ )h, g i.
hf , π(λ)g ihπ(λ)h, ki = vol(Λ)−1
λ◦ ∈Λ◦
λ∈Λ
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Janssen representation
Poisson summation formula for the symplectic Fourier
transform:
X
X
F (λ) = vol(Λ)−1
Fb (λ◦ ).
λ∈Λ
λ◦ ∈Λ◦
Fundamental Identity of Gabor analysis:
X
X
hf , π(λ◦ )kihπ(λ◦ )h, g i.
hf , π(λ)g ihπ(λ)h, ki = vol(Λ)−1
λ◦ ∈Λ◦
λ∈Λ
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Janssen representation
Theorem:
For f , g , h ∈ Ms1 (Rd ) or in S(Rd ) we have that
X
X
hf , π(λ)g ihπ(λ)h = vol(Λ)−1
hh, π(λ◦ )g iπ(λ◦ )f ,
λ◦ ∈Λ◦
λ∈Λ
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Rieffel’s Theorem
A deep result of Rieffel relates these different noncommutative
tori.
Theorem:
C ∗ (Λ, c) and C ∗ (Λ◦ , c) are Morita-Rieffel equivalent.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Rieffel’s Theorem
A deep result of Rieffel relates these different noncommutative
tori.
Theorem:
C ∗ (Λ, c) and C ∗ (Λ◦ , c) are Morita-Rieffel equivalent.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Gabor frames – traditional point of view
Let G(g , Λ) = {π(λ)g : λ ∈ Λ} be a Gabor system.
analysis operator: Cg f = (hf , π(λ)g i)λ∈Λ
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Gabor frames – traditional point of view
Let G(g , Λ) = {π(λ)g : λ ∈ Λ} be a Gabor system.
analysis operator: Cg f = (hf , π(λ)g i)λ∈Λ
P
synthesis operator: Dg a = λ∈Λ a(λ)π(λ)f
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Gabor frames – traditional point of view
Let G(g , Λ) = {π(λ)g : λ ∈ Λ} be a Gabor system.
analysis operator: Cg f = (hf , π(λ)g i)λ∈Λ
P
synthesis operator: Dg a = λ∈Λ a(λ)π(λ)f
P
frame operator: Sg ,Λ f = λ∈Λ hf , π(λ)g iπ(λ)g
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Gabor frames – traditional point of view
Let G(g , Λ) = {π(λ)g : λ ∈ Λ} be a Gabor system.
analysis operator: Cg f = (hf , π(λ)g i)λ∈Λ
P
synthesis operator: Dg a = λ∈Λ a(λ)π(λ)f
P
frame operator: Sg ,Λ f = λ∈Λ hf , π(λ)g iπ(λ)g
G(g , Λ) is a frame for L2 (Rd ) if Sg ,Λ is invertible on
L2 (Rd ).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Gabor frames – traditional point of view
Let G(g , Λ) = {π(λ)g : λ ∈ Λ} be a Gabor system.
analysis operator: Cg f = (hf , π(λ)g i)λ∈Λ
P
synthesis operator: Dg a = λ∈Λ a(λ)π(λ)f
P
frame operator: Sg ,Λ f = λ∈Λ hf , π(λ)g iπ(λ)g
G(g , Λ) is a frame for L2 (Rd ) if Sg ,Λ is invertible on
L2 (Rd ).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Gabor frames – traditional point of view
Let G(g , Λ) = {π(λ)g : λ ∈ Λ} be a Gabor system.
analysis operator: Cg f = (hf , π(λ)g i)λ∈Λ
P
synthesis operator: Dg a = λ∈Λ a(λ)π(λ)f
P
frame operator: Sg ,Λ f = λ∈Λ hf , π(λ)g iπ(λ)g
G(g , Λ) is a frame for L2 (Rd ) if Sg ,Λ is invertible on
L2 (Rd ).
X
Akf k22 ≤
|hf , π(λ)g i|2 ≤ Bkf k22
λ∈Λ
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Gabor frames – traditional point of view
Let G(g , Λ) = {π(λ)g : λ ∈ Λ} be a Gabor system.
analysis operator: Cg f = (hf , π(λ)g i)λ∈Λ
P
synthesis operator: Dg a = λ∈Λ a(λ)π(λ)f
P
frame operator: Sg ,Λ f = λ∈Λ hf , π(λ)g iπ(λ)g
G(g , Λ) is a frame for L2 (Rd ) if Sg ,Λ is invertible on
L2 (Rd ).
X
Akf k22 ≤
|hf , π(λ)g i|2 ≤ Bkf k22
λ∈Λ
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Discrete reconstruction
f
= S −1 Sf =
X
hf , π(λ)S −1 g iπ(λ)g
λ∈Λ
= SS
−1
f =
X
hf , π(λ)g iπ(λ)S −1 g
λ∈Λ
= S
−1/2
SS
−1/2
f =
X
hf , π(λ)S −1/2 g iπ(λ)S −1/2 g .
λ∈Λ
canonical dual atom g̃ := (SgΛ,g )−1 g
canonical tight atom h0 := (SgΛ,g )−1/2 g
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Observation
Left action of A1s (Λ, c) on Ms1 (Rd ) by
X
Da g = πΛ (a)·g =
a(λ)π(λ) g for a ∈ `1s (Λ), g ∈ Ms1 (Rd )
λ∈Λ
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Observation
Left action of A1s (Λ, c) on Ms1 (Rd ) by
X
Da g = πΛ (a)·g =
a(λ)π(λ) g for a ∈ `1s (Λ), g ∈ Ms1 (Rd )
λ∈Λ
Λ hf , g i
=
X
hf , π(λ)g iπ(λ)
λ∈Λ
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Observation
Left action of A1s (Λ, c) on Ms1 (Rd ) by
X
Da g = πΛ (a)·g =
a(λ)π(λ) g for a ∈ `1s (Λ), g ∈ Ms1 (Rd )
λ∈Λ
Λ hf , g i
=
X
hf , π(λ)g iπ(λ)
λ∈Λ
For f , g ∈ Ms1 (Rd ) define
P
Λ hf , g i = πΛ (Vg f ) =
λ∈Λ hf , π(λ)g iπ(λ)
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Observation
Left action of A1s (Λ, c) on Ms1 (Rd ) by
X
Da g = πΛ (a)·g =
a(λ)π(λ) g for a ∈ `1s (Λ), g ∈ Ms1 (Rd )
λ∈Λ
Λ hf , g i
=
X
hf , π(λ)g iπ(λ)
λ∈Λ
For f , g ∈ Ms1 (Rd ) define
P
Λ hf , g i = πΛ (Vg f ) =
λ∈Λ hf , π(λ)g iπ(λ)
Λ hf , g i
= Λ hg , f i∗
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Observation
Left action of A1s (Λ, c) on Ms1 (Rd ) by
X
Da g = πΛ (a)·g =
a(λ)π(λ) g for a ∈ `1s (Λ), g ∈ Ms1 (Rd )
λ∈Λ
Λ hf , g i
=
X
hf , π(λ)g iπ(λ)
λ∈Λ
For f , g ∈ Ms1 (Rd ) define
P
Λ hf , g i = πΛ (Vg f ) =
λ∈Λ hf , π(λ)g iπ(λ)
Λ hf , g i
= Λ hg , f i∗
Λ hπΛ (a)f , g i
= πΛ (a) Λ hf , g i
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Observation
Left action of A1s (Λ, c) on Ms1 (Rd ) by
X
Da g = πΛ (a)·g =
a(λ)π(λ) g for a ∈ `1s (Λ), g ∈ Ms1 (Rd )
λ∈Λ
Λ hf , g i
=
X
hf , π(λ)g iπ(λ)
λ∈Λ
For f , g ∈ Ms1 (Rd ) define
P
Λ hf , g i = πΛ (Vg f ) =
λ∈Λ hf , π(λ)g iπ(λ)
Λ hf , g i
= Λ hg , f i∗
Λ hπΛ (a)f , g i
Λ hf , f i
= πΛ (a) Λ hf , g i
is positive in C ∗ (Λ, c), acutally in A1s (Λ, c)
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Observation
Left action of A1s (Λ, c) on Ms1 (Rd ) by
X
Da g = πΛ (a)·g =
a(λ)π(λ) g for a ∈ `1s (Λ), g ∈ Ms1 (Rd )
λ∈Λ
Λ hf , g i
=
X
hf , π(λ)g iπ(λ)
λ∈Λ
For f , g ∈ Ms1 (Rd ) define
P
Λ hf , g i = πΛ (Vg f ) =
λ∈Λ hf , π(λ)g iπ(λ)
Λ hf , g i
= Λ hg , f i∗
Λ hπΛ (a)f , g i
Λ hf , f i
= πΛ (a) Λ hf , g i
is positive in C ∗ (Λ, c), acutally in A1s (Λ, c)
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Observation
Left action of A1s (Λ, c) on Ms1 (Rd ) by
X
Da g = πΛ (a)·g =
a(λ)π(λ) g for a ∈ `1s (Λ), g ∈ Ms1 (Rd )
λ∈Λ
Λ hf , g i
=
X
hf , π(λ)g iπ(λ)
λ∈Λ
For f , g ∈ Ms1 (Rd ) define
P
Λ hf , g i = πΛ (Vg f ) =
λ∈Λ hf , π(λ)g iπ(λ)
Λ hf , g i
= Λ hg , f i∗
Λ hπΛ (a)f , g i
Λ hf , f i
= πΛ (a) Λ hf , g i
is positive in C ∗ (Λ, c), acutally in A1s (Λ, c)
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Hilbert C ∗ (Λ, c)-module
Theorem:
Ms1 (Rd ) becomes a full left Hilbert C ∗ (Λ, c)-module Λ V w.r.t
to right action on Ms1 (Rd ) and the inner product h., .iΛ when
1/2
completed w.r.t. kf kΛ = khf , f iΛ kop .
There is an analogous result for the opposite C ∗ -algebra of
C ∗ (Λ◦ , c), i.e. C ∗ (Λ◦ , c).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Hilbert C ∗ (Λ, c)-module
Theorem:
Ms1 (Rd ) becomes a full left Hilbert C ∗ (Λ, c)-module Λ V w.r.t
to right action on Ms1 (Rd ) and the inner product h., .iΛ when
1/2
completed w.r.t. kf kΛ = khf , f iΛ kop .
There is an analogous result for the opposite C ∗ -algebra of
C ∗ (Λ◦ , c), i.e. C ∗ (Λ◦ , c).
What is the appropriate right action of C ∗ (Λ◦ , c) on Ms1 (Rd )?
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Hilbert C ∗ (Λ, c)-module
Theorem:
Ms1 (Rd ) becomes a full left Hilbert C ∗ (Λ, c)-module Λ V w.r.t
to right action on Ms1 (Rd ) and the inner product h., .iΛ when
1/2
completed w.r.t. kf kΛ = khf , f iΛ kop .
There is an analogous result for the opposite C ∗ -algebra of
C ∗ (Λ◦ , c), i.e. C ∗ (Λ◦ , c).
What is the appropriate right action of C ∗ (Λ◦ , c) on Ms1 (Rd )?
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Hilbert C ∗ (Λ◦ , c)-module
FIGA = vol(Λ)−1
X
hπ(λ◦ )∗ f , kihπ(λ◦ )h, g i
λ◦ ∈Λ◦
D
E
X
=
vol(Λ)−1
hπ(λ◦ )h, g iπ(λ◦ )∗ f , k
λ◦ ∈Λ◦
g · πΛ◦ (b) =
X
π(λ◦ )∗ g b(λ◦ ) b ∈ `1s (Λ◦ ), g ∈ Ms1 (R)
λ◦ ∈Λ◦
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Hilbert C ∗ (Λ◦ , c)-module
FIGA = vol(Λ)−1
X
hπ(λ◦ )∗ f , kihπ(λ◦ )h, g i
λ◦ ∈Λ◦
D
E
X
=
vol(Λ)−1
hπ(λ◦ )h, g iπ(λ◦ )∗ f , k
λ◦ ∈Λ◦
g · πΛ◦ (b) =
X
π(λ◦ )∗ g b(λ◦ ) b ∈ `1s (Λ◦ ), g ∈ Ms1 (R)
λ◦ ∈Λ◦
hf , g iΛ◦ =
X
π(λ◦ )∗ hg , π(λ◦ )f i f , g ∈ Ms1 (Rd )
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Hilbert C ∗ (Λ◦ , c)-module
FIGA = vol(Λ)−1
X
hπ(λ◦ )∗ f , kihπ(λ◦ )h, g i
λ◦ ∈Λ◦
D
E
X
=
vol(Λ)−1
hπ(λ◦ )h, g iπ(λ◦ )∗ f , k
λ◦ ∈Λ◦
g · πΛ◦ (b) =
X
π(λ◦ )∗ g b(λ◦ ) b ∈ `1s (Λ◦ ), g ∈ Ms1 (R)
λ◦ ∈Λ◦
hf , g iΛ◦ =
X
π(λ◦ )∗ hg , π(λ◦ )f i f , g ∈ Ms1 (Rd )
Theorem:
Ms1 (Rd ) becomes a full right Hilbert C ∗ (Λ◦ , c)-module VΛ◦
w.r.t to right action on Ms1 (Rd ) and the inner product h., .iΛ◦
1/2
when completed w.r.t. kf kΛ◦ = khf , f iΛ◦ kop .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Hilbert C ∗ (Λ◦ , c)-module
FIGA = vol(Λ)−1
X
hπ(λ◦ )∗ f , kihπ(λ◦ )h, g i
λ◦ ∈Λ◦
D
E
X
=
vol(Λ)−1
hπ(λ◦ )h, g iπ(λ◦ )∗ f , k
λ◦ ∈Λ◦
g · πΛ◦ (b) =
X
π(λ◦ )∗ g b(λ◦ ) b ∈ `1s (Λ◦ ), g ∈ Ms1 (R)
λ◦ ∈Λ◦
hf , g iΛ◦ =
X
π(λ◦ )∗ hg , π(λ◦ )f i f , g ∈ Ms1 (Rd )
Theorem:
Ms1 (Rd ) becomes a full right Hilbert C ∗ (Λ◦ , c)-module VΛ◦
w.r.t to right action on Ms1 (Rd ) and the inner product h., .iΛ◦
1/2
when completed w.r.t. kf kΛ◦ = khf , f iΛ◦ kop .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Adjointable C ∗ (Λ, c)-module operators
By definition T is adjointable if there exists a T ? such that
?
Λ hf , Tg i = Λ hf , T g i
X
hTf , π(λ)g iπ(λ) =
λ∈Λ
X
hf , π(λ)T ? g iπ(λ)
λ∈Λ
T ? exists only if T π(λ) = π(λ)T for all λ ∈ Λ, such an
operator T is Λ-invariant.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Adjointable C ∗ (Λ, c)-module operators
By definition T is adjointable if there exists a T ? such that
?
Λ hf , Tg i = Λ hf , T g i
X
hTf , π(λ)g iπ(λ) =
λ∈Λ
X
hf , π(λ)T ? g iπ(λ)
λ∈Λ
T ? exists only if T π(λ) = π(λ)T for all λ ∈ Λ, such an
operator T is Λ-invariant.
Therefore, T ? is the adjoint T ∗ of a Λ-invariant operator.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Adjointable C ∗ (Λ, c)-module operators
By definition T is adjointable if there exists a T ? such that
?
Λ hf , Tg i = Λ hf , T g i
X
hTf , π(λ)g iπ(λ) =
λ∈Λ
X
hf , π(λ)T ? g iπ(λ)
λ∈Λ
T ? exists only if T π(λ) = π(λ)T for all λ ∈ Λ, such an
operator T is Λ-invariant.
Therefore, T ? is the adjoint T ∗ of a Λ-invariant operator.
Most important example, ΘΛg ,h f = Λ hf , g i · h and its adjoint is
ΘΛh,g .
ΘΛg ,h are known as Gabor frame-type operator
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Adjointable C ∗ (Λ, c)-module operators
By definition T is adjointable if there exists a T ? such that
?
Λ hf , Tg i = Λ hf , T g i
X
hTf , π(λ)g iπ(λ) =
λ∈Λ
X
hf , π(λ)T ? g iπ(λ)
λ∈Λ
T ? exists only if T π(λ) = π(λ)T for all λ ∈ Λ, such an
operator T is Λ-invariant.
Therefore, T ? is the adjoint T ∗ of a Λ-invariant operator.
Most important example, ΘΛg ,h f = Λ hf , g i · h and its adjoint is
ΘΛh,g .
ΘΛg ,h are known as Gabor frame-type operator
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Invertible adjointable C ∗ (Λ, c)-module operators
If ΘΛg ,g is invertible, then G (g , Λ) is a Gabor frame for L2 (Rd ).
Equivalently, Λ V is singly generated.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Invertible adjointable C ∗ (Λ, c)-module operators
If ΘΛg ,g is invertible, then G (g , Λ) is a Gabor frame for L2 (Rd ).
Equivalently, Λ V is singly generated.
There are various criteria about the invertibility of the Gabor
frame operator.
For g ∈ M 1 (Rd ) the Gabor frame operator ΘΛg ,g is
invertible if and only if the analysis operator CgΛ is
one-to-one from M 1 (Rd )0 to `∞ (Λ).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Invertible adjointable C ∗ (Λ, c)-module operators
If ΘΛg ,g is invertible, then G (g , Λ) is a Gabor frame for L2 (Rd ).
Equivalently, Λ V is singly generated.
There are various criteria about the invertibility of the Gabor
frame operator.
For g ∈ M 1 (Rd ) the Gabor frame operator ΘΛg ,g is
invertible if and only if the analysis operator CgΛ is
one-to-one from M 1 (Rd )0 to `∞ (Λ).
For g ∈ M 1 (Rd ) the Gabor frame operator ΘΛg ,g is
invertible if and only if the synthesis operator DgΛ is
one-to-one from `∞ (Λ) to M 1 (Rd )0 .
For g ∈ M 1 (RdP
) the Gabor frame operator has as Weyl
symbol σg ,Λ = λ∈Λ Tλ W (g , g ), where W (g , g ) is the
Wigner distribution.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Invertible adjointable C ∗ (Λ, c)-module operators
If ΘΛg ,g is invertible, then G (g , Λ) is a Gabor frame for L2 (Rd ).
Equivalently, Λ V is singly generated.
There are various criteria about the invertibility of the Gabor
frame operator.
For g ∈ M 1 (Rd ) the Gabor frame operator ΘΛg ,g is
invertible if and only if the analysis operator CgΛ is
one-to-one from M 1 (Rd )0 to `∞ (Λ).
For g ∈ M 1 (Rd ) the Gabor frame operator ΘΛg ,g is
invertible if and only if the synthesis operator DgΛ is
one-to-one from `∞ (Λ) to M 1 (Rd )0 .
For g ∈ M 1 (RdP
) the Gabor frame operator has as Weyl
symbol σg ,Λ = λ∈Λ Tλ W (g , g ), where W (g , g ) is the
Wigner distribution.
Then ΘΛg ,g is invertible if and only if σg ,Λ is invertible in
M ∞,1 (R2d ) with respect to twisted convolution.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Invertible adjointable C ∗ (Λ, c)-module operators
If ΘΛg ,g is invertible, then G (g , Λ) is a Gabor frame for L2 (Rd ).
Equivalently, Λ V is singly generated.
There are various criteria about the invertibility of the Gabor
frame operator.
For g ∈ M 1 (Rd ) the Gabor frame operator ΘΛg ,g is
invertible if and only if the analysis operator CgΛ is
one-to-one from M 1 (Rd )0 to `∞ (Λ).
For g ∈ M 1 (Rd ) the Gabor frame operator ΘΛg ,g is
invertible if and only if the synthesis operator DgΛ is
one-to-one from `∞ (Λ) to M 1 (Rd )0 .
For g ∈ M 1 (RdP
) the Gabor frame operator has as Weyl
symbol σg ,Λ = λ∈Λ Tλ W (g , g ), where W (g , g ) is the
Wigner distribution.
Then ΘΛg ,g is invertible if and only if σg ,Λ is invertible in
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 call
the algebras A0 and B0 Morita-Rieffel equivalent.
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 call
the algebras A0 and B0 Morita-Rieffel equivalent.
The inner products Λ h., .i and h., .iΛ◦ satisfy Rieffel’s
associativity condition
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 call
the algebras A0 and B0 Morita-Rieffel equivalent.
The inner products Λ h., .i and h., .iΛ◦ satisfy Rieffel’s
associativity condition
Λ hf , g i
Luef, Franz
· h = f hg , hiΛ◦
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 call
the algebras A0 and B0 Morita-Rieffel equivalent.
The inner products Λ h., .i and h., .iΛ◦ satisfy Rieffel’s
associativity condition
Λ hf , g i
· h = f hg , hiΛ◦
rank-one operators on the Hilbert C ∗ (Λ, c)-module:
ΘΛf ,g h = Λ hf , g i · h = SgΛ,h f are Gabor frame-type operators
and the associativity condition is the Janssen representation for
these Gabor frame-type operators.
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 call
the algebras A0 and B0 Morita-Rieffel equivalent.
The inner products Λ h., .i and h., .iΛ◦ satisfy Rieffel’s
associativity condition
Λ hf , g i
· h = f hg , hiΛ◦
rank-one operators on the Hilbert C ∗ (Λ, c)-module:
ΘΛf ,g h = Λ hf , g i · h = SgΛ,h f are Gabor frame-type operators
and the associativity condition is the Janssen representation for
these Gabor frame-type operators.
Theorem:
Ms1 (Rd ) is an equivalence bimodule between C ∗ (Λ, c) and
C ∗ (Λ◦ , c). Moreover Ms1 (Rd ) is an equivalence bimodule
between A1s (Λ, c) and A1s (Λ◦ , c).
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 call
the algebras A0 and B0 Morita-Rieffel equivalent.
The inner products Λ h., .i and h., .iΛ◦ satisfy Rieffel’s
associativity condition
Λ hf , g i
· h = f hg , hiΛ◦
rank-one operators on the Hilbert C ∗ (Λ, c)-module:
ΘΛf ,g h = Λ hf , g i · h = SgΛ,h f are Gabor frame-type operators
and the associativity condition is the Janssen representation for
these Gabor frame-type operators.
Theorem:
Ms1 (Rd ) is an equivalence bimodule between C ∗ (Λ, c) and
C ∗ (Λ◦ , c). Moreover Ms1 (Rd ) is an equivalence bimodule
between A1s (Λ, c) and A1s (Λ◦ , c).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Projective modules over noncommutative tori
Theorem:
Ms1 (Rd ) is a finitely generated projective right A1 (Λ◦ , c),i.e.
there exist g1 , ..., gn in Ms1 (Rd )
f =
n
X
Λ hf , gi i
i=1
Luef, Franz
· gi =
n
X
f hgi , gi iΛ◦ .
i=1
Hilbert C ∗ -modules over noncommutative tori
Projective modules over noncommutative tori
Theorem:
Ms1 (Rd ) is a finitely generated projective right A1 (Λ◦ , c),i.e.
there exist g1 , ..., gn in Ms1 (Rd )
f =
n
X
Λ hf , gi i
· gi =
i=1
Λ hf , f i
n
X
f hgi , gi iΛ◦ .
i=1
=
n
X
Λ hgi , f iΛ hf , gi i
i=1
for all f in Ms1 (Rd ) (or S(Rd )).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Projective modules over noncommutative tori
Theorem:
Ms1 (Rd ) is a finitely generated projective right A1 (Λ◦ , c),i.e.
there exist g1 , ..., gn in Ms1 (Rd )
f =
n
X
Λ hf , gi i
· gi =
i=1
Λ hf , f i
n
X
f hgi , gi iΛ◦ .
i=1
=
n
X
Λ hgi , f iΛ hf , gi i
i=1
for all f in Ms1 (Rd ) (or S(Rd )).In particular
kf k22 =
n X
X
|hf , π(λ)gi i|2 .
i=1 λ∈Λ
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Projective modules over noncommutative tori
Theorem:
Ms1 (Rd ) is a finitely generated projective right A1 (Λ◦ , c),i.e.
there exist g1 , ..., gn in Ms1 (Rd )
f =
n
X
Λ hf , gi i
· gi =
i=1
Λ hf , f i
n
X
f hgi , gi iΛ◦ .
i=1
=
n
X
Λ hgi , f iΛ hf , gi i
i=1
for all f in Ms1 (Rd ) (or S(Rd )).In particular
kf k22 =
n X
X
|hf , π(λ)gi i|2 .
i=1 λ∈Λ
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Generalizations
b , where G is a
All our results hold for Λ a lattice in G × G
locally compact abelian group.
There exist weighted variants of our statements:
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Generalizations
b , where G is a
All our results hold for Λ a lattice in G × G
locally 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 .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Generalizations
b , where G is a
All our results hold for Λ a lattice in G × G
locally 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 .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Generalizations
b , where G is a
All our results hold for Λ a lattice in G × G
locally 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.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Generalizations
b , where G is a
All our results hold for Λ a lattice in G × G
locally 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.
Gröchenig: 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 all
λ ∈ Λ.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Generalizations
b , where G is a
All our results hold for Λ a lattice in G × G
locally 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.
Gröchenig: 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 all
λ ∈ Λ.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Projections in a C ∗ -algebra
Lemma:
• Let g be in A VB . Then Pg := A hg , g i is a projection in A
if and only if g hg , g iB = g .
• Any element g in the unit sphere S(VB ) of the Hilbert
B-module VB , i.e. of all g ∈ VB such that hg , g iB = IB ,
gives a projection Pg in A.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Projections in a C ∗ -algebra
Lemma:
• Let g be in A VB . Then Pg := A hg , g i is a projection in A
if and only if g hg , g iB = g .
• Any element g in the unit sphere S(VB ) of the Hilbert
B-module VB , i.e. of all g ∈ VB such that hg , g iB = IB ,
gives a projection Pg in A.
−1/2
In particular g0 = g hg , g iB
is in S(VB ), therefore Pg0 is
a (canonical) projection in A.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Projections in a C ∗ -algebra
Lemma:
• Let g be in A VB . Then Pg := A hg , g i is a projection in A
if and only if g hg , g iB = g .
• Any element g in the unit sphere S(VB ) of the Hilbert
B-module VB , i.e. of all g ∈ VB such that hg , g iB = IB ,
gives a projection Pg in A.
−1/2
In particular g0 = g hg , g iB
is in S(VB ), therefore Pg0 is
a (canonical) projection in A.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Projections in noncommutative tori
Recall p is called a projection if p = p ∗ = p 2 .
Theorem:
Let G(g , Λ) be a Gabor system on L2 (Rd ).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Projections in noncommutative tori
Recall p is called a projection if p = p ∗ = p 2 .
Theorem:
Let G(g , Λ) be a Gabor system on L2 (Rd ).Then pg = Λ hg , g i is
a projection in C ∗ (Λ, c) if and only if one of the following
condition holds:
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Projections in noncommutative tori
Recall p is called a projection if p = p ∗ = p 2 .
Theorem:
Let G(g , Λ) be a Gabor system on L2 (Rd ).Then pg = Λ hg , g i is
a projection in C ∗ (Λ, c) if and only if one of the following
condition holds:
G(g , Λ) is a tight Gabor frame for L2 (Rd ).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Projections in noncommutative tori
Recall p is called a projection if p = p ∗ = p 2 .
Theorem:
Let G(g , Λ) be a Gabor system on L2 (Rd ).Then pg = Λ hg , g i is
a projection in C ∗ (Λ, c) if and only if one of the following
condition holds:
G(g , Λ) is a tight Gabor frame for L2 (Rd ).
G(g , Λ◦ ) is an orthogonal system.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Projections in noncommutative tori
Recall p is called a projection if p = p ∗ = p 2 .
Theorem:
Let G(g , Λ) be a Gabor system on L2 (Rd ).Then pg = Λ hg , g i is
a projection in C ∗ (Λ, c) if and only if one of the following
condition holds:
G(g , Λ) is a tight Gabor frame for L2 (Rd ).
G(g , Λ◦ ) is an orthogonal system.
hg , g iΛ◦ = I .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Projections in noncommutative tori
Recall p is called a projection if p = p ∗ = p 2 .
Theorem:
Let G(g , Λ) be a Gabor system on L2 (Rd ).Then pg = Λ hg , g i is
a projection in C ∗ (Λ, c) if and only if one of the following
condition holds:
G(g , Λ) is a tight Gabor frame for L2 (Rd ).
G(g , Λ◦ ) is an orthogonal system.
hg , g iΛ◦ = I .
hg , π(λ◦ )g i = vol(Λ)δλ◦ ,0 for all λ◦ ∈ Λ◦ .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Projections in noncommutative tori
Recall p is called a projection if p = p ∗ = p 2 .
Theorem:
Let G(g , Λ) be a Gabor system on L2 (Rd ).Then pg = Λ hg , g i is
a projection in C ∗ (Λ, c) if and only if one of the following
condition holds:
G(g , Λ) is a tight Gabor frame for L2 (Rd ).
G(g , Λ◦ ) is an orthogonal system.
hg , g iΛ◦ = I .
hg , π(λ◦ )g i = vol(Λ)δλ◦ ,0 for all λ◦ ∈ Λ◦ .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Projections in noncommutative tori
Recall p is called a projection if p = p ∗ = p 2 .
Theorem:
Let G(g , Λ) be a Gabor system on L2 (Rd ).Then pg = Λ hg , g i is
a projection in C ∗ (Λ, c) if and only if one of the following
condition holds:
G(g , Λ) is a tight Gabor frame for L2 (Rd ).
G(g , Λ◦ ) is an orthogonal system.
hg , g iΛ◦ = I .
hg , π(λ◦ )g i = vol(Λ)δλ◦ ,0 for all λ◦ ∈ Λ◦ .
In particular the canonical tight Gabor atom h0 := (SgΛ,g )−1/2 g
yields a projection ph0 in C ∗ (Λ, c).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Projections in noncommutative tori
Recall p is called a projection if p = p ∗ = p 2 .
Theorem:
Let G(g , Λ) be a Gabor system on L2 (Rd ).Then pg = Λ hg , g i is
a projection in C ∗ (Λ, c) if and only if one of the following
condition holds:
G(g , Λ) is a tight Gabor frame for L2 (Rd ).
G(g , Λ◦ ) is an orthogonal system.
hg , g iΛ◦ = I .
hg , π(λ◦ )g i = vol(Λ)δλ◦ ,0 for all λ◦ ∈ Λ◦ .
In particular the canonical tight Gabor atom h0 := (SgΛ,g )−1/2 g
yields a projection ph0 in C ∗ (Λ, c).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Projections in noncommutative tori – continued
Lema:
Let g be in Λ VΛ◦ . Then Pg := Λ hg , g i is a projection in
C ∗ (Λ, c) if and only if g hg , g iΛ◦ = g . If g ∈ Ms1 (R) or S (R),
then Pg gives a projection in A1s (Λ, c) or A∞ (Λ, c),
respectively.
First we assume that g hg , g iΛ◦ = g for some g in Λ VΛ◦ . Then
we have that
Pg2 = Λ hg , g iΛ hg , g i = Λ Λ hg , g ig , g = Λ hg hg , g iΛ◦ , g i = Λ hg , g i
and Pg∗ = Pg .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Projections in noncommutative tori – continued
Lema:
Let g be in Λ VΛ◦ . Then Pg := Λ hg , g i is a projection in
C ∗ (Λ, c) if and only if g hg , g iΛ◦ = g . If g ∈ Ms1 (R) or S (R),
then Pg gives a projection in A1s (Λ, c) or A∞ (Λ, c),
respectively.
First we assume that g hg , g iΛ◦ = g for some g in Λ VΛ◦ . Then
we have that
Pg2 = Λ hg , g iΛ hg , g i = Λ Λ hg , g ig , g = Λ hg hg , g iΛ◦ , g i = Λ hg , g i
and Pg∗ = Pg .
Now we suppose that Λ hg , g i is a projection in C ∗ (Λ, c). Then
an elementary computation yields the assertion:
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Projections in noncommutative tori – continued
Lema:
Let g be in Λ VΛ◦ . Then Pg := Λ hg , g i is a projection in
C ∗ (Λ, c) if and only if g hg , g iΛ◦ = g . If g ∈ Ms1 (R) or S (R),
then Pg gives a projection in A1s (Λ, c) or A∞ (Λ, c),
respectively.
First we assume that g hg , g iΛ◦ = g for some g in Λ VΛ◦ . Then
we have that
Pg2 = Λ hg , g iΛ hg , g i = Λ Λ hg , g ig , g = Λ hg hg , g iΛ◦ , g i = Λ hg , g i
and Pg∗ = Pg .
Now we suppose that Λ hg , g i is a projection in C ∗ (Λ, c). Then
an elementary computation yields the assertion:
Λ g hg , g iΛ◦ − g , g hg , g iΛ◦ − g = 0.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Projections in noncommutative tori – continued
Lema:
Let g be in Λ VΛ◦ . Then Pg := Λ hg , g i is a projection in
C ∗ (Λ, c) if and only if g hg , g iΛ◦ = g . If g ∈ Ms1 (R) or S (R),
then Pg gives a projection in A1s (Λ, c) or A∞ (Λ, c),
respectively.
First we assume that g hg , g iΛ◦ = g for some g in Λ VΛ◦ . Then
we have that
Pg2 = Λ hg , g iΛ hg , g i = Λ Λ hg , g ig , g = Λ hg hg , g iΛ◦ , g i = Λ hg , g i
and Pg∗ = Pg .
Now we suppose that Λ hg , g i is a projection in C ∗ (Λ, c). Then
an elementary computation yields the assertion:
Λ g hg , g iΛ◦ − g , g hg , g iΛ◦ − g = 0.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Consequences
The unit sphere of the Hilbert C ∗ (Λ◦ , c)-module VΛ◦ is
defined by S(VΛ◦ ) = {g ∈ VΛ◦ : hg , g iΛ◦ = I }, which is the set
of all tight Gabor frames.
2
1
g1 (t) = (2)1/4 e −πt a Gaussian, g2 (t) = ( π2 )1/2 cosh(πt)
the
hyperbolic secant and g3 (t) = e −π|t| the two-sided exponential.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Consequences
The unit sphere of the Hilbert C ∗ (Λ◦ , c)-module VΛ◦ is
defined by S(VΛ◦ ) = {g ∈ VΛ◦ : hg , g iΛ◦ = I }, which is the set
of all tight Gabor frames.
2
1
the
g1 (t) = (2)1/4 e −πt a Gaussian, g2 (t) = ( π2 )1/2 cosh(πt)
hyperbolic secant and g3 (t) = e −π|t| the two-sided exponential.
Theorem:
Let Λ = αZ × βZ. Then pgi = Λ hgi , gi i is a projection in
C ∗ (Λ, c) if and only if αβ < 1.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Consequences
The unit sphere of the Hilbert C ∗ (Λ◦ , c)-module VΛ◦ is
defined by S(VΛ◦ ) = {g ∈ VΛ◦ : hg , g iΛ◦ = I }, which is the set
of all tight Gabor frames.
2
1
the
g1 (t) = (2)1/4 e −πt a Gaussian, g2 (t) = ( π2 )1/2 cosh(πt)
hyperbolic secant and g3 (t) = e −π|t| the two-sided exponential.
Theorem:
Let Λ = αZ × βZ. Then pgi = Λ hgi , gi i is a projection in
C ∗ (Λ, c) if and only if αβ < 1.
The case of the Gaussian g1 is known as Boca’s projection. In
Manin’s work pg1 = Λ hg1 , g1 i appear as quantum theta
functions.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Consequences
The unit sphere of the Hilbert C ∗ (Λ◦ , c)-module VΛ◦ is
defined by S(VΛ◦ ) = {g ∈ VΛ◦ : hg , g iΛ◦ = I }, which is the set
of all tight Gabor frames.
2
1
the
g1 (t) = (2)1/4 e −πt a Gaussian, g2 (t) = ( π2 )1/2 cosh(πt)
hyperbolic secant and g3 (t) = e −π|t| the two-sided exponential.
Theorem:
Let Λ = αZ × βZ. Then pgi = Λ hgi , gi i is a projection in
C ∗ (Λ, c) if and only if αβ < 1.
The case of the Gaussian g1 is known as Boca’s projection. In
Manin’s work pg1 = Λ hg1 , g1 i appear as quantum theta
functions.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Extensions
Feichtinger-Kaiblinger showed in particular that the set of
Gabor frames G(g0 , Λ) with g0 ∈ Ms1 (Rd ) is open, i.e. there
exists a ε > 0 such that for kg − g0 kMs1 < ε then G(g , Λ) is
also a Gabor frame.
Consequently, we have an open set of projections Λ hg , g i close
to Λ hg0 , g0 i.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Extensions
Feichtinger-Kaiblinger showed in particular that the set of
Gabor frames G(g0 , Λ) with g0 ∈ Ms1 (Rd ) is open, i.e. there
exists a ε > 0 such that for kg − g0 kMs1 < ε then G(g , Λ) is
also a Gabor frame.
Consequently, we have an open set of projections Λ hg , g i close
to Λ hg0 , g0 i.
Actually, Feichtinger and Kaiblinger proved that one can also
vary the lattice.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Extensions
Feichtinger-Kaiblinger showed in particular that the set of
Gabor frames G(g0 , Λ) with g0 ∈ Ms1 (Rd ) is open, i.e. there
exists a ε > 0 such that for kg − g0 kMs1 < ε then G(g , Λ) is
also a Gabor frame.
Consequently, we have an open set of projections Λ hg , g i close
to Λ hg0 , g0 i.
Actually, Feichtinger and Kaiblinger proved that one can also
vary the lattice.
Therefore, a formulation in our setting amounts to incorporate
the fact that noncommutative tori are also give rise to a
continuous field of C ∗ -algebras.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Extensions
Feichtinger-Kaiblinger showed in particular that the set of
Gabor frames G(g0 , Λ) with g0 ∈ Ms1 (Rd ) is open, i.e. there
exists a ε > 0 such that for kg − g0 kMs1 < ε then G(g , Λ) is
also a Gabor frame.
Consequently, we have an open set of projections Λ hg , g i close
to Λ hg0 , g0 i.
Actually, Feichtinger and Kaiblinger proved that one can also
vary the lattice.
Therefore, a formulation in our setting amounts to incorporate
the fact that noncommutative tori are also give rise to a
continuous field of C ∗ -algebras.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Extensions
Feichtinger-Kaiblinger showed in particular that the set of
Gabor frames G(g0 , Λ) with g0 ∈ Ms1 (Rd ) is open, i.e. there
exists a ε > 0 such that for kg − g0 kMs1 < ε then G(g , Λ) is
also a Gabor frame.
Consequently, we have an open set of projections Λ hg , g i close
to Λ hg0 , g0 i.
Actually, Feichtinger and Kaiblinger proved that one can also
vary the lattice.
Therefore, a formulation in our setting amounts to incorporate
the fact that noncommutative tori are also give rise to a
continuous field of C ∗ -algebras.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Extensions
Rieffel demonstrated that viewed as quantum metric spaces one
is dealing with a continuous field of quantum metric spaces.
Another problem fitting into this framework is the
approximation of continuous Gabor frames by
finite-dimensional Gabor frames.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Extensions
Rieffel demonstrated that viewed as quantum metric spaces one
is dealing with a continuous field of quantum metric spaces.
Another problem fitting into this framework is the
approximation of continuous Gabor frames by
finite-dimensional Gabor frames.
In other words one is looking to control the changes of
projective modules over “close” noncommutative tori.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Extensions
Rieffel demonstrated that viewed as quantum metric spaces one
is dealing with a continuous field of quantum metric spaces.
Another problem fitting into this framework is the
approximation of continuous Gabor frames by
finite-dimensional Gabor frames.
In other words one is looking to control the changes of
projective modules over “close” noncommutative tori.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Rotation algebras
Theorem:
L Let α be an irrational number. Then C ∗ (αZ, R/Z) and
C ∗ (Z, R/αZ) are Morita-Rieffel equivalent.
C ∗ (αZ, R/Z) is the C ∗ -algebra for Z acting on the
circle by 2πα, and it is the completion of the ∗-algebra
Cc (T × Z) where the convolution product is given by
X
F ∗ G (t, n) =
F (m, t)G (n − m, e −2πimθ t)
m∈Z
and involution is given by
F ∗ (n, t) = F (−n, −e −2πiθ )t
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Rotation algebras
Theorem:
L Let α be an irrational number. Then C ∗ (αZ, R/Z) and
C ∗ (Z, R/αZ) are Morita-Rieffel equivalent.
C ∗ (αZ, R/Z) is the C ∗ -algebra for Z acting on the
circle by 2πα, and it is the completion of the ∗-algebra
Cc (T × Z) where the convolution product is given by
X
F ∗ G (t, n) =
F (m, t)G (n − m, e −2πimθ t)
m∈Z
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 by
2π/α and it is isomorphic to C ∗ (α−1 Z, R/Z).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Rotation algebras
Theorem:
L Let α be an irrational number. Then C ∗ (αZ, R/Z) and
C ∗ (Z, R/αZ) are Morita-Rieffel equivalent.
C ∗ (αZ, R/Z) is the C ∗ -algebra for Z acting on the
circle by 2πα, and it is the completion of the ∗-algebra
Cc (T × Z) where the convolution product is given by
X
F ∗ G (t, n) =
F (m, t)G (n − m, e −2πimθ t)
m∈Z
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 by
2π/α and it is isomorphic to C ∗ (α−1 Z, R/Z).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Morita-Rieffel equivalence for rotation algebras
Let A = C ∗ (αZ, R/Z) and B = C ∗ (Z, R/αZ). Then we define
on Cc (R) the structure of a right pre A Hilbert module and a
left pre B Hilbert module.
For f , g ∈ Cc (R), F ∈ C ∗ (αZ, R/Z) and G ∈ C ∗ (Z, R/αZ) we
define:
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Morita-Rieffel equivalence for rotation algebras
Let A = C ∗ (αZ, R/Z) and B = C ∗ (Z, R/αZ). Then we define
on Cc (R) the structure of a right pre A Hilbert module and a
left pre B Hilbert module.
For f , g ∈ Cc (R), F ∈ C ∗ (αZ, R/Z) and G ∈ C ∗ (Z, R/αZ) we
define:
X
(f F )(t) =
f (t − nα)F (nα, t − nα)
n∈Z
hf , g iA (mα, t) =
X
f (r − n)g (r − n + mα)
n∈Z
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Morita-Rieffel equivalence for rotation algebras
Let A = C ∗ (αZ, R/Z) and B = C ∗ (Z, R/αZ). Then we define
on Cc (R) the structure of a right pre A Hilbert module and a
left pre B Hilbert module.
For f , g ∈ Cc (R), F ∈ C ∗ (αZ, R/Z) and G ∈ C ∗ (Z, R/αZ) we
define:
X
(f F )(t) =
f (t − nα)F (nα, t − nα)
n∈Z
hf , g iA (mα, t) =
X
f (r − n)g (r − n + mα)
n∈Z
(G f )(t) =
X
G (n, t)f (t − n)
n∈Z
B hf , g i(m, t)
=
X
f (t − nα)g (t − nα − m).
n∈Z
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Morita-Rieffel equivalence for rotation algebras
Let A = C ∗ (αZ, R/Z) and B = C ∗ (Z, R/αZ). Then we define
on Cc (R) the structure of a right pre A Hilbert module and a
left pre B Hilbert module.
For f , g ∈ Cc (R), F ∈ C ∗ (αZ, R/Z) and G ∈ C ∗ (Z, R/αZ) we
define:
X
(f F )(t) =
f (t − nα)F (nα, t − nα)
n∈Z
hf , g iA (mα, t) =
X
f (r − n)g (r − n + mα)
n∈Z
(G f )(t) =
X
G (n, t)f (t − n)
n∈Z
B hf , g i(m, t)
=
X
f (t − nα)g (t − nα − m).
n∈Z
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Wiener amalgam spaces
In his work on Generalized Harmonic Analysis and Tauberian
Theorems Norbert Wiener introduced the space W (L∞ , `1 ).
Later H.-G. Feichtinger generalized Wiener’s space in his work
on function spaces which allows one to measure global and
local information W (B1 , B2 ).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Wiener amalgam spaces
In his work on Generalized Harmonic Analysis and Tauberian
Theorems Norbert Wiener introduced the space W (L∞ , `1 ).
Later H.-G. Feichtinger generalized Wiener’s space in his work
on function spaces which allows one to measure global and
local information W (B1 , B2 ).
These spaces found applications in approximation theory,
sampling theory, symbol classes of pseudo-differential
operators, stochastic processes, etc.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Wiener amalgam spaces
In his work on Generalized Harmonic Analysis and Tauberian
Theorems Norbert Wiener introduced the space W (L∞ , `1 ).
Later H.-G. Feichtinger generalized Wiener’s space in his work
on function spaces which allows one to measure global and
local information W (B1 , B2 ).
These spaces found applications in approximation theory,
sampling theory, symbol classes of pseudo-differential
operators, stochastic processes, etc.
We restrict our discussion to the case of weighted analogs of
W (L∞ , `1 ).
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Wiener amalgam spaces
In his work on Generalized Harmonic Analysis and Tauberian
Theorems Norbert Wiener introduced the space W (L∞ , `1 ).
Later H.-G. Feichtinger generalized Wiener’s space in his work
on function spaces which allows one to measure global and
local information W (B1 , B2 ).
These spaces found applications in approximation theory,
sampling theory, symbol classes of pseudo-differential
operators, stochastic processes, etc.
We restrict our discussion to the case of weighted analogs of
W (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.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Wiener amalgam spaces
In his work on Generalized Harmonic Analysis and Tauberian
Theorems Norbert Wiener introduced the space W (L∞ , `1 ).
Later H.-G. Feichtinger generalized Wiener’s space in his work
on function spaces which allows one to measure global and
local information W (B1 , B2 ).
These spaces found applications in approximation theory,
sampling theory, symbol classes of pseudo-differential
operators, stochastic processes, etc.
We restrict our discussion to the case of weighted analogs of
W (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.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Wiener amalgam spaces - continued
For α > 0 we set Iα = [0, α) and denote by χα the
characteristic function of Iα .
A function f belongs to W (L∞ , `1v ) if
X
kf kW (L∞ ,`1v ) =
kf · Tkα χα kL∞ v (k) < ∞.
k∈Z
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Wiener amalgam spaces - continued
For α > 0 we set Iα = [0, α) and denote by χα the
characteristic function of Iα .
A function f belongs to W (L∞ , `1v ) if
X
kf kW (L∞ ,`1v ) =
kf · Tkα χα kL∞ v (k) < ∞.
k∈Z
The definition is independent of α and we have the following
embeddings:
S ⊂ W (L∞ , `1v ) ⊂ W (L∞ , `1 ) ⊂ L2 .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Wiener amalgam spaces - continued
For α > 0 we set Iα = [0, α) and denote by χα the
characteristic function of Iα .
A function f belongs to W (L∞ , `1v ) if
X
kf kW (L∞ ,`1v ) =
kf · Tkα χα kL∞ v (k) < ∞.
k∈Z
The definition is independent of α and we have the following
embeddings:
S ⊂ W (L∞ , `1v ) ⊂ W (L∞ , `1 ) ⊂ L2 .
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Walnut representation of Gabor frame operator
Recall B = C ∗ (Z, R/αZ).
Lemma:
Let g , h be in W (L∞ , `1v ). Then hg , hiB is in L∞ (R).
Theorem:
If f , g , h are in W (L∞ , `1v ), then
X
hg , hiB f =
hf , Tαk Ml g iTαk Ml h =: Sg ,h f .
k,n∈Z
In time-frequency analysis the representation in the last
Theorem is called the Walnut representation of a Gabor
frame operator.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori
Walnut representation of Gabor frame operator
Recall B = C ∗ (Z, R/αZ).
Lemma:
Let g , h be in W (L∞ , `1v ). Then hg , hiB is in L∞ (R).
Theorem:
If f , g , h are in W (L∞ , `1v ), then
X
hg , hiB f =
hf , Tαk Ml g iTαk Ml h =: Sg ,h f .
k,n∈Z
In time-frequency analysis the representation in the last
Theorem is called the Walnut representation of a Gabor
frame operator.
Luef, Franz
Hilbert C ∗ -modules over noncommutative tori