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arXiv:math/0603091v1 [math.FA] 3 Mar 2006
Structured Parseval Frames in Hilbert
C∗-modules
Wu Jing, Deguang Han, and Ram N. Mohapatra
Abstract. We investigate the structured frames for Hilbert C∗-
modules. In the case that the underlying C∗-algebra is a com-
mutative W∗-algebra, we prove that the set of the Parseval frame
generators for a unitary operator group can be parameterized by
the set of all the unitary operators in the double commutant of
the group. Similar result holds for the set of all the general frame
generators where the unitary operators are replaced by invertible
and adjointable operators. Consequently, the set of all the Parseval
frame generators is path-connected. We also obtain the existence
and uniqueness results for the best Parseval multi-frame approxi-
mations for multi-frame generators of unitary operator groups on
Hilbert C∗-modules when the underlying C∗-algebra is commuta-
tive.
1. Introduction
Frames (modular frames) for Hilbert C∗-modules were introduced
by Frank and Larson and some basic properties were also investigated in
a series of their papers [2, 3, 4]. It should be remarked that although
(at the first glance) some of the definitions and result statements of
modular frames may appear look like similar to their Hilbert space
frame counterparts, these are not simple generalizations of the Hilbert
space frames due to the complexity of the Hilbert C∗-module struc-
tures and to the fact that many useful techniques in Hilbert spaces are
either not available or not known in Hilbert C∗-modules. For example,
it is well-known that that every Hilbert space has an orthonormal basis
1991 Mathematics Subject Classification. Primary 46L99; Secondary 42C15,
46H25.
Key words and phrases. Frames, Parseval frame vectors, mutil-frame approxi-
mations, unitary groups, Hilbert C∗-modules.
1
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2WU JING, DEGUANG HAN, AND RAM N. MOHAPATRA
which can be simply obtained by applying the Gram-Schmidt orthonor-
malization process to a linearly independent generating subset of the
Hilbert space. However, it is well-known that not every Hilbert C∗-
module has an “orthonormal basis”. This makes frames particulary
relevant to Hilbert C∗-modules. Remarkably every countably gener-
ated Hilbert C∗-module admits a (countable) frame. It requires a very
deep Hilbert C*-module result (Kasparov’s Stabilization Theorem) to
prove this fact which is a trivial fact in Hilbert space setting (cf [4, 18].
In fact, it is still an interesting question whether there exists a alter-
native proof for this fact without using the Kasparov’s Stabilization
Theorem. Another still open problem is whether every uncountably
generated Hilbert C∗-module admits a (uncountably indexed) Parseval
frame (again, a trivial fact for Hilbert spaces). Equivalently, for ev-
ery Hilbert C*-module over a unital C*-algebra A, does there exist an
isometric embedding into a standard Hilbert C*-module l2(A,I) as an
orthogonal direct summand for some index set I? (see [4]).
In recent years, there have been growing evidence indicating that
modular frames are also closely related to some other areas of research
such as the area of wavelet frame constructions (cf. [14, 15, 16, 22]).
Considering the fact that the theory and applications of structured
frames (such as Gabor frame, wavelet frames and frames induced from
group unitary representations) for Hilbert spaces have been the main
focus of the Hilbert space frame theory, we believe that structure mod-
ular frames may well be suitable for some applications either in theo-
retical or applied nature. The purpose of this paper is to initiate the
study of structured modular frames. It is reasonable that we should
first take a close look at those existed results for structured Hilbert
space frames and make an effort to check whether they are still valid
for structured modular frames. The two results (Theorems 3.2 and 4.3)
presented in this paper are generalizations of the corresponding Hilbert
space frame results obtained in [9] and [6]. Theorem 3.2 states that
all the Parseval frame generators for a unitary group can be parame-
terized in terms of the unitary elements in the double commutant of
the group under the commutativity condition on the underlying C∗-
algebras. This is slightly different from the Hilbert space setting since
the the double commutant theorem for von Neumann algebras is not
always available for the Hilbert C∗-module setting. For the similar
reason, the “finiteness ” of the involved “commutant” algebras need
to be verified, and each step needs to be carefully checked to make
sure it is valid in the C∗-algebra context. Theorem 4.3 deals with the
best approximations of modular frame generators by Parseval frame
generators. The difficulty arises when comes to compare two positive
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STRUCTURED PARSEVAL FRAMES IN HILBERT C∗-MODULES3
elements in the underlying C∗-algebras which is not an issue in the
scalar case. We are not able to prove these results when the underlying
C∗-algebras are non-commutative.
2. Preliminaries
This section contains some basic definitions about Hilbert C∗-modules
and some simple properties for Hilbert C∗-module frames that will be
needed in the next two sections. Let A be a C∗-algebra and H be a
(left) A-module. Suppose that the linear structures given on A and H
are compatible, i.e. λ(ax) = a(λx) for every λ ∈ C,a ∈ A and x ∈ H.
If there exists a mapping ?·,·? : H × H → A with the properties
(1) ?x,x? ≥ 0 for every x ∈ H,
(2) ?x,x? = 0 if and only if x = 0,
(3) ?x,y? = ?y,x?∗for every x,y ∈ H,
(4) ?ax,y? = a?x,y? for every a ∈ A, every x,y ∈ H,
(5) ?x + y,z? = ?x,z? + ?y,z? for every x,y,z ∈ H.
Then the pair {H,?·,·?} is called a (left)- pre-Hilbert A-module.
The map ?·,·? is said to be an A-valued inner product. If the pre-
Hilbert A-module {H,?·,·?} is complete with respect to the norm ?x? =
??x,x??
A Hilbert A-module H is (algebraically) finitely generated if there
exists a finite set {x1,...,xn} ⊆ H such that every element x ∈ H
can be expressed as an A-linear combination x =?n
Hilbert A-module is countably generated if there exists a countable set
of generator.
It should be mentioned that by no means all results of Hilbert
space theory can be simply generalized to the situation of Hilbert C∗-
modules. Fist of all, the analogue of the Riesz representation theo-
rem for bounded A-linear mapping is not valid for H. Secondly, the
bounded A-linear operator on H may not have an adjoint operator.
Thirdly, the Hilbert A-submodule I of the Hilbert A-module H is not
a direct summand. Let H be a Hilbert A-module over a unital C∗-
algebra A. The set of all bounded A-linear operators on H is denoted
by EndA(H), and the set of all adjointable bounded A-linear operators
on H is denoted by End∗
A C∗-algebra M is called a W∗-algebra if it is a dual space as
a Banach space, i.e. if there exists a Banach space M∗ such that
(M∗)∗= M. We also call M∗the predual of M. It should mention
here that End∗
M is said to be finite if its identity is finite. Equivalently, M is finite
if and only if every isometry in M is unitary.
1
2 then it is called a Hilbert A-module.
i=1ai,ai∈ A. A
A(H).
A(l2(A)) is not a W∗-algebra in general. A W∗-algebra
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4 WU JING, DEGUANG HAN, AND RAM N. MOHAPATRA
Definition 2.1. Let A be a unital C∗-algebra and J be a finite or
countable index set. A sequence {xj}j∈J of elements in a Hilbert A-
module H is said to be a (standard) frame if there exist two constants
C,D > 0 such that
C · ?x,x? ≤
?
j∈J
?x,xj??xj,x? ≤ D · ?x,x?
(2.1)
for every x ∈ H, where the sum in the middle of the inequality is
convergent in norm. The optimal constants (i.e. maximal for C and
minimal for D) are called frame bounds.
The frame {xj}j∈Jis said to be tight frame if C = D, and said to
be Parseval if C = D = 1.
Note that not every Hilbert C∗-module has an orthonormal basis.
Though any countably generated Hilbert C∗-module admits a frame,
there are countably generated Hilbert C∗-modules that contain no or-
thonormal basis even no orthogonal Riesz basis (see Example 3.4 in
[4]).
The main property of frames for Hilbert spaces is the existence of
the reconstruction formula that allows a simple standard decomposition
of every element of the spaces with respect to the frame. For standard
frames we have the following reconstruction formula.
Theorem 2.2. ([4]) Let {xj}j∈Jbe a standard frame in a finitely
or countably generated Hilbert A-module H over a unital C∗-algebra A.
Then there exists a unique operator S ∈ End∗
A(H) such that
x =
?
j∈J
?x,S(xj)?xj
for every x ∈ H. The operator can be explicitly given by the formula
S = T∗T for any adjointable invertible bounded operator T mapping H
onto some other Hilbert A-module K and realizing {T(xj) : j ∈ J} to
be a standard Parseval frame in K.
Let S ⊆ EndA(H), we denote its commutant {A ∈ EndA(H) :
AS = SA,S ∈ S} by S′. For a non-empty set U and a unital C∗-
algebra A, let l2
U(A) be the Hilbert A-module defined by
?
U∈U
Let {χU}U∈U denote the standard orthonormal basis of l2
χUtakes value 1Aat U and 0Aat everywhere else. In the case when U
is a group, we define for each U ∈ U,
l2
U(A) = {{aU}U∈U⊆ A :
aUa∗
Uconvergences in ? · ?}.
U(A), where
LUχV = χUV and RUχV = χV U−1.
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STRUCTURED PARSEVAL FRAMES IN HILBERT C∗-MODULES5
Note that L−1
the left and right regular representations of U.
Let H be a Hilbert A-module over a unital C∗-algebra A. A vector
ψ in H is called a wandering vector for a unitary group U on H if Uψ =
{Uψ : U ∈ U} is an orthonormal set. If Uψ is an orthonormal basis
for H, then ψ is called a complete wandering vector for U. Similarly,
a vector ψ is called a Parseval frame vector (resp. frame vector with
bounds C and D, or Bessel sequence vector with bound D) for a unitary
group U if Ux forms a Parseval frame (resp. frame with bounds C and
D, or Bessel sequence with bound D) for span(Ux). Moreover, x is
called a complete Parseval frame vector (resp. complete frame vector
with bounds C and D, or complete Bessel sequence with bound D) when
Ux is a Parseval frame (resp. frame with bounds C and D, or Bessel
sequence with bound D) for H.
The following simple lemma will be used in the proof of Theorem
3.2.
U= L∗
U= LU∗ and R−1
U= R∗
U= RU∗. Here L and R are
Lemma 2.3. Let G be a unitary group on a finitely or countably
generated Hilbert A-module H over a unital C∗-algebra A. If G admits
a complete Parseval frame vector η, then G is unitarily equivalent to
{LU|K: U ∈ U}, where K = T(H) and T : H → l2
operator defined by T(x) =?
Proof. It is easy to check that T is an adjointable isometry. By
Theorem 15.3.5 and Theorem 15.3.8 in [21], we have that
G(A) be the analysis
U∈G?x,Uη?χU.
l2
G(A) = (T(H))⊥⊕ T(H).
Hence we have the orthogonal projection P from l2
For each V ∈ G, we have
G(A) onto T(H).
LUT(V η) = LU(
?
W∈G
?V η,Wη?χW) =
?
W∈G
?
W∈G
?V η,Wη?χUW
=
?
W∈G
?UV η,UWη?χUW=
?UV η,Wη?χW
= TU(V η).
Thus LUT = TU.
?
We remark that the orthogonal projection from l2
is in the commutant of {LU : U ∈ G} and satisfies Tη = PχI. This
also implies the so-called dilation property meaning that there exists a
Hilbert A-module˜H ⊇ H and a unitary group˜G on˜H such that˜G has
complete wandering vectors in˜H, H is an invariant subspace of˜G such
G(A) onto T(H)
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6 WU JING, DEGUANG HAN, AND RAM N. MOHAPATRA
that˜G|H= G, and the map G ?→ G|His a group isomorphism from˜G
onto G.
3. Frame Vector Parameterizations
In [1] the set of all wandering vectors for a unitary group was pa-
rameterized by the set of unitary operators in its commutant. However,
unlike the wandering vector case, it was shown in [9] that the set of all
the Parseval frame vectors for a unitary group can not be parameter-
ized by the set of all the unitary operators in the communtant of the
unitary group. This means that the Parseval frame vectors for a repre-
sentation of a countable group are not necessarily unitarily equivalent.
However, this set can be parameterized by the set of all the unitary
operators in the von Neumann algebra generated by the representation
([6, 9]). This turns out to be a very useful result in Gabor analysis
(cf. [6, 8]). Although it remains a question whether this result is still
valid in the Hilbert C∗-module setting, we will prove in this section
that this result holds when the underlying C∗-algebra is a commuta-
tive W∗-algebra. Even in this commutative case, a lot more extra work
and care are needed in order to prove this generalization.
Lemma 3.1. Let G be a unitary group on a Hilbert A-module over
a commutative unital C∗-algebra A, then
M = N′= {RU: U ∈ G}′and N = M′= {LU: U ∈ G}′,
where M = {LU: U ∈ G}′′and N = {RU: U ∈ G}′′.
Proof. Note that RULV = LVRUholds for any U,V ∈ G. There-
fore to prove this lemma it suffices to show that TS = ST for arbitrary
T ∈ M′and S ∈ N′.
Suppose that
TχI=
?
U∈G
aUχU and SχI=
?
U∈G
bUχU
for some aU,bU∈ A.
Now for any V ∈ G, on one hand, we have
STχV
= STLVχI= SLVTχI
?
U∈G
?
U∈G
?
U∈G
= SLV(
aUχU) = S(
?
U∈G
aUχV U)
= S(
aUR(V U)−1χI) =
?
U∈G
aUR(V U)−1SχI
=
aUR(V U)−1(
?
W∈G
bWχW) =
?
U,W∈G
aUbWχWV U.
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STRUCTURED PARSEVAL FRAMES IN HILBERT C∗-MODULES7
On the other hand
TSχV
= TSRV−1χI= TRV−1SχI
?
W∈G
?
W∈G
?
W∈G
= TRV−1(
bWχW) = T(
?
W∈G
bWχWV)
= T(
bWLWVχI) =
?
W∈G
bWLWVTχI
=
bWLWV(
?
U∈G
aUχU) =
?
U,W∈G
bWaUχWV U.
Since A is commutative, it follows that STχV = TSχV, and so ST =
TS.
?
We now define a natural conjugate A-linear isomorphism π from
M onto M′= N by
π(A)BχI= BA∗χI, ∀A,B ∈ M.
In particular, π(A)χI= A∗χI.
Now we are in a position to prove the parameterization of complete
Parseval frame vectors for unitary groups.
Theorem 3.2. Let G be a unitary group on a finitely or countably
generated Hilbert A-module H over a commutative W∗-algebra A such
that l2
frame vector for G. For ξ in H we have
(1) ξ is a complete Parseval frame vector for G if and only if there
exists a unitary operator A ∈ G′′such that ξ = Aη.
(2) ξ is a complete frame vector for G if and only if there exists an
invertible and adjointable operator A ∈ G′′such that ξ = Aη.
(3) ξ is a complete Bessel sequence vector for G if and only if there
exists an adjointable operator A ∈ G′′such that ξ = Aη.
G(A) is self-dual. Suppose that η ∈ H be a complete Parseval
Proof. We will prove (1). The proof of (2) and (3) is similar and
we leave it to the interested readers.
By Lemma 2.3, we can assume that G = {LU|Rang(P),U ∈ G} and
η = PχI, where P is an orthogonal projection in the commutant of
{LU: U ∈ G}
Let M = {LU: U ∈ G}′′
First assume that there exists a unitary operator A ∈ G′′such that
ξ = Aη.
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8WU JING, DEGUANG HAN, AND RAM N. MOHAPATRA
We now show that Aη is a complete Parseval frame vector for G.
For any x ∈ Rang(P), we have
?
U∈G
?
U∈G
?
U∈G
?
U∈G
?
U∈G
?
U∈G
?x,UAη??UAη,x? =
?
U∈G
?x,LUPAη??LUPAη,x?
=
?x,LUPAPχI??LUPAPχI,x? =
?
U∈G
?x,LUPAχI??LUPAχI,x?
=
?x,PLUAχI??PLUAχI,x? =
?
U∈G
?Px,LUAχI??LUAχI,Px?
=
?x,LUπ(A∗)χI??LUπ(A∗)χI,x?
=
?x,π(A∗)LUχI??π(A∗)LUχI,x?
=
?(π(A∗))∗x,χU??χU,(π(A∗))∗x?
= ?(π(A∗))∗x,(π(A∗))∗x? = ?x,x?,
where in the seventh equality we use that fact π(A∗)LU = LUπ(A∗),
and in the last equality we use that fact that π(A∗) is unitary. Therefore
Aη is a complete Parseval frame vector for G.
Now let ξ ∈ Rang(P) be a complete Parseval frame vector for G.
We want to find a unitary operator A ∈ G′′such that ξ = Aη.
To this aim, we first define an operator B : l2
G(A) → l2
G(A) by
χU?−→ LUξ, U ∈ G.
One can check that B is an adjointable operator and B∗χV =
W∈G?LW−1LVη,ξ?χWfor any V ∈ G.
Now for any U,V ∈ G, we see that
?
?(BB∗− P)χU,χV?
?
W∈G
?
W∈G
?
W∈G
= ?LUη,LVη? − ?Uη,V η? = ?LUPχI,LVPχI? − ?Uη,V η?
= ?LUTηη,LVTηη? − ?Uη,V η? = ?TηUη,TηV η? − ?Uη,V η? = 0,
= ??LW−1LUη,ξ?χW,
?
S∈G
?LS−1LVη,ξ?χS? − ?TηUη,χV?
=
?LW−1LUη,ξ??ξ,LW−1LVη? − ?
?
W∈G
?Uη,Wη?χW,χV?
=
?LUη,LWξ??LWξ,LVη? − ?Uη,V η?
this leads to the fact that P = BB∗.
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STRUCTURED PARSEVAL FRAMES IN HILBERT C∗-MODULES9
From
BLUχV = BχUV = LUVξ = LULVξ = LUBχV,
we see that B ∈ M′. Hence B is a partial isometry in M′.
Let Q = B∗B, then P and Q are equivalent projections in M′.
Since l2
(End∗
σ(End∗
M and M′are W∗-algebras (see [20]).
G(A) is self-dual, by [17], End∗
G(A)))∗be its predual. One can check that M and M′are
A(l2
A(l2
G(A)) is a W∗-algebra. Let
A(l2
G(A)),(End∗
A(l2
G(A)))∗)-closed in End∗
A(l2
G(A)), and so both
Claim. M and M′are finite W∗-algebras.
We now define φ : M → A by
φ(A) = ?AχI,χI?, ∀A ∈ M.
We want to show that φ is a faithful A-valued trace for M.
Since span{LUχI,U ∈ G} = l2
G(A), for any A,B ∈ M, we have
AχI= lim
nAnχI and BχI= lim
nBnχI,
where
AnχI=
kn
?
i=1
a(n)
iLVi(n)χI and BnχI=
ln
?
j=1
b(n)
jLWj(n)χI
for some a(n)
Then
i,b(n)
j
∈ A and Vi(n),Wj(n)∈ G.
φ(AB) = ?ABχI,χI? = lim
mlim
n?
lm
?
j=1
kn
?
i=1
b(m)
j
a(n)
i LWj(m)LVi(n)χI,χI?.
While
φ(BA) = lim
nlim
m?
kn
?
i=1
lm
?
j=1
a(n)
ib(m)
j
LVi(n)LWj(m)χI,χI?.
Note that
?LWj(m)LVi(n)χI,χI? = ?LVi(n)LWj(m)χI,χI?.
Therefore φ(AB) = φ(BA).
If A ∈ M is positive and φ(A) = 0, then
?A
1
2χI,A
1
2χI? = ?AχI,χI? = φ(A) = 0.
Thus A
Now for any U ∈ G, we have
1
2χI= 0.
A
1
2χU= A
1
2RUχI= RUA
1
2χI= 0.
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10 WU JING, DEGUANG HAN, AND RAM N. MOHAPATRA
Therefore A
can prove that M′is also finite.
It follows from Proposition 2.4.2 in [20] that I − P and I − Q are
equivalent projections in M′. Therefore there exists a partial isometry
C ∈ M′such that CC∗= I − P and C∗C = I − Q.
Let T = B + C. Then T is a unitary operator in M′, and so
A = (π−1(T))∗is a unitary operator in M.
In order to complete the proof it remains to prove that A˜ η =˜ξ.
In fact,
1
2 = 0, and so A = 0. Similarly, by using Lemma 3.1 we
Aη = (π−1(T))∗PχI= P(π−1(T))∗χI
= Pπ(π−1(T))χI= PTχI
= P(B + C)χI= PBχI+ PCχI
= Pξ = ξ,
which completes the proof.
?
The following result follows immediately from Theorem 3.2 and the
fact the set of all the unitary elements in any W∗-algebra is path-
connected in norm.
Corollary 3.3. Let G be a unitary group on a finitely or countably
generated Hilbert A-module H over a commutative W∗-algebra A such
that l2
is path-connected.
G(A) is self-dual, then the set of all Paserval frame vectors for G
4. Parseval Frame Approximations
In the Hilbert space frame setting, the original work on symmetric
orthogonalization was done by L¨ owidin [12] in the late 1970’s. The
concept of symmetric approximation of frames by Parseval frame was
introduced in [5] to extend the symmetric orthogonalization of bases
by orthogonal bases in Hilbert spaces. The existence and the unique-
ness results for the symmetric approximation of frames by Parseval
frames were obtained in [5]. Following their definition, a Parseval frame
{yj}∞
Hilbert space H if it is similar to {xj}∞
j=1is said to be a symmetric approximation of frame {xj}∞
j=1in
j=1and
∞
?
j=1
?zj− xj?2≥
∞
?
j=1
?yj− xj?2
(4.1)
is valid for all Parseval frames {zj}∞
Observed by the first author ([6], [7]) that in some situations the
symmetric approximation fails to work when the underlying Hilbert
j=1of H that are similar to {xj}∞
j=1.
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STRUCTURED PARSEVAL FRAMES IN HILBERT C∗-MODULES11
space is infinite dimensional since if we restrict ourselves to the frames
induced by a unitary system then the summation in (4.1) is always
infinite when the given frame is not Parseval. Instead of using the
symmetric approximations to consider the frames generated by a col-
lection of unitary transformations and some window functions, it was
proposed to approximate the frame generator by Parseval frame gener-
ators. Existence and uniqueness results for such a best approximation
were obtained in [6, 7]. We will extend this result to Hilbert C∗-
module frames when the underlying C∗-algebra is commutative. It
remains open whether this is true when the underlying C∗-algebra is
non-commutative.
Following ([7]) we first give the following definition.
Definition 4.1. Let Φ = (φ1,...,φN) be a multi-frame generator
for a unitary system U. Then a Parseval multi-frame generator Ψ =
(ψ1,...,ψN) for U is called a best Parseval multi-frame approximation
for Φ if the inequality
N
?
k=1
?φk− ψk,φk− ψk? ≤
N
?
k=1
?φk− ξk,φk− ξk?
is valid for all the Parseval multi-frame generator Ξ = (ξ1,...,ξN) for
U.
Let Φ = {φ1,φ2,...,φN} be a multi-frame generator for a unitary
system U on a finitely or countably generated Hilbert A-module H over
a unital C∗-algebra A. We use TΦto denote the analysis operator from
H to l2
U×{1,2,...,N}(A) defined by
TΦx =
N
?
j=1
?
U∈U
?x,Uφj?χ(U,j), ∀x ∈ H,
where {χ(U,j): U ∈ U,j = 1,2,...,N} is the standard orthonormal
basis for l2
Note that TΦis adjointable and its adjoint operator satisfying
T∗
Φχ(U,j)= Uφj, U ∈ U, j = 1,2,...,N.
U×{1,2,...,N}(A).
Lemma 4.2. Let G be a unitary group on a Hilbert A-module H
over a commutative C∗-algebra A. Suppose that Φ = {φ1,φ2,...,φN}
and Ψ = {ψ1,ψ2,...,ψN} be two multi-frame generators for G, then
N
?
k=1
?φk,φk? =
N
?
k=1
?ψk,ψk?.
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12 WU JING, DEGUANG HAN, AND RAM N. MOHAPATRA
Proof. We compute
N
?
k=1
?φk,φk? =
N
?
k=1
N
?
j=1
?
U∈G
?φk,Uψj??Uψj,φk?
=
N
?
k=1
N
?
j=1
?
U∈G
?U∗φk,ψj??ψj,U∗φk?
=
N
?
j=1
N
?
k=1
?
U∈G
?ψj,U∗φk??U∗φk,ψj?
=
N
?
j=1
?ψj,ψj?.
?
Theorem 4.3. Let G be a unitary group on a finitely or countably
generated Hilbert A-module H over a commutative unital C∗-algebra
A. Suppose that Φ = {φ1,φ2,...,φN} is a multi-frame generator for
G. Then S
for Φ, where S is the frame operator for the multi-frame {Uφj : j =
1,...,N,U ∈ G}.
Proof. We first show that S ∈ G′.
For arbitrary V ∈ G and x ∈ H we have
1
2Φ is the unique best Parseval multi-frame approximation
SV x =
N
?
k=1
?
U∈G
?V x,Uφk?Uφk
=
N
?
k=1
?
U∈G
?x,V∗Uφk?Uφk
= V (
N
?
k=1
?
U∈G
?x,V∗Uφk?V∗Uφk)
= V (
N
?
k=1
?
U∈G
?x,Uφk?Uφk)
= V Sx.
This shows that S ∈ G′.
Since End∗
positive elements in C∗-algebra, we can infer that S−1
Therefore {S−1
frame generator for G.
A(H) is a C∗-algebra, by the spectral decomposition for
2,S−1
4 ∈ G′.
2φ1,S−1
2φ2,...,S−1
2φN} is a complete Parseval multi-
Page 13
STRUCTURED PARSEVAL FRAMES IN HILBERT C∗-MODULES13
Let Ψ = {ψ1,ψ2,...,ψN} be any Parseval multi-frame generator
for G. We claim that
N
?
k=1
?TS−1
2ΦS−1
4φk,TΨS−1
4φk? =
N
?
k=1
?ψk,φk?,
where TS−1
Parseval multi-frame generators S−1
We compute
2Φand TΨ are the analysis operators with respect to the
2Φ and Ψ respectively.
N
?
k=1
?TS−1
2ΦS−1
4φk,TΨS−1
4φk?
=
N
?
k=1
?
N
?
j=1
?
U∈G
?S−1
4φk,US−1
2φj?χ(U,j),
N
?
i=1
?
V ∈G
?S−1
4φk,V ψi?χ(V,i)?
=
N
?
k=1
N
?
j=1
?
U∈G
?S−1
4φk,US−1
2φj??Uψj,S−1
4φk?
=
N
?
k=1
N
?
j=1
?
U∈G
?Uψj,S−1
4φk??S−1
4φk,US−1
2φj?
=
N
?
j=1
N
?
k=1
?
U∈G
?S
1
4ψj,U∗S−1
2φk??U∗S−1
2φk,S−1
4φj?
=
N
?
j=1
?S
1
4ψj,S−1
4φj? =
N
?
j=1
?ψj,φj?.
We now prove that S
tion for Φ. We need to show that
1
2Φ is a best Parseval multi–frame approxima-
N
?
k=1
?ψk− φk,ψk− φk? ≥
N
?
k=1
?S−1
2φk− φk,S−1
2φk− φk?.
By Lemma 4.2, it suffices to prove that
N
?
k=1
(?S−1
2φk,φk? + ?φk,S−1
2φk) − ?ψk,φk? − ?φk,ψk?) ≥ 0.
Page 14
14 WU JING, DEGUANG HAN, AND RAM N. MOHAPATRA
In fact, we have
N
?
k=1
(?S−1
2φk,φk? + ?φk,S−1
2φk? − ?ψk,φk? − ?φk,ψk?)
=
N
?
k=1
(?S−1
4φk,S−1
4φk? + ?S−1
4φk,S−1
4φk)?
−?TS−1
N
?
k=1
2ΦS−1
4φk,TΨS−1
4φk? − ?TΨS−1
4φk,TS−1
2ΦS−1
4φk?)
=
(?TS−1
2ΦS−1
4φk,TS−1
2ΦS−1
4φk? + ?TΨS−1
4φk,TΨS−1
4φk?
−?TS−1
N
?
k=1
2ΦS−1
4φk,TΨS−1
4φk? − ?TΨS−1
4φk,TS−1
2ΦS−1
4φk?)
=
?(TS−1
2Φ− TΨ)S−1
4φk,(TS−1
2Φ− TΨ)S−1
4φk? ≥ 0.
This implies that S−1
for Φ.
For the uniqueness, assume that Ξ = {ξ1,ξ2,...,ξN} be another
best Parseval multi-frame approximation for Φ. Then we have
2Φ is a best Parseval multi-frame approximation
(4.2)
N
?
k=1
?ξk− φk,ξk− φk? =
N
?
k=1
?S−1
2φk− φk,S−1
2φk− φk?.
By Lemma 4.2, we also have
(4.3)
N
?
k=1
?ξk,ξk? =
N
?
k=1
?S−1
2φk,S−1
2φk?.
Identities (4.2) and (4.3) imply that
N
?
k=1
(?ξk,φk? + ?φk,ξk?) =
?
k=1
(?S−1
2φk,φk? + ?φk,S−1
2φk?)
= 2
N
?
k=1
?S−1
4φk,S−1
4φk?.
We claim that
N
?
k=1
?S
1
4ξk,S
1
4ξk? =
N
?
k=1
?S−1
4φk,S−1
4φk?.
Page 15
STRUCTURED PARSEVAL FRAMES IN HILBERT C∗-MODULES15
In fact,
N
?
k=1
?S
1
4ξk,S
1
4ξk?
=
N
?
k=1
N
?
j=1
?
U∈G
?S
1
4ξk,US−1
2φj??US−1
2φj,S
1
4ξk?
=
N
?
j=1
N
?
k=1
?
U∈G
?S−1
4φj,U∗ξk??U∗ξk,S−1
4φj?
=
N
?
j=1
?S−1
4φj,S−1
4φj?.
Then we have
N
?
k=1
?S
1
4ξk− S−1
4φk,S
1
4ξk− S−1
4φk?
=
N
?
k=1
−?S−1
(?S
1
4ξk,S
1
4ξk? − ?S
1
4ξk,S−1
4φk?
4φk,S
1
4ξk? + ?S−1
4φk,S−1
4φk?)
=
N
?
k=1
(2?S−1
4φk,S−1
4φk? − ?ξk,φk? − ?φk,ξk?)
= 0.
This implies that
S
1
4ξk= S−1
4φk, k = 1,2,...,N.
Therefore
ξk= S−1
2φk, k = 1,2,...,N.
i.e. Ξ = S−1
2Φ, as expected.
?
Acknowledgement. The first author is grateful to Professor M.
Frank for many useful communications and help. The authors thank
the referee for some valuable suggestions in improving the presentation
of the paper.
References
1. X. Dai, D. Larson, Wandering vectors for unitary systems and orthogonal
wavelets, Memoirs Amer. Math. Soc., 134 (1998), No. 640.
Page 16
16 WU JING, DEGUANG HAN, AND RAM N. MOHAPATRA
2. M. Frank, D. Larson, A module frame concept for Hilbert C∗-modules, in
Functional and Harmonic Analysis of wavelets (San Antonio, TX, Jan. 1999),
Contemp. Math., 247 (1999), 207–233.
3. M. Frank, D. Larson, Modular frames for Hilbert C∗-modules and symmetric
approximation of frames, Proc. SPIE, 4119 (2000), 325–336.
4. M. Frank, D. Larson, Frames in Hilbert C∗-modules and C∗-algebras, J. Op-
erator Theory, 48 (2002), 273–314.
5. M. Frank, V. Paulsen and T. Tiballi, Symmetric approximation of frames and
bases in Hilbert spaces, Trans. Amer. Math. Soc., 354 (2002), 777–793.
6. D. Han, Approximation for Gabor and wavelet frames, Trans. Amer. Math.
Soc., 355 (2004), 3329–3342.
7. D. Han, Tight frame approximation for multi-frames and super-frames, J. Ap-
prox. Theory, 129 (2004), 78–93.
8. D. Han, J-P. Gabardo and D. Larson, Gabor frames and operator algebras,
Wavelet Applications in Signal and Image Processing, Proc. SPIE., 4119
(2000), 337–345.
9. D. Han, D. Larson, Frames, bases and group representations, Memoirs Amer.
Math. Soc., 147 (2000), No. 697.
10. G. Kasparov, Hilbert C∗-modules: The theorem of Stinespring and Voiculescu,
J. Operator Theory, 4 (1980), 133–150.
11. E. Lance, Hilbert C∗-modules - a Toolkit for Operator Algebraists, London
Mathematical Society lecture Note Series v. 210, Cambridge University Press,
Cambridge, England, 1995.
12. P.-O. L¨ owidin, On the nonorthogonality problem, Adv. Quantum Chem., 5
(1970), 185–199.
13. V. Manuilov, E. Troisky, Hilbert C∗-Modules, American Mathematical Society,
Providene, Rhode island, 2005.
14. J. Packer, Applications of the work of Stone and von Neumann to wavelets,
Contemp. Math., 365 2004, 253–279.
15. J. Packer and M. Rieffel, Wavelet filter functions, the matrix completion prob-
lem, and projective modules over C(Tn), J. Fourier Anal. Appl., 9 (2003),
101–116
16. J. Packer and M. Rieffel, Projective multi-resolution analyses for L2(R2), J.
Fourier Anal. Appl., 10 (2004), 439–464.
17. W. Paschke, Inner product modules over B∗-algebras, Trans. Amer. Math. Soc.,
182 (1973), 443–468.
18. I. Raeburn, S. Thompson, Countably generated Hilbert modules, the Kasparov
stabilization theorem, and frames in Hilbert modules, Proc. Amer. Math. Soc.,
131 (2003), 1557–1564.
19. M. Rieffel, Morita equivalence for C∗-algebra and W∗-algebra, J. Pure Applied
Algebra, 5 (1974), 51–96.
20. S. Sakai, C∗-algebrs and W∗-algebras, Springer-Verlag, New York, 1998.
21. N. Wegge-Olsen, K-theory and C∗-algebras - a Friendly Approach, Oxford Uni-
versity Press, Oxford, England, 1993.
22. P. Wood, Wavelets and Hilbert modules, J. Fourier Anal. Appl., 10 (2004),
573–598.
Page 17
STRUCTURED PARSEVAL FRAMES IN HILBERT C∗-MODULES17
Department of Mathematics, University of central Florida, Or-
lando, FL 32816
E-mail address: mathjing@gmail.com
Department of Mathematics, University of central Florida, Or-
lando, FL 32816
E-mail address: dhan@pegasus.cc.ucf.edu
Department of Mathematics, University of central Florida, Or-
lando, FL 32816
E-mail address: ramm@mail.ucf.edu
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