https://doi.org/10.4134/CKMS.c200179 pISSN: 1225-1763 / eISSN: 2234-3024
THE WOVEN FRAME OF MULTIPLIERS IN HILBERT
C∗-MODULES
Mona Naroei Irani and Akbar Nazari
Abstract. In this paper, by using the sequence of adjointable opera-tors from C∗-algebra A into Hilbert A-module E, the woven frames of
multipliers in Hilbert C∗-modules are introduced. Meanwhile, we study
the effect of operators on these frames and, also we construct the new woven frame of multipliers in Hilbert A-module A. Finally, compositions of woven frames of multipliers in Hilbert C∗-modules are studied.
1. Introduction
The frame theory was first introduced by Duffin and Scheaffer [4], to deal with some problems in nonharmonic Fourier series in 1952. They were reintro-duced and developed in 1986 by Daubechies et al. [3], and popularized from then on.
The notion of frames in Hilbert C∗-modules has been investigated in [5]. Frank and Larson [5, 6] defined the standard frames in Hilbert C∗-modules over unital C∗-algebras.
Recently, Bemrose et al. in [2] introduced a new concept in frame theory, so-called weaving frames, which are motivated by a problem regarding distributed signal processing. Two frames {fi}i∈I and {gi}i∈I for a Hilbert space H are called woven if there exist constants 0 < A ≤ B so that the weaving {fi}i∈σ∪ {gi}i∈σc is a frame with bounds A, B for every σ ⊂ I.
F. Ghobadzadeh et al. in [7] extend the concept of weaving frames in Hilbert C∗-modules. The ∗-frame of multipliers in Hilbert pro-C∗-algebras are inves-tigated in [10] by Naroei and Nazari. Authors, by using the sequence of ad-jointable operators from pro-C∗-algebra A into Hilbert A-module E, defined the ∗-frame of multipliers. In this paper, we reconsider the ideas of articles [7, 10] and introduce woven frame of multipliers in Hilbert C∗-modules.
The paper is organized as follows. In the next section, we give a brief survey of the fundamental definitions and basic properties of Hilbert C∗-modules. In
Received May 27, 2020; Revised August 27, 2020; Accepted September 17, 2020. 2010 Mathematics Subject Classification. 42C15, 46L08.
Key words and phrases. Hilbert C∗-modules, woven frame of multipliers, the frame of
multipliers.
c
2021 Korean Mathematical Society 257
Section 3, the woven frame of multipliers and example are introduced. Finally, in Section 4, a new woven frame of multipliers in Hilbert A-module A is also constructed and the combination of woven frames and frames of multipliers in Hilbert C∗-modules are studied.
Throughout this manuscript, let A be an unital C∗-algebra and E, F be finitely or countably generated Hilbert A-modules. We use I to denote finite or countably infinite index set. For every σ ⊂ I, we show the complement of σ by σc.
2. Preliminaries
In this section, we recall the fundamental definitions and basic properties of Hilbert C∗-modules and introduce a frame of multipliers in Hilbert C∗-modules. For more information about Hilbert C∗-modules, we refer to [8].
Definition. Let A be a C∗-algebra. A pre-Hilbert A-module is a complex vector space E which is also a right A-module equipped with an A-valued inner product h·, ·i : E × E → A such that:
(1) hx, xi ≥ 0, x ∈ E;
(2) hx, xi = 0 ⇔ x = 0, x ∈ E;
(3) hz, αx + βyi = hz, xi α + hz, yi β, α, β ∈ C and x, y, z ∈ E; (4) hx, ayi = hx, yi a, x, y ∈ E, a ∈ A;
(5) hx, yi = hy, xi, x, y ∈ E.
The action of A on E is C- and A-linear, i.e., (xa)λ = x(aλ) = (xλ)a for every λ ∈ C, a ∈ A and x ∈ E. If the pre-Hilbert A-module {E, h·, ·i} is complete concerning the norm kxk = khx, xik12, then it is called a Hilbert A-module. For every a in C∗-algebra A, we have |a| = (a∗a)1
2.
A Hilbert A-module E is countably generated if there exists a countable set {xn}n in E such that the submodule of E generated by {xna : a ∈ A}n∈N is dense in E. The C∗-algebra A itself can be recognized as a Hilbert A-module with the inner product ha, bi = ab∗.
The standard Hilbert A-module l2(A) is defined by l2(A) := {{aj}j∈N⊆ A :
X
j∈N
aja∗j converges in A},
with A-inner product D{aj}j∈N, {bj}j∈N E
= P
j∈N ajb∗j.
Let E and F be two Hilbert A-modules. A linear map T : E → F is said to be adjointable, if there exists a map T∗: F → E satisfying hT x, yi = hx, T∗yi whenever x ∈ E and y ∈ F. The set of all adjointable maps from E into F is denoted by L(E, F ) and the set of all bounded A-module maps from E into F is denoted by B(E, F ). It is known that L(E, F ) ⊆ B(E, F ). We denote L(E, E) and B(E, E) with L(E) and B(E), respectively.
For C∗-algebra A and Hilbert A-module E, the set L(A, E) is a Hilbert L(A)-module with the action of L(A) on L(A, E) defined by t.s = tos, for t ∈ L(A, E) and s ∈ L(A), also the L(A)-valued inner product defined by ht, si = t∗os. Since L(A) can be identified with the multiplier algebra M (A) of A (see [9]), L(A, E) becomes a Hilbert M (A)-module, called the multiplier module of E, and it is denoted by M (E). For all h ∈ M (E) and x ∈ E, we denote hh, xiM (E)= h∗(x).
Throughout the paper, we need the following lemma that it will illustrate lower and upper bounds of operators corresponding to a given operator T concerning A-valued inner products.
Lemma 2.1 ([1]). Let E and F be two Hilbert C∗-modules and T ∈ L(E, F ). Then
(i) If T is injective and T has closed range, then the adjointable map T∗T is invertible and (T ∗T )−1 −1 ≤ T∗T ≤ kT k2 .
(ii) If T is surjective, then the adjointable map T T∗ is invertible and (T T ∗)−1 −1 ≤ T T∗≤ kT k2 .
Definition. Let E be a Hilbert C∗-module. The sequence {hi}i∈I in M (E) is called a standard frame of multipliers in E if for each x ∈ E, the series P
i∈Ihx, hiiM (E)hhi, xiM (E) is convergent in A, and there exist two positive constants C and D such that
Chx, xiE≤X i∈I
hx, hiiM (E)hhi, xiM (E) ≤ Dhx, xiE for all x ∈ E .
The frame of multipliers is called tight if D = C, and normalized if D = C = 1. If {hi}i∈I possess an upper bound but not necessarily a lower bound, it is called a Bessel sequence. If {hi}i∈I is a Bessel sequence, then the pre-frame operator
T : E → l2(A), T (x) = {hhi, xiM (E)}i∈I,
is bounded. Its adjoint operator T∗({ai}i∈I) = Pi∈Ihiai is called the syn-thesis operator. The frame operator S(x) = T∗T (x) =P
i∈Ihihhi, xiM (E) is bounded, positive and if {hi}i∈I is a standard frame of multipliers, operator S is invertible.
Let {hi}i∈I be a standard frame of multipliers in E with lower and upper frame bounds C and D, respectively. For all x ∈ E we have
C hx, xiE≤ hSx, xiE ≤ D hx, xiE. Now, we define ehi= S−1hi. Then for each x ∈ E,
x =X i∈I hi D e hi, x E M (E)= X i∈I e hihhi, xiM (E).
Also { ehi}i∈I is a standard frame of multipliers for E, called the canonical dual frame of {hi}i∈I.
3. The woven frame of multipliers
In this section, the woven frame of multipliers is introduced and example of these frames are investigated.
Definition. For C∗-algebra A and Hilbert A-module E, two standard frames of multipliers {hi}i∈I and {ti}i∈I for E are woven if there exist constants 0 < A ≤ B < ∞ such that for every σ ⊂ I, the family {hi}i∈σ∪ {ti}i∈σc is a standard frame of multipliers for E, with bounds A, B.
Definition. For C∗-algebra A and Hilbert A-module E, two standard frames of multipliers {hi}i∈I and {ti}i∈I for E are the weakly woven if for every σ ⊂ I, the family {hi}i∈σ∪ {ti}i∈σc is a frame of multipliers for E, and each {hi}i∈σ∪ {ti}i∈σc is called a weaving.
Obviously, if {hi}i∈I and {ti}i∈I are the woven frame of multipliers in E, then they are a weakly woven frame of multipliers in E.
Proposition 3.1 ([7]). Suppose that {hi}i∈I and {ti}i∈I are Bessel sequences of multipliers in E with Bessel bounds B1 and B2, respectively. Then for any subset σ of I, the family {hi}i∈σ∪ {ti}i∈σc is a Bessel sequence of multipliers with Bessel bound B1+ B2.
Example 3.2. Let A be a C∗-algebra and l2(A) be a Hilbert A-module with inner product h{am}m, {am}mi = P amam. We define hi : A → l2(A) by hi(a) = (0, 0, 0, . . . , a, 0, . . .) and we have h∗i({am}m) = ai.
For D ∈ N define families f ≡ {fi}i∈I and g ≡ {gi}i∈I in M (l
2(A)) as follows: f ≡ {fi}i∈I= {h1, Dh1, h2, D 2h2, h3, D 3h3, . . .}, g ≡ {gi}i∈I= {h1, Dh1, h2, D 2h2, h3, D 2h3, h4, D 3h4, h5, D 3h5, . . .}. Then, for every {am}m∈ l2(A) sequences {fi}i∈I and {gi}i∈I are frames of multipliers in l2(A) with lower and upper bounds 1 and (1 + D), respectively. For every subset σ of I, then {fi}i∈σ ∪ {gi}i∈σc is a frames of multipliers in l2(A) with lower and upper bounds 1 and (1 + D), respectively.
4. Main results
In this section, some results are presented for the woven frame of multipli-ers. The following result characterizes woven frames in terms of the action of operators on frame of multipliers.
Proposition 4.1. Let {hji}i∈I, for j = 1, 2, be the (C, D) woven frame of multipliers in E. If T is a surjective element in L(E), then {T hji}i∈I, for j = 1, 2 is also (C (T T ∗)−1 −1 , DkT k2) woven in E.
Proof. Since T is surjective, by Lemma 2.1 for every σ ⊂ I and each x ∈ E we have X i∈σ hx, T h1iiM (E)hT h1i, xiM (E)+ X i∈σc hx, T h2iiM (E)hT h2i, xiM (E) = X i∈σ hT∗x, h1iiM (E)hh1i, T∗xiM (E)+ X i∈σc hT∗x, h2iiM (E)hh2i, T∗xiM (E) ≤ DhT∗x, T∗xiE ≤ DkT k2hx, xiE. Similarly X i∈σ hx, T h1iiM (E)hT h1i, xiM (E)+ X i∈σc hx, T h2iiM (E)hT h2i, xiM (E) = X i∈σ hT∗x, h 1iiM (E)hh1i, T∗xiM (E)+ X i∈σc hT∗x, h 2iiM (E)hh2i, T∗xiM (E) ≥ ChT∗x, T∗xiE ≥ C (T T ∗)−1 −1 hx, xiE.
Hence the family {{T hji}i∈I : j = 1, 2} is a woven frame of multipliers in
E.
Corollary 4.2. Let {tji}i∈I for j = 1, 2 in M (A) be the (C, D) woven frame of multipliers for A. If h is a surjective multiplier in M (E), then {htji}i∈I for j = 1, 2 is also (C (hh ∗)−1 −1 , Dkhk2) woven in E.
Proposition 4.3. Let {hji}i∈I, for j = 1, 2, be the sequence in M (E). If there exists an invertible map V ∈ L(E) such that {V hji}i∈I for j = 1, 2, is a (A, B) woven frame of multipliers for E, then {hji}i∈I, for j = 1, 2, is a (AkV∗k−2, B V ∗−1 2
) woven frame of multipliers in E.
Proof. Since V is an invertible element in L(E) for any x ∈ E we have kV∗k−2hx, xiE≤DV∗−1x, V∗−1xE E≤ V ∗−1 2 hx, xiE. (1)
For any subset σ of I ADV∗−1x, V∗−1xE E ≤ X i∈σ D V∗−1x, V h1i E M (E) D V h1i, V∗ −1 xE M (E) +X i∈σc D V∗−1x, V h2i E M (E) D V h2i, V∗ −1 xE M (E) = X i∈σ x, V−1V h 1i M (E)V−1V h1i, x M (E)
+X i∈σc x, V−1V h 2i M (E)V−1V h2i, x M (E) = X i∈σ hx, h1iiM (E)hh1i, xiM (E) +X i∈σc hx, h2iiM (E)hh2i, xiM (E). Now (1) implies that
AkV∗k−2hx, xiE≤X i∈σ hx, h1iiM (E)hh1i, xiM (E)+ X i∈σc hx, h2iiM (E)hh2i, xiM (E). Similarly X i∈σ hx, h1iiM (E)hh1i, xiM (E)+ X i∈σc hx, h2iiM (E)hh2i, xiM (E)≤ B V ∗−1 2 hx, xiE.
Hence {hji}i∈I, for j = 1, 2, is a (AkV∗k−2, B V ∗−1 2
) woven frame of
multi-pliers in E.
Proposition 4.4. Let {hi}i∈I and {ti}i∈I be the (C, D) woven frames of mul-tipliers in E. If V : E → F is a co-isometry map, then {V hi}i∈I and {V ti}i∈I are the (C, D) woven frames of multipliers in F.
Proof. For every subset σ of I and each y ∈ F, since V is co-isometry, we have Chy, yiF = ChV∗y, V∗yiE ≤ X i∈σ hV∗y, h iiM (E)hhi, V∗yiM (E) +X i∈σc hV∗y, tiiM (E)hti, V∗yiM (E) ≤ DhV∗y, V∗yiE= Dhy, yiF. This shows that
Chy, yiF ≤X i∈σ hy, V hiiM (F )hV hi, yiM (F )+ X i∈σc hy, V tiiM (F )hV ti, yiM (F ) ≤ Dhy, yiF.
Hence {V hi}i∈Iand {V ti}i∈Iare (C, D) woven frame of multipliers in F. In the following, we are going to construct a new woven frame of multipliers in Hilbert A-module A.
Theorem 4.5. Let {hi}i∈I and {ti}i∈I be the (C, D) woven frames of multi-pliers in E. Suppose that x is an element in E such that hx, xiE is a nonzero element in the center of A. Then {hhi, xiM (E)}i∈I and {hti, xiM (E)}i∈I are the woven frames in A.
Proof. For every subset σ of I and a ∈ A, by the definition of woven aC hx, xiEa∗≤ a(X i∈σ hx, hiiM (E)hhi, xiM (E)+ X i∈σc hx, tiiM (E)hti, xiM (E))a ∗ ≤ aD hx, xiEa∗, and a(X i∈σ hx, hiiM (E)hhi, xiM (E)+ X i∈σc hx, tiiM (E)hti, xiM (E))a∗ = X i∈σ D a, hhi, xiM (E) E A D hhi, xiM (E), a E A +X i∈σc D a, hti, xiM (E) E A D hti, xiM (E), a E A.
Since hx, xiE is in the center of A, the following inequalities are valid for all a ∈ A, (Chx, xiE)ha, aiA≤ X i∈σ D a, hhi, xiM (E) E A D hhi, xiM (E), a E A +X i∈σc D a, hti, xiM (E) E A D hti, xiM (E), a E A ≤ (Dhx, xiE)ha, aiA.
The last inequality shows that {hhi, xiM (E)}i∈I and {hti, xiM (E)}i∈I are the
woven frames in A.
In the following, we show that the combination of woven frames of multipliers is a woven frame of multipliers.
Theorem 4.6. Suppose that {hi}∞i=1 and {ti}∞i=1 in M (E) are the (C, D) wo-ven frames of multipliers for E. Let {ϕik}k∈Ji⊂N and {ψik}k∈Qi⊂N in M (A) be (A, B) woven frames of multipliers for A. Then {hiϕik}i∈N
k∈Ji
and {tiψik}i∈N
k∈Qi in M (E) are (AC, BD) woven frames of multipliers in E.
Proof. For any subset σ of N and for any x ∈ E, we compute X i∈σ X k∈Ji hx, hiϕikiM (E)hhiϕik, xiM (E)+ X i∈σc X k∈Qi hx, tiψikiM (E)htiψik, xiM (E) = X i∈σ X k∈Ji hh∗ix, ϕikiM (A)hϕik, h∗ixiM (A)+ X i∈σc X k∈Qi ht∗ix, ψikiM (A)hψik, t∗ixiM (A) ≤ BX i∈σ hh∗ix, h∗ixiA+ BX i∈σc ht∗ix, t∗ixiA = BX i∈σ hx, hiiM (E)hhi, xiM (E)+ B X i∈σc hx, tiiM (E)hti, xiM (E) ≤ BD hx, xi .
This gives a universal upper bound for the family {hiϕik}i∈N
k∈Ji
and {tiψik}i∈N
k∈Qi . For the universal lower bound, we compute
X i∈σ X k∈Ji hx, hiϕikiM (E)hhiϕik, xiM (E)+ X i∈σc X k∈Qi hx, tiψikiM (E)htiψik, xiM (E) = X i∈σ X k∈Ji hh∗ix, ϕikiM (A)hϕik, h∗ixiM (A)+ X i∈σc X k∈Qi ht∗ix, ψikiM (A)hψik, t∗ixiM (A) ≥ AX i∈σ hh∗ix, h∗ixiA+ AX i∈σc ht∗ix, t∗ixiA = AX i∈σ hx, hiiM (E)hhi, xiM (E)+ A X i∈σc hx, tiiM (E)hti, xiM (E) ≥ AC hx, xi for all x ∈ E.
This shows that {hiϕik}i∈N
k∈Ji
and {tiψik}i∈N
k∈Qi
are (AC, BD) woven frames of
multipliers in E.
In the following, we study the combination of woven frames of multipliers and frames of multipliers.
Proposition 4.7. Suppose that H = {hi}∞i=1 and T = {ti}∞i=1 in M (E) are frames of multipliers for E. Let {ϕik}k∈Ji⊂N and {ψik}k∈Qi⊂N in M (A) be frames of multipliers for A with frame bounds α, β and α0, β0, respectively.
Then the following conditions are equivalent.
(i) H and T are woven frames of multipliers for E. (ii) {hiϕik}i∈N
k∈Ji
and {tiψik}i∈N
k∈Qi
are woven frames of multipliers in E. Proof. (i)⇒(ii) Let H and T be the (C, D) woven frames of multipliers in E. For any subset σ of N and for any x ∈ E, we compute
X i∈σ X k∈Ji hx, hiϕikiM (E)hhiϕik, xiM (E)+ X i∈σc X k∈Qi hx, tiψikiM (E)htiψik, xiM (E) = X i∈σ X k∈Ji hh∗ ix, ϕikiM (A)hϕik, h∗ixiM (A)+ X i∈σc X k∈Qi ht∗ ix, ψikiM (A)hψik, t∗ixiM (A) ≤ βX i∈σ hh∗ix, h∗ixiA+ β0 X i∈σc ht∗ix, t∗ixiA ≤ max{β, β0}(X i∈σ hx, hiiM (E)hhi, xiM (E)+ X i∈σc hx, tiiM (E)hti, xiM (E)) ≤ max{β, β0}D hx, xi .
This gives a universal upper bound for the family {hiϕik}i∈N
k∈Ji
and {tiψik}i∈N
k∈Qi . For the universal lower bound, we compute
X i∈σ X k∈Ji hx, hiϕikiM (E)hhiϕik, xiM (E)+ X i∈σc X k∈Qi hx, tiψikiM (E)htiψik, xiM (E)
= X i∈σ X k∈Ji hh∗ix, ϕikiM (A)hϕik, h∗ixiM (A)+ X i∈σc X k∈Qi ht∗ix, ψikiM (A)hψik, t∗ixiM (A) ≥ αX i∈σ hh∗ix, h∗ixiA+ α0X i∈σc ht∗ix, t∗ixiA ≥ min{α, α0}(X i∈σ hx, hiiM (E)hhi, xiM (E)+ X i∈σc hx, tiiM (E)hti, xiM (E)) ≥ min{α, α0}C hx, xi for all x ∈ E.
Hence {hiϕik}i∈N
k∈Ji
and {tiψik}i∈N
k∈Qi
are woven frames of multipliers for E. (ii)⇒(i) Let {hiϕik}i∈N
k∈Ji
and {tiψik}i∈N
k∈Qi
be (A, B) woven frames of multi-pliers in E. Then for any subset σ of N, we compute
X i∈σ hx, hiiM (E)hhi, xiM (E)+ X i∈σc hx, tiiM (E)hti, xiM (E) ≤ 1 α X i∈σ X k∈Ji hx, hiϕikiM (E)hhiϕik, xiM (E) + 1 α0 X i∈σc X k∈Qi hx, tiψikiM (E)htiψik, xiM (E) ≤ max{1 α, 1 α0}B hx, xi for all x ∈ E.
This gives a universal upper bound for the family H and T. Similarly, A min{β1,β10} is a universal lower frame bound for H and T. By using Proposition 4.7 in Hilbert spaces E ⊕ F and E ⊗ F the next result is obtained.
Corollary 4.8. Let E be a Hilbert A-module and F be a Hilbert B-module. If H = {hi}i∈Iand T = {ti}i∈Iare (A, B) woven frames of multipliers in E, also H0= {h0i}i∈I and T0= {t0i}i∈I are (A0, B0) woven frames of multipliers in F, then
(1) {hi⊕ h0i}i∈I and {ti⊕ t0i}i∈I are (min{A, A0}, max{B, B0}) woven frames of multipliers in Hilbert A ⊕ B-module E ⊕ F.
(2) {hi⊗ h0i}i∈I and {ti⊗ t0i}i∈I are (AA, BB0) woven frames of multi-pliers in Hilbert A ⊗ B-module E ⊗ F.
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Department of Mathematics Kerman Branch
Islamic Azad University Kerman, Iran
Email address: M.naroei.math@gmail.com Akbar Nazari
Department of Pure Mathematics Faculty of Mathematics and Computer Shahid Bahonar University of Kerman Kerman, Iran
