arXiv:1407.2466v1 [math.OA] 9 Jul 2014 [ n13,G rus[]soe htfrtoLbsu nerbefu integrable Lebesgue two for that showed Gr¨uss [9] G. 1934, In ednt by denote We es hti antb elcdb mle constant. smaller a by replaced be cannot it that sense hoe 1. Theorem [3]. known is h constant The bestand set able (1.1) holds inequality following the then hold, that provided audfntosi elo ope ibr pcsi ie n[2]. in given is spaces Hilbert complex or real in functions valued and e ohe . Bochner r equivalently, or, ,b a, ∈

h olwn nqaiyo rustp nra rcmlxinrproduc inner complex or real in Gr¨uss type of inequality following The 2000 Let ute xeso fG¨s yeieult o ohe integra Bochner for inequality Gr¨uss type of extension further A e od n phrases. and words Key b ] H, −∞ k → − 1 f h k GR A ahmtc ujc Classification. Subject Mathematics k a H e R in r logiven. also are tions apnswt ausi Hilbert in values with mappings Abstract. 2 2 k n < Z ,M ,N n, M, m, ,ρ , ; a h 1 = ,. ., b := ρ f ≤ ii 4 1

S YEIEULT O VECTOR-VALUE FOR INEQUALITY TYPE USS ( If . L ¨ Let R Ω : t x UCIN NHILBERT IN FUNCTIONS Ω sbs osbei (1.1). in possible best is ) 2 eara rcmlxHletsae Ω space, Hilbert complex or real a be g Re g ,ρ − ρ ( ≤ ,β ,µ λ, β, α, (Ω −→ nti ae epoeavrino rusitga inequalit Gr¨uss integral of version a prove we paper this In ( ( t h α s ) H αe dt < N ) r elnmeswt h property the with numbers real are H , k + |h ; 2 f h· [0 − ,y x, − β ( h e falsrnl esrbefunctions measurable strongly all of set the ) , , s ·i Hilbert e ,x x, b ∞ ∞ )

h − i MRGAE GHAZANFARI GHASEM AMIR ) k − 1 ∈ eegemaual ucinwith function measurable Lebesgue a ) 2 ≤ ea ne rdc pc over space product inner an be s< ds a − K ..o [ on a.e. 2 1 ,e x, Z 1. βe and | a C α b ∗ ∞ ≥ i h i introduction f mdls rusieult,Lna yeinequality, type Landau inequality, Gr¨uss -modules, − C ( . ,y e, ,y x, ∗ t β mdls oeapiain o uhfunc- such for applications Some -modules. ) 0 60,4H5 26D15. 46H25, 46L08, ,b a, dt | , Re , |≤ i| 1 ∈ b ] . −

1 H h constant The y 1 4 a h − r uhta conditions that such are | λe Z α a λ − b − C + g 2 β ∗ ( ,y y, -MODULES t µ || ) ⊂ λ dt e

− − R

≤ −∞ ≤ n µe µ 4 1 | eaLbsu measur- Lebesgue a be 1 2 . 4 1 sbs osbei the in possible best is ≥ i | K ( λ m < M − 0 − ( µ K ≤ | m R nctions Ω = f for y so vector- of ls )( f ρ N ≤ ( nΩsuch Ω on C s , ) − < M spaces t ds R n ) ,g f, 1. = ) and , ∞ : 2 A. G. GHAZANFARI

Theorem 2. Let hH; h., .ii be a real or complex , Ω ⊂ Rn be a Lebesgue measurable set and ρ :Ω −→ [0, ∞) a Lebesgue measurable function with

Ω ρ(s)ds =1. If f,g belong to L2,ρ(Ω,H) and there exist the vectors x,X,y,Y ∈ H such that R (1.2) ρ(t)RehX − f(t),f(t) − xidt ≥ 0, ZΩ ρ(t)RehY − g(t),g(t) − yidt ≥ 0, ZΩ or, equivalently, X + x 2 1 (1.3) ρ(t) f(t) − dt ≤ kX − xk2, 2 4 ZΩ 2 Y + y 1 ρ(t) g(t) − dt ≤ kY − yk2. 2 4 ZΩ

Then the following inequalities hold

ρ(t)hf(t),g(t)idt − ρ(t)f(t)dt, ρ(t)g(t)dt ZΩ ZΩ ZΩ 

1 (1.4) ≤ kX − xkkY − yk− ρ(t)RehX − f(t),f(t) − x idt 4 Ω Z 1 2 × ρ(t)RehY − g(t),g(t) − yidt ZΩ  1 ≤ kX − xkkY − yk. 4 1 The constant 4 is sharp in the sense mentioned above. The Gr¨uss inequality has been investigated in inner product modules over H∗- algebras and C∗-algebras [1, 8], completely bounded maps [12], n-positive linear maps [11] and semi-inner product C∗-modules [6]. Also Joci´cet.al. in [10] presented the following Gr¨uss type inequality kD − Ck.kF − Ek A XB dµ(t) − A dµ(t)X B dµ(t) ≤ |||X||| t t t t 4 ZΩ ZΩ ZΩ for all bounded self-adjoint fields satisfying C ≤ At ≤ D and E ≤ Bt ≤ F for all t ∈ Ω and some bounded self-adjoint operators C,D,E and F , and for all X ∈C|||.|||(H). The main aim of this paper is to obtain a generalization of Theorem 2 for vector- value functions in Hilbert C∗-modules. Some applications for such functions are also given.

2. Preliminaries Hilbert C∗-modules are used as the framework for Kasparov’s bivariant K-theory and form the technical underpinning for the C∗-algebraic approach to quantum groups. Hilbert C∗-modules are very useful in the following research areas: op- erator K-theory, index theory for operator valued conditional expectations, group representation theory, the theory of AW ∗-algebras, , and others. Hilbert C∗-modules form a category in between Banach spaces and Hilbert GRUSS¨ TYPE INEQUALITY 3 spaces and obey the same axioms as a Hilbert space except that the inner prod- uct takes values in a general C∗-algebra than the complex number C. This sim- ple generalization gives a lot of trouble. Fundamental and familiar Hilbert space properties like Pythagoras’ equality, self-duality and decomposition into orthogonal complements must be given up. Moreover, a bounded module map between Hilbert C∗-modules need not have an adjoint; not every adjointable operator need have a polar decomposition. Hence to get its applications, we have to use it with great care. Let A be a C∗-algebra. A semi-inner product module over A is a right module X over A together with a generalized semi-inner product, that is with a mapping h., .i on X × X, which is A-valued and having the following properties: (i) hx, y + zi = hx, yi + hx, zi for all x,y,z ∈ X, (ii) hx,yai = hx, yi a for x, y ∈ X,a ∈ A, (iii) hx, yi∗ = hy, xi for all x, y ∈ X, (iv) hx, xi≥ 0 for x ∈ X. We will say that X is a semi-inner product C∗-module. The absolute value of x ∈ X is defined as the square root of hx, xi and it is denoted by |x|. If, in addition, (v) hx, xi = 0 implies x = 0, then h., .i is called a generalized inner product and X is called an inner product module over A or an inner product C∗-module. An Inner product C∗-module which 1 is complete with respect to the norm kxk := khx, xik 2 (x ∈ X) is called a Hilbert C∗-module. As we can see, an inner product module obeys the same axioms as an ordinary inner product space, except that the inner product takes values in a more general structure than in the field of complex numbers. If A is a C∗-algebra and X is a semi-inner product A-module, then the following Schwarz inequality holds: (2.1) hx, yihy, xi≤khx, xik hy,yi (x, y ∈ X). (e.g. [14, Proposition 1.1]). It follows from the Schwarz inequality (2.1) that kxk is a semi-norm on X. Now let A be a ∗-algebra, ϕ a positive linear functional on A and X be a semi- inner A-module. We can define a sesquilinear form on X ×X by σ(x, y)= ϕ (hx, yi); the Schwarz inequality for σ implies that (2.2) |ϕhx, yi|2 ≤ ϕhx, xiϕhy,yi. In [7, Proposition 1, Remark 1] the authors present two other forms of the Schwarz inequality in semi-inner A-module X, one for positive linear functional ϕ on A: (2.3) ϕ(hx, yihy, xi) ≤ ϕhx, xirhy,yi, where r is spectral radius, another one for C∗-seminorm γ on A: (2.4) (γhx, yi)2 ≤ γhx, xiγhy,yi.

3. The main results Let A be a C∗-algebra, first we state some basic properties of of A-value functions with respect to a positive for Bochner integrability of functions which we need to use them in our discussion. For basic properties of integrals of 4 A. G. GHAZANFARI

vector value functions with respect to scalar measures and integrals of scalar value functions with respect to vector measures see chapter II in [4]. Lemma 1. Let A be a C∗-algebra, X a Hilbert C∗-module and (Ω, M,µ) be a measure space. If f :Ω → X is Bochner integrable and a ∈ A then (a) fa is Bochner integrable, where fa(t)= f(t)a (t ∈ Ω), (b) the function f ∗ : Ω → A defined by f ∗(t) = (f(t))∗ is Bochner integrable ∗ ∗ and Ω f dµ = Ω f dµ . Furtheremore, (c) If µ(Ω)R < ∞ andR f is positive, i.e., f(t) ≥ 0 for all t ∈ Ω, then

Ω f(t)dµ(t) ≥ 0. Proof. (a)R Suppose ϕ : Ω → X is a simple function with finite support i.e., ϕ = n i=1 χEi xi with µ(Ei) < ∞ for each non-zero xi ∈ X (i = 1, 2, 3, ..., n), then for every a ∈ A the function ϕa :Ω → X defined by ϕa(t)= ϕ(t)a (t ∈ Ω) is a simple P function and Ω ϕa dµ = Ω ϕdµ a. Since f : Ω → X is Bochner integrable and kf(t)ak≤kf(t)k kak thus fa is strongly measurable and fa dµ = fdµ a, R R  Ω Ω therefore fa is Bochner integrable. n ∗R n R ∗  (b) For every simple function ϕ = i=1 χEi ai we have ϕ = i=1 χEi ai conse- ∗ quently ϕ∗dµ = ϕ dµ . The result therefore follows. Ω Ω P P (c) Suppose that µ(Ω) < ∞ and f is a Bochner integrable function and positive, R R  i.e., f(t) ≥ 0 for all t ∈ Ω. Since f is Bochner integrable Ω kf(t)kdµ(t) < ∞ by Theorem 2 in [4, chapter II, section 2]. Using Holder inequality for Lebesgue R integrable functions we get 1 2 1 1 1 kf 2 (t)kdµ(t)= kf(t)k 2 dµ(t) ≤ kf(t)kdµ(t) µ(Ω) 2 < ∞. ZΩ ZΩ ZΩ  1 So f 2 is Bochner integrable thus there is a sequence of simple functions ϕn such 1 1 2 2 that ϕn(t) → f (t) for almost all t ∈ Ω and Ω ϕn(t)dµ(t) → Ω f (t)dµ(t) in norm topology in A. ∗ R R This implies that ϕn(t) ϕn(t) → f(t), i.e., for every positive Bochner integrable function f there is a sequence of positive simple functions ψn such that ψn(t) → f(t) for almost all t ∈ Ω and Ω ψn(t)dµ(t) → Ω f(t)dµ(t) in norm topology in A. By proposition (1.6.1) in [5] the set of positive elements in a C∗-algebra is a closed R R  convex cone, therefore Ω f(t)dµ(t) ≥ 0 since Ω ψn(t)dµ(t) ≥ 0.

If µ is a probability measureR on Ω. We denoteR by L2(Ω,X) the set of all strongly 2 2 measurable functions f on Ω such that kfk2 := Ω kf(s)k dµ(s) < ∞. For every a ∈ X, we define the constant function ea ∈ L2(Ω,X) by ea(t) = R a (t ∈ Ω). In the following lemma we show that a special kind of invariant property holds, which we will use in the sequel.

Lemma 2. If f,g ∈ L2(Ω,X), a,b ∈ X and ea,eb are measurable then (3.1)

hf(t)−ea(t),g(t)−eb(t)idµ(t)− (f(t) − ea(t))dµ(t), (g(t) − eb(t))dµ(t) ZΩ ZΩ ZΩ  = hf(t),g(t)idµ(t) − f(t)dµ(t), g(t)dµ(t) . ZΩ ZΩ ZΩ  GRUSS¨ TYPE INEQUALITY 5

In particular,

2 2 2 (3.2) |f(t)| dµ(t) − f(t)dµ(t) ≤ |f(t) − ea(t)| dµ(t). ZΩ ZΩ ZΩ

Proof. We must state that the functions under the integrals (3.1) are Bochner integrable on Ω since they are strongly measurable and we can state the following obvious results: ∗ For every Λ ∈ X we have Λ Ω f(t)dµ(t) = Ω Λ(f(t))dµ(t). Therefore R  R hf(t),bidµ(t)= f(t)dµ(t),b , ZΩ ZΩ  ha,g(t)idµ(t)= a, g(t)dµ(t) . ZΩ  ZΩ  Also for almost all t ∈ Ω we have

1 1 2 2 2 kf(t)kdµ(t) ≤ µ(Ω) kf(t)k dµ(t) = kfk2, Ω Ω Z   Z  1 1 2 2 2 kg(t)kdµ(t) ≤ µ(Ω) kg(t)k dµ(t) = kgk2, Ω Ω Z   Z  and

khf(t),g(t)ikdµ(t) ≤kfk2kgk2. ZΩ A simple calculation shows that

hf(t)−ea(t),g(t)−eb(t)idµ(t)− (f(t) − ea(t))dµ(t), (g(t) − eb(t))dµ(t) ZΩ ZΩ ZΩ  = hf(t),g(t)i − hf(t),bi − ha,g(t)i + ha,bi dµ(t) Ω Z   − f(t)dµ(t) − a, g(t)dµ(t) − b ZΩ ZΩ  = hf(t),g(t)idµ(t) − f(t)dµ(t), g(t)dµ(t) , ZΩ ZΩ ZΩ  and for f = g and a = b we deduce (3.2). 

The following result concerning a generalized semi-inner product on L2(Ω,X) may be stated:

′ ′ Lemma 3. If f,g ∈ L2(Ω,X), x, x ,y,y ∈ X, then (i) the following inequalities (3.3) and (3.4) are equivalent

(3.3) Rehx′ − f(t),f(t) − xidµ(t) ≥ 0, ZΩ Rehy′ − g(t),g(t) − yidµ(t) ≥ 0, ZΩ 6 A. G. GHAZANFARI

x′ + x 2 1 (3.4) f(t) − dµ(t) ≤ |x′ − x|2, 2 4 ZΩ 2 y′ + y 1 g(t) − dµ(t) ≤ |y′ − y|2. 2 4 ZΩ

(ii) The map [f,g]: L 2(Ω,X) × L2(Ω ,X) → A,

(3.5) [f,g] := hf(t),g(t)idµ(t) − f(t)dµ(t), g(t)dµ(t) , ZΩ ZΩ ZΩ  is a generalized semi-inner product on L2(Ω,X). Proof. ′ (i) If f ∈ L2(Ω,X), since for any y, x, x ∈ X x′ + x 2 1 y − − |x′ − x|2 = Rehy − x′,y − xi, 2 4

hence

Rehx′ − f(t),f(t) − xidµ(t) ZΩ 1 x′ + x 2 (3.6) = |x′ − x|2 − f(t) − dµ(t) 4 2 ZΩ " #

1 x′ + x 2 = |x′ − x|2 − f (t) − dµ(t) 4 2 ZΩ

showing that, indeed, the inequalities (3.3) and (3.4) are equivalent.

(ii) We note that the first integral in (3.5) is belong to A and later integrals are in X and the following Korkine type identity for Bochner integrals holds:

(3.7) hf(t),g(t)idµ(t) − f(t)dµ(t), g(t)dµ(t) ZΩ ZΩ ZΩ  1 = hf(t) − f(s),g(t) − g(s)idµ(t)dµ(s). 2 ZΩ ZΩ By an application of the identity (3.7), 2 1 (3.8) |f(t)|2dµ(t) − f(t)dµ(t) = |f(t) − f(s)|2dµ(t)dµ(s) ≥ 0. 2 ZΩ ZΩ ZΩ ZΩ

It is easy to show that [., .] is a generalized semi-inner product on L2(Ω,X). 

The following theorem is a generalization of Theorem 2 for Hilbert C∗-modules. Theorem 3. Let X be a Hilbert C∗-module, µ a probability measure on Ω. If f,g ′ ′ belong to L2(Ω,X) and there exist the vectors x, x ,y,y ∈ X such that

(3.9) Rehx′ − f(t),f(t) − xidµ(t) ≥ 0, ZΩ Rehy′ − g(t),g(t) − yidµ(t) ≥ 0, ZΩ GRUSS¨ TYPE INEQUALITY 7 or, equivalently, x′ + x 2 1 (3.10) f(t) − dµ(t) ≤ |x′ − x|2, 2 4 ZΩ 2 y′ + y 1 g(t) − dµ(t) ≤ |y′ − y|2. 2 4 ZΩ

Then the following inequalities hold

(3.11) hf(t),g(t)idµ(t) − f(t)dµ(t), g(t)dµ(t) ZΩ ZΩ ZΩ  1 1 2 2 2 2 ≤ |f(t)|2dµ(t) − f(t)dµ(t) |g(t)|2dµ(t) − g(t)dµ(t) Ω Ω Ω Ω Z Z Z Z 1 1 2 ≤ |x′ − x|2 − Re hx ′ − f(t),f(t) − xidµ( t) 4 ZΩ 1 1 2 × |y′ − y|2 − Rehy′ − g(t),g(t) − yidµ(t) 4 ZΩ 1 ≤ k x′ − xkky′ − yk. 4 1 The coefficient 1 in second inequality and constant 4 in the last inequality are sharp in the sense that they cannot be replaced by a smaller quantity.

Proof. Since (3.5) is a generalized semi-inner product on L2(Ω,X), so Schwarz inequality holds, i.e., 2 (3.12) [f,g] ≤ [f,f] [g,g] . ′ Using (3.2) with a = x+x and (3.6) we get 2 2 [f,f]= |f(t)|2dµ(t) − f(t)dµ(t) ZΩ ZΩ 2 x′ + x (3.13) ≤ f(t) − dµ(t) 2 ZΩ 1 = |x ′ − x|2 − Re hx′ − f(t),f(t) − xidµ(t) 4 ZΩ 1 ≤ |x′ − x|2. 4 Similarly, 2 [g,g]= |g(t)|2dµ(t) − g(t)dµ(t) ZΩ ZΩ 2 y′ + y (3.14) ≤ g(t) − dµ(t) 2 ZΩ 1 = |y ′ − y|2 − Re hy′ − g(t),g(t) − yidµ(t) 4 ZΩ 1 ≤ |y′ − y|2. 4 8 A. G. GHAZANFARI

By Schwarz inequality (3.12) and inequalities (3.13), (3.14) we deduce (3.11). Now, suppose that (3.11) holds with the constants C,D > 0 in the third and forth inequalities. That is,

(3.15) hf(t),g(t)idµ(t) − f(t)dµ(t), g(t)dµ(t) ZΩ ZΩ ZΩ  1 1 2 ≤ C |x′ − x|2 − Rehx′ − f(t),f(t) − xidµ (t) 4 ZΩ 1 1 2 × |y′ − y|2 − Rehy′ − g(t),g(t) − yidµ(t) 4 ZΩ ′ ′ ≤ Dk x − xkky − yk.

Every Hilbert space H can be regarded as a Hilbert C-module. If we choose Ω = [0, 1] ⊆ R,X = C,f,g : [0, 1] → R ⊆ X, 1 −1 if 0 ≤ t ≤ 2 (3.16) f(t)= g(t)= 1 ( 1 if 2 ≤ t ≤ 1 then for x′ = y′ = 1, x = y = −1 and µ Lebesgue measure on Ω the conditions (3.10) holds. By (3.15) we deduce 1 ≤ C ≤ 4D 1 giving C ≥ 1 and D ≥ 4 , and the theorem is proved. 

4. Applications 1. Let X be a Hilbert C∗-module and B(X) the set of all adjointable operators on X. We recall that if A ∈B(X) then its operator norm is defined by kAk = sup{kAxk : x ∈ X, kxk≤ 1}, with this norm B(X) is a C∗-algebra. Let Ω = [0, 1] and f(t) = etA for t ∈ Ω, where A is an invertible element in B(X). Since for each t ∈ [0, 1] one has ketAk≤ etkAk ≤ ekAk, then an application of first inequality in (3.13) for x′ =2eA, x = −eA gives: 1 1 2 2 9 2 0 ≤ etA dt − etAdt ≤ eA . 4 Z0 Z0

This implies that 1 2 9 2 2 etA dt ≤ eA + A−1(eA − I) . 4 Z0

2. For square integrable functions f and g on [0, 1] and 1 1 1 D(f,g)= f(t)g(t)dt − f(t)dt g(t)dt Z0 Z0 Z0 Landau proved (see [13]) |D(f,g)|≤ D(f,f) D(g,g). p p GRUSS¨ TYPE INEQUALITY 9

Joci´cet.al. in [10] have proved for a probability measure µ and for square integrable fields (At) and (Bt) (t ∈ Ω) of commuting normal operators the following Landau type inequality holds

AtXBtdµ(t) − Atdµ(t)X Btdµ(t) ZΩ ZΩ ZΩ

2 2 2 2 ≤ |At| dµ(t) − Atdµ(t) X |Bt| dµ(t) − Btdµ(t) s Ω Ω s Ω Ω Z Z Z Z

for all X ∈ B (H) and for all unitarily invariant norms |||.|||.

Every C∗-algebra can be regarded as a Hilbert C∗-modules over itself with the inner product defined by ha,bi = a∗b. If we apply the first inequality in (3.11) of Theorem 3, we obtain the following result. Corollary 1. Let A be a C∗-algebra, µ a probability measure on Ω. If f,g belong to L2(Ω, A) Then the following inequality holds

(4.1) f(t)g(t)dµ(t) − f(t)dµ(t) g(t)dµ(t) ZΩ ZΩ ZΩ 1 1 2 2 2 2 ≤ |f(t)|2dµ(t) − f(t)dµ(t) |g(t) |2dµ(t) − g(t)dµ(t) . Ω Ω Ω Ω Z Z Z Z

Acknowledgment The author would like to thank the referee for some useful comments and sug- gestions.

References

[1] S. Bani´c, D. Iliˇsevi´cand S. Varoˇsanec, Bessel- and Gr¨uss-type inequalities in inner product modules, Proc. Edinb. Math. Soc. (2) 50 (2007), no. 1, 23-36. [2] C. Bu¸se, P. Cerone, S. S. Dragomir and J. Roumelitos, A refinement of Gr¨uss type inequality for the Bochner integral of vector-valued functions in Hilbert spaces and applications, J. Korean Math. Soc. 43 (2006), No. 5, pp. 911-929. [3] S. S. Dragomir, Advances in Inequalities of the Schwarz, Gr¨uss and Bessel Type in Inner Product Spaces, Nova Science puplishers Inc., New York, 2005. [4] J. Diestel, J. J. Uhl, Jr. Vector Measures, Mathematical Surveys, Number 15, Amer. Math. Soc. 1977. [5] J. Dixmier, C∗-algebras, North-Holland publishing company, 1982. [6] J.I. Fujii, M. Fujii, M.S. Moslehian and Y. Seo, Cauchy-Schwarz inequality in semi-inner product C∗-modules via polar decomposition, J. Math. Anal. Appl. 394 (2012), no. 2, 835– 840. [7] A. G. Ghazanfari, S. S. Dragomir, Schwarz and Gr¨uss type inequalities for C*-seminorms and positive linear functionals on Banach ∗-modules, Linear Algebra and Appl. 434 (2011), 944-956. [8] A. G. Ghazanfari, S. S. Dragomir, Bessel and Gr¨uss type inequalities in inner product modules over Banach ∗-algebras, J. Inequal. Appl. Vol(2011), Article ID 562923. ¨ 1 b − [9] G. Gr¨uss, Uber das Maximum des absoluten Betrages von b−a Ra f(x)g(x)dx b b 1 f x dx g x dx (b−a)2 Ra ( ) Ra ( ) , Math. Z. 39(1934), 215-226. [10] D. Joci´c, D. Krtini´cand M. S. Moslehian, Landau and Gruss type inequalities for inner product type integral transformers in norm ideals, Math. Inequal. Appl. vol 16, no 1 (2013), 109-125. 10 A. G. GHAZANFARI

[11] M. S. Moslehian and R. Raji´c, A Gr¨uss inequality for n-positive linear maps, Linear Algebra Appl. 433 (2010), no. 8-10, 1555–1560. [12] I. Peri´cand R. Raji´c, Gr¨uss inequality for completely bounded maps, Linear Algebra Appl. 390 (2004), 287–292. [13] E. Landau, Uber¨ mehrfach monotone Folgen, Prace Mat.-Fiz. XLIV (1936), 337-351. [14] E.C. Lance, Hilbert C∗-Modules, London Math. Soc. Lecture Note Series 210, Cambridge Univ. Press, 1995.

Department of Mathematics, Lorestan University, P. O. Box 465, Khoramabad, Iran. E-mail address: [email protected]