




































©2022 Ada Academica https://adac.eeEur. J. Math. Anal. 2 (2022) 4doi: 10.28924/ada/ma.2.4
∗-K-Operator Frame for Hom∗A(X )

Mohamed Rossafi1,∗, Roumaissae El Jazzar2 and Ali Kacha2
1LaSMA Laboratory Department of Mathematics, Faculty of Sciences Dhar El Mahraz,

University Sidi Mohamed Ben Abdellah, B. P. 1796 Fes Atlas, Morocco
mohamed.rossafi@usmba.ac.ma

2Laboratory of Partial Differential Equations, Spectral Algebra and Geometry Department of Mathematics,
Faculty of Sciences, University Ibn Tofail, Kenitra, Morocco

roumaissae.eljazzar@uit.ac.ma, ali.kacha@yahoo.fr
∗Correspondence: rossafimohamed@gmail.com

Abstract. In this work, we introduce the concept of ∗-K-operator frames in Hilbert pro-C∗-modules,which is a generalization of K-operator frame. We present the analysis operator, the synthesisoperator and the frame operator. We also give some properties and we study the tensor product of
∗-K-operator frame for Hilbert pro-C∗-modules.

1. Introduction
Duffin and Schaeffer introduced the notion of frame in nonharmonic Fourier analysis in 1952 [3].In 1986 the work of Duffin and Schaeffer were reintroduced and developed by Grossman andMeyer [7]. The concept of frame on Hilbert space has already been successfully extended to pro-

C∗-algebras and Hilbert modules. Many properties of frames in Hilbert C∗-modules are valid forframes of multipliers in Hilbert modules over pro-C∗-algebras [9].Operator frames for B(H) is a new notion of frames that Li and Cio introduced in [11] andgeneralized by Rossafi in [16]. In this work we introduce the notion of ∗-K-operator frame for thespace Hom∗A(X ) of all adjointable operators on a Hilbert pro-C∗-module for X .This paper is divided into three sections. In section 2 we recall some fundamental definitionsand notations of Hilbert pro-C∗-modules. In section 3 we introduce the ∗-K-operator Frame andwe give some of its properties. Lastly we investigate tensor product of Hilbert pro-C∗-modules, weshow that tensor product of ∗-K-operator frames for Hilbert pro-C∗-modules X and Y , present an
∗-K-operator frames for X ⊗ Y , and tensor product of their frame operators is the frame operatorof their tensor product of ∗-K-operator frames.

Received: 18 Nov 2021.
Key words and phrases. frame; ∗-K-operator frame; K-operator frame pro-C∗-algebra; Hilbert pro-C∗-modules; tensorproduct. 1

https://adac.ee
https://doi.org/10.28924/ada/ma.2.4


Eur. J. Math. Anal. 10.28924/ada/ma.2.4 22. Preliminaries
The basic information about pro-C∗-algebras can be found in the works [4–6,8, 12,14,15].
C∗-algebra whose topology is induced by a family of continuous C∗-seminorms instead of a

C∗-norm is called pro-C∗-algebra. Hilbert pro-C∗-modules are generalizations of Hilbert spacesby allowing the inner product to take values in a pro-C∗-algebra rather than in the field of complexnumbers.Pro-C∗-algebra is defined as a complete Hausdorff complex topological ∗-algebra A whosetopology is determined by its continuous C∗-seminorms in the sens that a net {aα} converges to 0if and only if p(aα) converges to 0 for all continuous C∗-seminorm p on A [8,10,15], and we have:1) p(ab) ≤ p(a)p(b)2) p(a∗a) = p(a)2for all a, b ∈ AIf the topology of pro-C∗-algebra is determined by only countably many C∗-seminorms, then it iscalled a σ-C∗-algebra.We denote by sp(a) the spectrum of a such that: sp(a) = {λ ∈ C : λ1A − a is not invertible } forall a ∈ A. Where A is unital pro-C∗-algebra with unite 1A.The set of all continuous C∗-seminorms on A is denoted by S(A). If A+ denotes the set of allpositive elements of A, then A+ is a closed convex C∗-seminorms on A.
Example 2.1. Every C∗-algebra is a pro-C∗-algebra.
Proposition 2.2. [8] Let A be a unital pro-C∗-algebra with an identity 1A. Then for any p ∈ S(A),

we have:(1) p(a) = p(a∗) for all a ∈ A(2) p (1A) = 1(3) If a, b ∈ A+ and a ≤ b, then p(a) ≤ p(b)(4) If 1A ≤ b, then b is invertible and b−1 ≤ 1A(5) If a, b ∈ A+ are invertible and 0 ≤ a ≤ b, then 0 ≤ b−1 ≤ a−1(6) If a, b, c ∈ A and a ≤ b then c∗ac ≤ c∗bc(7) If a, b ∈ A+ and a2 ≤ b2, then 0 ≤ a ≤ b

Definition 2.3. [15] A pre-Hilbert module over pro-C∗-algebra A, is a complex vector space Ewhich is also a left A-module compatible with the complex algebra structure, equipped with an
A-valued inner product 〈., .〉 E×E → A which is C-and A-linear in its first variable and satisfiesthe following conditions:1) 〈ξ, η〉∗ = 〈η, ξ〉 for every ξ, η ∈ E2) 〈ξ, ξ〉 ≥ 0 for every ξ ∈ E

https://doi.org/10.28924/ada/ma.2.4


Eur. J. Math. Anal. 10.28924/ada/ma.2.4 33) 〈ξ, ξ〉 = 0 if and only if ξ = 0for every ξ, η ∈ E. We say E is a Hilbert A-module (or Hilbert pro-C∗-module over A ). If E iscomplete with respect to the topology determined by the family of seminorms
p̄E(ξ) =

√
p(〈ξ, ξ〉) ξ ∈ E, p ∈ S(A)

Let A be a pro-C∗-algebra and let X and Y be Hilbert A-modules and assume that I and J becountable index sets. A bounded A-module map from X to Y is called an operators from X to Y .We denote the set of all operator from X to Y by HomA(X ,Y).
Definition 2.4. [1] An A-module map T : X −→ Y is adjointable if there is a map T ∗ : Y −→ Xsuch that 〈Tξ, η〉 = 〈ξ, T ∗η〉 for all ξ ∈ X , η ∈ Y , and is called bounded if for all p ∈ S(A), thereis Mp > 0 such that p̄Y(Tξ) ≤ Mpp̄X (ξ) for all ξ ∈ X .We denote by Hom∗A(X ,Y), the set of all adjointable operator from X to Y and Hom∗A(X ) =

Hom∗A(X ,X )

Definition 2.5. [1] Let A be a pro-C∗-algebra and X ,Y be two Hilbert A-modules. The operator
T : X → Y is called uniformly bounded below, if there exists C > 0 such that for each p ∈ S(A),

p̄Y(Tξ) 6 Cp̄X (ξ), for all ξ ∈ X
and is called uniformly bounded above if there exists C′ > 0 such that for each p ∈ S(A),

p̄Y(Tξ) > C′p̄X (ξ), for all ξ ∈ X
‖T‖∞ = inf{M : M is an upper bound for T}
p̂Y(T ) = sup {p̄Y(T (x)) : ξ ∈ X , p̄X (ξ) 6 1}It’s clear to see that, p̂(T ) 6 ‖T‖∞ for all p ∈ S(A).

Proposition 2.6. [2]. Let X be a Hilbert module over pro-C∗-algebra A and T be an invertible
element in Hom∗A(X ) such that both are uniformly bounded. Then for each ξ ∈ X ,∥∥T−1∥∥−2∞ 〈ξ, ξ〉 ≤ 〈Tξ, Tξ〉 ≤ ‖T‖2∞〈ξ, ξ〉.

3. ∗-K-operator frame for Hom∗A(X )

We begin this section with the definition of a K-operator frame.
Definition 3.1. Let {Ti}i∈I be a family of adjointable operators on a Hilbert A-module X over aunital pro-C∗-algebra, and let K ∈ Hom∗A(X ). {Ti}i∈I is called a K-operator frame for Hom∗A(X ),if there exist two positive constants A,B > 0 such that

A〈K∗ξ,K∗ξ〉 ≤
∑
i∈I
〈Tiξ, Tiξ〉 ≤ B〈ξ, ξ〉,∀ξ ∈ X . (3.1)

https://doi.org/10.28924/ada/ma.2.4


Eur. J. Math. Anal. 10.28924/ada/ma.2.4 4The numbers A and B are called lower and upper bound of the K-operator frame, respectively. If
A〈K∗ξ,K∗ξ〉 =

∑
i∈I
〈Tiξ, Tiξ〉,

the K-operator frame is an A-tight. If A = 1, it is called a normalized tight K-operator frame or aParseval K-operator frame.
We will now move to define the ∗-K-operator frame for Hom∗A(X ).

Definition 3.2. Let {Ti}i∈I be a family of adjointable operators on a Hilbert A-module X overa unital pro-C∗-algebra, and let K ∈ Hom∗A(X ). {Ti}i∈I is called a ∗-K-operator frame for
Hom∗A(H), if there exists two nonzero elements A and B in A such that

A〈K∗ξ,K∗ξ〉A∗ ≤
∑
i∈I
〈Tiξ, Tiξ〉 ≤ B〈ξ, ξ〉B∗,∀ξ ∈ X . (3.2)

The elements A and B are called lower and upper bounds of the ∗-K-operator frame, respectively.If
A〈K∗ξ,K∗ξ〉∗ =

∑
i∈I
〈Tiξ, Tiξ〉,

the ∗-K-operator frame is an A-tight. If A = 1, it is called a normalized tight ∗-K-operator frameor a Parseval ∗-K-operator frame.
Example 3.3. Let l∞ be the set of all bounded complex-valued sequences. For any u = {uj}j∈N, v =

{vj}j∈N ∈ l∞, we define
uv = {ujvj}j∈N, u∗ = {ūj}j∈N, ‖u‖ = sup

j∈N
|uj |.

Then A = {l∞, ‖.‖} is a C∗-algebra. Then A is pro-C∗-algebra.Let X = C0 be the set of all null sequences. For any u, v ∈ X we define
〈u, v〉 = uv∗ = {uj ūj}j∈N.

Therefore X is a Hilbert A-module.Define fj = {f ji }i∈N∗ by f ji = 1
2 + 1

i if i = j and f ji = 0 if i 6= j ∀j ∈ N∗.Now define the adjointable operator Tj : X → X , Tj{(ξi)i} = (ξi f
j
i )i .Then for every x ∈ X we have∑

j∈N
〈Tjξ, Tjξ〉 = {

1

2
+

1

i
}i∈N∗〈ξ, ξ〉{

1

2
+

1

i
}i∈N∗ .

So {Tj}j is a {12 + 1
i }i∈N∗-tight ∗-operator frame.Let K : H → H defined by Kξ = { ξii }i∈N∗ .Then for every ξ ∈ X we have

〈K∗ξ,K∗ξ〉 ≤
∑
j∈N
〈Tjξ, Tjξ〉 = {

1

2
+

1

i
}i∈N∗〈ξ, ξ〉{

1

2
+

1

i
}i∈N∗ .

https://doi.org/10.28924/ada/ma.2.4


Eur. J. Math. Anal. 10.28924/ada/ma.2.4 5This shows that {Tj}j∈N is an ∗-K-operator frame with bounds 1, {12 + 1
i }i∈N∗ .

Remark 3.4. (1) Every ∗-operator frame for Hom∗A(X ) is an ∗-K-operator frame, for any K ∈
Hom∗A(X ): K 6= 0.(2) If K ∈ Hom∗A(X ) is a surjective operator, then every ∗-K-operator frame for Hom∗A(X ) isan ∗-operator frame.

Example 3.5. Let X be a finitely or countably generated Hilbert A-module. Hom∗A(X ). Let
K ∈ Hom∗A(X ) an invertible element such that both are uniformly bounded and K 6= 0. Let
{Ti}i∈I be an ∗-operator frame for X with bounds A and B, respectively. We have

A〈ξ, ξ〉A∗ ≤
∑
i∈I
〈Tiξ, Tiξ〉 ≤ B〈ξ, ξ〉B∗,∀ξ ∈ X .

Or
〈K∗ξ,K∗ξ〉 ≤ ‖K‖2∞〈ξ, ξ〉,∀ξ ∈ X .Then

‖K‖−1∞ A〈K∗ξ,K∗ξ〉(‖K‖−1∞ A)∗ ≤
∑
i∈I
〈Tiξ, Tiξ〉 ≤ B〈ξ, ξ〉B∗,∀ξ ∈ X .

So {Ti}i∈I is ∗-K-operator frame for X with bounds ‖K‖−1∞ A and B, respectively.
In what follows, we introduce the analysis, the synthesis and the frame operator. We alsoestablish some properties.Let {Ti}i∈I be an ∗-K-operator frame for Hom∗A(X ). Define an operator R : X → l2(X ) by

Rξ = {Tiξ}i∈I ,∀ξ ∈ X , then R is called the analysis operator. The adjoint of the analysis operator
R, R∗ : l2(X )→ X is given by R∗({ξi}i) =

∑
i∈I T

∗
i ξi ,∀{ξi}i ∈ l2(X ). The operator R∗ is calledthe synthesis operator. By composing R and R∗, the frame operator S : X → X is given by

Sξ = R∗Rξ =
∑
i∈I T

∗
i Tiξ.Note that S need not be invertible in general. But under some condition S will be invertible.

Theorem 3.6. Let K be a surjective operators in Hom∗A(X ). If {Ti}i∈I is an ∗-K-operator frame
for Hom∗A(X ), then the frame operator S is positive, invertible and adjointable. In addition we
have the reconstruction formula, ξ =

∑
i∈I T

∗
i TiS

−1ξ, ∀ξ ∈ X .

Proof. We start by showing that, S is a self-adjoint operator. By definition we have ∀ξ, η ∈ H
〈Sξ, η〉 =

〈∑
i∈I

T ∗i Tiξ, η

〉
=
∑
i∈I
〈T ∗i Tiξ, η〉

=
∑
i∈I
〈ξ, T ∗i Tiη〉

https://doi.org/10.28924/ada/ma.2.4


Eur. J. Math. Anal. 10.28924/ada/ma.2.4 6

=

〈
ξ,
∑
i∈I

T ∗i Tiη

〉
= 〈ξ, Sη〉.

Then S is a selfadjoint.The operator S is clearly positive.By (2) in Remark 3.4 {Ti}i∈I is an ∗-operator frame for Hom∗A(X ).The definition of an ∗-operator gives
A1〈ξ, ξ〉A∗1 ≤

∑
i∈I
〈Tiξ, Tiξ〉 ≤ B〈ξ, ξ〉B∗.

Thus by the definition of norm in l2(X )

p̄X (Rξ)2 = p̄X (
∑
i∈I
〈Tiξ, Tiξ〉) ≤ p̄X (B)2p(〈ξ, ξ〉),∀ξ ∈ X . (3.3)

Therefore R is well defined and p̄X (R) ≤ p̄X (B). It’s clear that R is a linear A-module map. Wewill then show that the range of R is closed. Let {Rξn}n∈N be a sequence in the range of R suchthat limn→∞Rξn = η. For n,m ∈ N, we have
p(A〈ξn − ξm, ξn − ξm〉A∗) ≤ p(〈R(ξn − ξm), R(ξn − ξm)〉) = p̄X (R(ξn − ξm))2.

Seeing that {Rξn}n∈N is Cauchy sequence in X , then
p(A〈ξn − ξm, ξn − ξm〉A∗)→ 0, as n,m →∞.Note that for n,m ∈ N,

p(〈ξn − ξm, ξn − ξm〉) = p(A−1A〈ξn − ξm, ξn − ξm〉A∗(A∗)−1)

≤ p(A−1)2p(A〈ξn − ξm, ξn − ξm〉A∗).

Thus the sequence {ξn}n∈N is Cauchy and hence there exists ξ ∈ X such that ξn → ξ as n →∞.Again by (3.3), we have
p̄X (R(ξn − ξm))2 ≤ p̄X (B)2p(〈ξn − ξ, ξn − ξ〉).

Thus p(Rξn − Rξ) → 0 as n → ∞ implies that Rξ = η. It is therefore concluded that therange of R is closed. We now show that R is injective. Let ξ ∈ X and Rξ = 0. Note that
A〈ξ, ξ〉A∗ ≤ 〈Rξ,Rξ〉 then 〈ξ, ξ〉 = 0 so ξ = 0 i.e. R is injective.For ξ ∈ X and {ξi}i∈I ∈ l2(X ) we have

〈Rξ, {ξi}i∈I〉 = 〈{Tiξ}i∈I , {ξi}i∈I〉 =
∑
i∈I
〈Tiξ, ξi〉 =

∑
i∈I
〈ξ, T ∗i ξi〉 = 〈ξ,

∑
i∈I

T ∗i ξi〉.

Then R∗({ξi}i∈I) =
∑
i∈I T

∗
i ξi . Since R is injective, then the operator R∗ has closed range and

X = range(R∗), therefore S = R∗R is invertible
�

https://doi.org/10.28924/ada/ma.2.4


Eur. J. Math. Anal. 10.28924/ada/ma.2.4 7Let K ∈ Hom∗A(X ), in the following theorem we constructed an ∗-K-operator frame by using an
∗-operator frame.
Theorem 3.7. Let {Ti}i∈I be an ∗-K-operator frame in X with bounds A, B and K ∈ Hom∗A(X ) be
an invertible element such that both are uniformly bounded. Then {TiK}i∈I is an ∗-K∗-operator
frame in X with bounds A, ‖K‖∞B. The frame operator of {TiK}i∈I is S′ = K∗SK, where S is
the frame operator of {Ti}i∈I .

Proof. From
A〈ξ, ξ〉A∗ ≤

∑
i∈I
〈Tiξ, Tiξ〉 ≤ B〈ξ, ξ〉B∗,∀ξ ∈ X .

We get for all ξ ∈ X ,
A〈Kξ,Kξ〉A∗ ≤

∑
i∈I
〈TiKξ, TiKξ〉 ≤ B〈Kξ,Kξ〉B∗ ≤ ‖K‖∞B〈ξ, ξ〉(‖K‖∞B)∗.

Then {TiK}i∈I is an ∗-K∗-operator frame in X with bounds A, ‖K‖∞B.By definition of S,we have SKξ =
∑
i∈I T

∗
i TiKξ. Then

K∗SK = K∗
∑
i∈I

T ∗i TiKξ =
∑
i∈I

K∗T ∗i TiKξ.

Hence S′ = K∗SK. �

Corollary 3.8. Let K ∈ Hom∗A(X ) and {Ti}i∈I be an ∗-operator frame. Then {TiS−1K}i∈I is an
∗-K∗-operator frame, where S is the frame operator of {Ti}i∈I .

Proof. Result of the Theorem 3.7 for the ∗-operator frame {TiS−1}i∈I . �

4. Tensor Product
We denote by A⊗B, the minimal or injective tensor product of the pro-C∗-algebras A and B, itis the completion of the algebraic tensor product A⊗alg B with respect to the topology determinedby a family of C∗-seminorms. Suppose that X is a Hilbert module over a pro-C∗-algebra A and

Y is a Hilbert module over a pro-C∗-algebra B. The algebraic tensor product X ⊗alg Y of X and
Y is a pre-Hilbert A⊗ B-module with the action of A⊗ B on X ⊗alg Y defined by

(ξ ⊗ η)(a ⊗ b) = ξa ⊗ ηb for all ξ ∈ X , η ∈ Y, a ∈ A and b ∈ B
and the inner product

〈·, ·〉 :
(
X ⊗alg Y)× (X ⊗alg Y)→ A⊗alg B. defined by
〈ξ1 ⊗ η1, ξ2 ⊗ η2〉 = 〈ξ1, ξ2〉 ⊗ 〈η1, η2〉And we know that for z =

∑n
i=1 ξi⊗ηi in X⊗algY we have 〈z, z〉A⊗B =

∑
i ,j〈ξi , ξj〉A⊗〈ηi , ηj〉B ≥ 0and 〈z, z〉A⊗B = 0 iff z = 0.

https://doi.org/10.28924/ada/ma.2.4


Eur. J. Math. Anal. 10.28924/ada/ma.2.4 8The external tensor product of X and Y is the Hilbert module X ⊗Y over A⊗B obtained by thecompletion of the pre-Hilbert A⊗ B-module X ⊗alg Y .If P ∈ M(X ) and Q ∈ M(Y) then there is a unique adjointable module morphism P ⊗ Q :

A⊗B → X ⊗Y such that (P ⊗Q)(a⊗ b) = P (a)⊗Q(b) and (P ⊗Q)∗(a⊗ b) = P ∗(a)⊗Q∗(b)for all a ∈ A and for all b ∈ B (see, for example, cite The minimal or injective tensor product ofthe pro-C∗-algebras A and B, denoted by A⊗B, is the completion of the algebraic tensor product
A⊗alg B with respect to the topology determined by a family of C∗-seminorms. Suppose that Xis a Hilbert module over a pro-C∗-algebra A and Y is a Hilbert module over a pro-C∗-algebra B.The algebraic tensor product X ⊗alg Y of X and Y is a pre-Hilbert A⊗B-module with the actionof A⊗ B on X ⊗alg Y defined by

(ξ ⊗ η)(a ⊗ b) = ξa ⊗ ηb for all ξ ∈ X , η ∈ Y, a ∈ A and b ∈ B
and the inner product

〈·, ·〉 :
(
X ⊗alg Y)× (X ⊗alg Y)→ A⊗alg B. defined by
〈ξ1 ⊗ η1, ξ2 ⊗ η2〉 = 〈ξ1, ξ2〉 ⊗ 〈η1, η2〉

We also know that for z =
∑n
i=1 ξi⊗ηi in X⊗algY we have 〈z, z〉A⊗B =

∑
i ,j〈ξi , ξj〉A⊗〈ηi , ηj〉B ≥ 0and 〈z, z〉A⊗B = 0 iff z = 0.The external tensor product of X and Y is the Hilbert module X ⊗Y over A⊗B obtained by thecompletion of the pre-Hilbert A⊗ B-module X ⊗alg Y .If P ∈ M(X ) and Q ∈ M(Y) then there is a unique adjointable module morphism P ⊗ Q :

A⊗B → X ⊗Y such that (P ⊗Q)(a⊗ b) = P (a)⊗Q(b) and (P ⊗Q)∗(a⊗ b) = P ∗(a)⊗Q∗(b)for all a ∈ A and for all b ∈ B (see, for example, [9])Let I and J be countable index sets.
Theorem 4.1. Let X and Y be two Hilbert pro-C∗-modules over unitary pro-C∗-algebras A and
B, respectively. Let {Ti}i∈I ⊂ Hom∗A(X ) be an ∗-K-operator frame for X with bounds A and B
and frame operators ST and {Pj}j∈J ⊂ Hom∗B(Y) be an ∗-L-operator frame for K with bounds
C and D and frame operators SL. Then {Ti ⊗ Lj}i∈I,j∈J is an ∗-K⊗L-operator frame for Hibert
A⊗ B-module X ⊗ Y with frame operator ST ⊗ SP and bounds A⊗ C and B ⊗D.

Proof. The defintion of ∗-K-operator frame {Ti}i∈I and ∗-L-operator frame {Pj}j∈J gives
A〈K∗ξ,K∗ξ〉AA∗ ≤

∑
i∈I
〈Tiξ, Tiξ〉A ≤ B〈ξ, ξ〉AB∗,∀ξ ∈ X .

C〈L∗η, L∗η〉BC∗ ≤
∑
j∈J
〈Pjη, Pjη〉B ≤ D〈η, η〉BD∗,∀η ∈ Y.

https://doi.org/10.28924/ada/ma.2.4


Eur. J. Math. Anal. 10.28924/ada/ma.2.4 9Therefore
(A〈K∗ξ,K∗ξ〉AA∗)⊗ (C〈L∗η, L∗η〉BC∗)

≤
∑
i∈I
〈Tiξ, Tiξ〉A ⊗

∑
j∈J
〈Pjη, Pjη〉B

≤ (B〈ξ, ξ〉AB∗)⊗ (D〈η, η〉BD∗),∀ξ ∈ X ,∀η ∈ Y.Then
(A⊗ C)(〈K∗ξ,K∗ξ〉A ⊗ 〈L∗η, L∗η〉B)(A∗ ⊗ C∗)

≤
∑
i∈I,j∈J

〈Tiξ, Tiξ〉A ⊗ 〈Pjη, Pjη〉B

≤ (B ⊗D)(〈ξ, ξ〉A ⊗ 〈η, η〉B)(B∗ ⊗D∗),∀ξ ∈ X ,∀η ∈ Y.Consequently we have
(A⊗ C)〈K∗ξ ⊗ L∗η,K∗ξ ⊗ L∗η〉A⊗B(A⊗ C)∗

≤
∑
i∈I,j∈J

〈Tiξ ⊗ Pjη, Tiξ ⊗ Pjη〉A⊗B

≤ (B ⊗D)〈ξ ⊗ η, ξ ⊗ η〉A⊗B(B ⊗D)∗,∀ξ ∈ X ,∀η ∈ Y.

Then for all ξ ⊗ η in X ⊗ Y we have
(A⊗ C)〈(K ⊗ L)∗(ξ ⊗ η), (K ⊗ L)∗(ξ ⊗ η)〉A⊗B(A⊗ C)∗

≤
∑
i∈I,j∈J

〈(Ti ⊗ Pj)(ξ ⊗ η), (Ti ⊗ Pj)(ξ ⊗ η)〉A⊗B

≤ (B ⊗D)〈ξ ⊗ η, ξ ⊗ η〉A⊗B(B ⊗D)∗.

The last inequality is true for every finite sum of elements in X ⊗alg Y and then it’s true for all
z ∈ X ⊗K. It shows that {Ti ⊗ Pj}i∈I,j∈J is an ∗-K ⊗ L-operator frame for Hilbert A⊗B-module
X ⊗ Y with lower and upper bounds A⊗ C and B ⊗D, respectively.By the definition of frame operator ST and SP we have

ST ξ =
∑
i∈I

T ∗i Tiξ, ∀ξ ∈ X .

SPη =
∑
j∈J

P ∗j Pjη, ∀η ∈ Y.

Therefore
(ST ⊗ SP )(ξ ⊗ η) = ST ξ ⊗ SPη

=
∑
i∈I

T ∗i Tiξ ⊗
∑
j∈J

P ∗j Pjη

=
∑
i∈I,j∈J

T ∗i Tiξ ⊗ P ∗j Pjη

=
∑
i∈I,j∈J

(T ∗i ⊗ P ∗j )(Tiξ ⊗ Pjη)

https://doi.org/10.28924/ada/ma.2.4


Eur. J. Math. Anal. 10.28924/ada/ma.2.4 10

=
∑
i∈I,j∈J

(T ∗i ⊗ P ∗j )(Ti ⊗ Pj)(ξ ⊗ η)

=
∑
i∈I,j∈J

(Ti ⊗ Pj)∗(Ti ⊗ Pj)(ξ ⊗ η).

Then by the uniqueness of frame operator, the last expression is equal to ST⊗P (ξ⊗η). Consequentlywe have (ST ⊗SP )(ξ⊗η) = ST⊗P (ξ⊗η). The last equality is true for every finite sum of elementsin X ⊗alg Y and then it’s true for all z ∈ X ⊗ Y . It follows that (ST ⊗ SP )(z) = ST⊗P (z). Thus
ST⊗P = ST ⊗ SP . �

References
[1] N. Haddadzadeh, G-frames in Hilbert modules over pro-C*-algebras, Int. J. Ind. Math. 9(4) (2017) 259-267.[2] M. Azhini and N. Haddadzadeh, Fusion frames in Hilbert modules over pro-C∗-algebras, Int. J. Ind. Math. 5(2)(2013) Article ID IJIM-00211.[3] R. J. Duffin and A. C. Schaeffer, A class of nonharmonic fourier series, Trans. Amer. Math. Soc. 72 (1952) 341-366.[4] M. Fragoulopoulou, An introduction to the representation theory of topological ∗-algebras, Schriftenreihe, Univ.Münster, 48 (1988) 1-81.[5] M. Fragoulopoulou, Tensor products of enveloping locally C∗-algebras, Schriftenreihe, Univ. Münster (1997) 1-81.[6] M. Fragoulopoulou, Topological algebras with involution, North Holland, Amsterdam, 2005.[7] A. Grossman, and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986) 1271-1283.[8] A. Inoue, Locally C∗-algebra, Mem. Fac. Sci. Kyushu Univ. Ser. A, Math. 25(2) (1972) 197-235.[9] M. Joita, On frames in Hilbert modules over pro-C∗-algebras, Topol. Appl. 156 (2008) 83-92.[10] E. C. Lance, Hilbert C∗-modules, A toolkit for operator algebraists, London Math. Soc. Lecture Note Series 210.Cambridge Univ. Press, Cambridge, 1995.[11] C. Y. Li and H. X. Cao, Operator Frames for B(H), Wavelet Analysis and Applications. Birkhäuser Basel, 2006.67-82.[12] A. Mallios, Topological algebras: Selected Topics, North Holland, Amsterdam, 1986.[13] M. Naroei and A. Nazari, Some properties of –frames in Hilbert modules over pro-C∗-algebras, Sahand Commun.Math. Anal. 16 (2019) 105–117.[14] N. C. Phillips, Inverse limits of C*-algebras, J. Oper. Theory. 19 (1988) 159-195.[15] N. C. Phillips, Representable K-theory for σ -C∗-algebras, K-Theory, 3 (1989) 441-478.[16] M. Rossafi and S. Kabbaj, Operator frame for End∗A(H), J. Linear Topol. Algebra. 8 (2019) 85-95.

https://doi.org/10.28924/ada/ma.2.4

	1. Introduction
	2. Preliminaries
	3. -K-operator frame for HomA(X)
	4. Tensor Product
	References

