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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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2 WU 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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4WU 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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6WU 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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10WU 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∗-MODULES 13

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.