Orthogonality preserving property for pairs of operators on Hilbert $ C ...

ORTHOGONALITY PRESERVING PROPERTY FOR PAIRS OF OPERATORS ON HILBERT C ∗ -MODULES

arXiv:1711.04724v1 [math.OA] 10 Nov 2017

MICHAEL FRANK1 , MOHAMMAD SAL MOSLEHIAN2 AND ALI ZAMANI3 Abstract. We investigate the orthogonality preserving property for pairs of mappings on inner product C ∗ -modules extending existing results for a single orthogonality-preserving mapping. Guided by the point of view that the C ∗ valued inner product structure of a Hilbert C ∗ -module is determined essentially by the module structure and by the orthogonality structure, pairs of linear and local orthogonality-preserving mappings are investigated, not a priori bounded. The intuition is that most often C ∗ -linearity and boundedness can be derived from the settings under consideration. In particular, we obtain that if A is a C ∗ -algebra and T, S : E −→ F are two bounded A -linear mappings

between full Hilbert A -modules, then hx, yi = 0 implies hT (x), S(y)i = 0 for all x, y ∈ E if and only if there exists an element γ of the center Z(M (A ))

of the multiplier algebra M (A ) of A such that hT (x), S(y)i = γhx, yi for all x, y ∈ E . In addition, we give some applications.

1. Introduction and preliminaries
Let H , [·, ·] be an inner product space. Recall that vectors η, ζ ∈ H are said to be orthogonal, and write η ⊥ ζ, if [η, ζ] = 0. For inner product spaces H , K and two functions T, S : H → K , the orthogonality preserving property η ⊥ ζ =⇒ T (η) ⊥ S(ζ)

(η, ζ ∈ H )

was introduced in [5]. The following characterization was proved. Theorem 1.1. [5, Theorem 3.9] Let H , K be inner product spaces, and let T, S : H → K be linear mappings. The following conditions are equivalent: (i) η ⊥ ζ =⇒ T (η) ⊥ S(ζ) for all η, ζ ∈ H . (ii) There exists γ ∈ C such that [T (η), S(ζ)] = γ[η, ζ] for all η, ζ ∈ H . Notice that orthogonality preserving functions may be nonlinear and discontinuous, i.e. far from linear; see [4, Example 2]. For a given θ ∈ [0, 1) two vectors 2010 Mathematics Subject Classification. 47B49, 46L05, 46L08. Key words and phrases. Orthogonality preserving property, local mapping, inner product ∗ C -module. 1

2

M. FRANK, M. S. MOSLEHIAN, A. ZAMANI

η, ζ ∈ H are approximately orthogonal or θ-orthogonal, denoted by η ⊥θ ζ, if [η, ζ] ≤ θkηk kζk. Two mappings S, T : H → K are approximately orthogonality preserving mappings if for given δ, ε ∈ [0, 1) one has η ⊥δ ζ =⇒ T (η) ⊥ε S(ζ)

(η, ζ ∈ H ).

Often δ = 0 has been considered. The approximate orthogonality preserving mappings and the orthogonality equations have been investigated recently in [20, 21, 22]. J. Chmieli´ nski [4] and A. Turnˇsek [19] studied the approximate orthogonality preserving property for one linear mapping with δ = 0. Also, J. Chmieli´ nski et al. [6] have been recently verified the approximate orthogonality preserving property for two linear mappings. An inner product module over a C ∗ -algebra A is a (left) A -module E equipped with an A -valued inner product h·, ·i, which is C-linear and A -linear in the first variable and has the properties hx, yi∗ = hy, xi as well as hx, xi ≥ 0 with equality if and only if x = 0. E is called a Hilbert A -module if it is complete with respect 1 to the norm kxk = khx, xik 2 . An inner product A -module E has an “A -valued 1 norm” | · |, defined by |x| = hx, xi 2 . A mapping T : E −→ F , where E and F are inner product A -modules, is called A -linear if it is linear and T (ax) = aT (x) for all x ∈ E , a ∈ A . Further, T is called local if ax = 0 =⇒ aT (x) = 0

(a ∈ A , x ∈ E ).

Examples of local mappings include multiplication and differential operators. Note, that every A -linear mapping is local, but the converse is not true, in general (take linear differential operators into account). Moreover, every bounded local mapping between inner product modules is A -linear (see [11]). Although inner product C ∗ -modules generalize inner product spaces by allowing inner products to take values in a certain C ∗ -algebra instead of the C ∗ -algebra of complex numbers, some fundamental properties of inner product spaces are no longer valid in inner product C ∗ -modules in their full generality. Therefore, when we are studying inner product C ∗ -modules, it is always of interest under which conditions the results analogous to those for inner product spaces can be reobtained, as well as which more general situations might appear. We refer the reader to [13] for more information on the basic theory of Hilbert C ∗ -modules. It is natural to explore the (approximate) orthogonality preserving property between inner product C ∗ -modules. Elements x, y in an inner product C ∗ -module
E , h·, ·i are said to be orthogonal, and write x ⊥ y, if hx, yi = 0. Analogously to

ORTHOGONALITY PRESERVING PROPERTY FOR PAIRS OF OPERATORS

3

the Hilbert space situation, for a given θ ∈ [0, 1) two elements x, y ∈ E are approximately orthogonal or θ-orthogonal, denoted by x ⊥θ y, if khx, yik ≤ θkxkkyk. A mapping T : E → F between inner product C ∗ -modules is approximately orthogonality preserving if for given δ, ε ∈ [0, 1) one has x ⊥δ y =⇒ T (x) ⊥ε T (y)

(x, y ∈ E ).

This definition was introduced and investigated in [9, 14]. Two natural problems are to describe such a class of approximately orthogonality preserving mappings and to determine the stability of the orthogonality preserving property. Let K(H ) and B(H ) be the C ∗ -algebras of all compact linear operators and of all bounded linear operators on a Hilbert space H , respectively. Recall that A is a standard C ∗ -algebra on a Hilbert space H if K(H ) ⊆ A ⊆ B(H ). This includes the case of finite-dimensional Hilbert spaces. In the case when A is a standard C ∗ -algebra and δ = 0, D. Iliˇsevi´c and A. Turnˇsek [9] studied the approximate orthogonality preserving property on A modules. In [14], the authors gave some sufficient conditions for a linear mapping between Hilbert C ∗ -modules to be approximate orthogonality preserving. Also, it was obtained in [14], whenever A is a standard C ∗ -algebra and T : E −→ F is a nonzero A -linear (δ, ε)-orthogonality preserving mapping between A -modules, then

4(ε − δ)

hT (x), T (y)i − kT k2 hx, yi ≤ kT (x)k kT (y)k (x, y ∈ E ). (1 − δ)(1 + ε) Now, we will concentrate our investigations on the following condition, x ⊥ y =⇒ T (x) ⊥ S(y)

(x, y ∈ E ),

which we call the orthogonality preserving property for two linear mappings T, S : E → F. In the case when S = T , the orthogonality preserving property has been treated by M. Frank et al. [8], by C.-W. Leung et al. [11, 12], and others. In the present paper we show that if A is a standard C ∗ -algebra and T, S : E −→ F are two nonzero local mappings between inner product A -modules, then x ⊥ y =⇒ T (x) ⊥ S(y) for all x, y ∈ E if and only if there exists γ ∈ C such that hT (x), S(y)i = γhx, yi for all x, y ∈ E . In fact, this result is one generalization of Theorem 1.1. We then apply it to prove that if A is a C ∗ algebra and T, S : E → F are two nonzero bounded A -linear mappings between full Hilbert A -modules such that x ⊥ y =⇒ T (x) ⊥ S(y) for all x, y ∈ E , then there exists an element γ of the center Z(M(A )) of the multiplier algebra M(A )

4

M. FRANK, M. S. MOSLEHIAN, A. ZAMANI

of A such that hT (x), S(y)i = γhx, yi for all x, y ∈ E . In case of pairs of merely bounded linear operators S, T the invertibility of S implies the A -linearity and adjointability of both these operators, and the set of such pairs can be described in a very convenient way. 2. Linear and local orthogonality-preserving mappings The aim of this section is to prove an analogue of Theorem 1.1 for two unknown linear mappings in inner product C ∗ -modules, and subsequently, a generalization of Theorem 1.1 for A -linear mappings between Hilbert A -modules. Let us start with some observations. The following result is a combination of Theorem 3.1 of [2] and Lemma 4.1 of [21]. Lemma 2.1. Let E be an inner product A -module and x, y ∈ E . The following statements are mutually equivalent: (i) (ii) (iii) (iv) (vi) (v)

x ⊥ y. |x − λy| = |x + λy| for all λ ∈ C. |x − ay| = |x + ay| for all a ∈ A . |x|2 ≤ |x + λy|2 for all λ ∈ C. |x|2 ≤ |x + ay|2 for all a ∈ A . |x| ≤ |x + ay| for all a ∈ A .

As a consequence of Lemma 2.1, we have the following result. Proposition 2.2. Let E and F be two inner product A -modules and x, y ∈ E . Let T, S : E −→ F be two nonzero linear mappings. The following statements are mutually equivalent: (i) (ii) (iii) (iv) (vi) (v)

x ⊥ y =⇒ T (x) ⊥ S(y). |x − λy| = |x + λy| =⇒ |T (x) − λS(y)| = |T (x) + λS(y)| for all λ ∈ C. |x − ay| = |x + ay| =⇒ |T (x) − aS(y)| = |T (x) + aS(y)| for all a ∈ A . |x|2 ≤ |x + λy|2 =⇒ |T (x)|2 ≤ |T (x) + λS(y)|2 for all λ ∈ C. |x|2 ≤ |x + ay|2 =⇒ |T (x)|2 ≤ |T (x) + aS(y)|2 for all a ∈ A . |x| ≤ |x + ay| =⇒ |T (x)| ≤ |T (x) + aS(y)| for all a ∈ A .

Remark 2.3. Consider the C ∗ -algebra M2 (C) of all complex 2 × 2 matrices, as an inner product C ∗ -module over itself. Let A, B ∈ M2 (C) and let T, S : M2 (C) −→ M2 (C) be two nonzero linear mappings. Then, by [2, Proposition 3.6], the following statements are mutually equivalent: (i) A ⊥ B =⇒ T (A) ⊥ S(B).

ORTHOGONALITY PRESERVING PROPERTY FOR PAIRS OF OPERATORS

5

(ii) |A| ≤ |A + λB| =⇒ |T (A)| ≤ |T (A) + λS(B)| for all λ ∈ C. Proposition 2.4. Let E and F be two inner product A -modules. Let T, S : E −→ F be two nonzero linear mappings such that hT (x), S(x)i = γ|x|2 for all x ∈ E and for some γ ∈ C. Then x ⊥ y =⇒ T (x) ⊥ S(y)

(x, y ∈ E ).

Proof. Let x, y ∈ E . By the polarization identity, we obtain 3

hT (x), S(y)i =

3

1X k 1X k i hT (x + ik y), S(x + ik y)i = i γ|x + ik y|2 = γhx, yi. 4 k=0 4 k=0

Thus, the assertion follows.



Notice that the converse of the above proposition is not true, even in the case T = S (see [14, Example 3.14]). In the next theorem, we prove that the converse of the above proposition is true if A is a standard C ∗ -algebra. We state some prerequisites for the next theorem. Given two vectors η, ζ in a Hilbert space H , we shall denote the one-rank operator defined by (η ⊗ ζ)(ξ) = [ξ, ζ]η by η ⊗ ζ ∈ K(H ). Observe that η ⊗ η is a minimal projection. Recall that a projection e in a C ∗ -algebra A is called minimal if eA e = Ce. Now let A be a standard C ∗ -algebra on a Hilbert space H and let E be an inner product (respectively, Hilbert) A -module. Let e = η ⊗ η for some unit vector η ∈ H , be any minimal projection. Then Ee = {ex : x ∈ E }, is a complex inner product (respectively, Hilbert) space contained in E with respect to the inner product [x, y] = tr(hx, yi), x, y ∈ Ee . Note that if x, y ∈ Ee , then hx, yi = [x, y]e and kxkEe = kxkE , where the norm k.kEe comes from the inner product [·, ·]. This enables us to apply Hilbert space theory by lifting results from the Hilbert space Ee to the whole A -module E . Theorem 2.5. Let A be a standard C ∗ -algebra on a Hilbert space H and let E , F be two inner product A -modules. Suppose, T, S : E −→ F are two nonzero A -linear mappings such that x ⊥ y =⇒ T (x) ⊥ S(y)

(x, y ∈ E ).

Then there exists γ ∈ C such that hT (x), S(y)i = γhx, yi

(x, y ∈ E ).

Proof. The following proof is a modification of that one given by D. Iliˇsevi´c and A. Turnˇsek in [9, Theorem 3.1]. Let e = ζ ⊗ ζ, f = η ⊗ η be minimal projections

6

M. FRANK, M. S. MOSLEHIAN, A. ZAMANI

and let u = ζ ⊗ η. For linear mappings Te , Se : Ee −→ Fe we have [x, y] = 0 =⇒ [Te (x), Se (y)] = 0 for all x, y ∈ Ee . Hence, by Theorem 1.1, there exists γe ∈ C such that [T (ex), S(ex)] = γe kexk2 = γe [ex, ex] (x ∈ Ee ). This yields [T (ex), S(ex)]e = γe [ex, ex]e, thus hT (ex), S(ex)i = γe hex, exi = γe |ex|2 , or equivalently ehT (x), S(x)ie = γe e|x|2 e

(x ∈ E ).

(2.1)

Similarly, there exists γf ∈ C such that f hT (x), S(x)if = γf f |x|2 f

(x ∈ E ).

(2.2)

Since uf u∗ = e, from (2.1) and (2.2) it follows that γe [ex, ex]e = γe hex, exi = γe ehx, xie

= ehT (x), S(x)ie = uf u∗hT (x), S(x)iuf u∗

= uf hT (u∗x), S(u∗x)if u∗ = uγf f |u∗x|2 f u∗ = γf uf u∗|x|2 uf u∗ = γf ehx, xie = γf hex, exi = γf [ex, ex]e.

Thus γe [ex, ex] = γf [ex, ex] Replacing x with

x , kexk

(x ∈ E ).

we conclude γe = γf = γ. Hence, by (2.1), we get

ehT (x), S(x)ie = eγ|x|2 e

(x ∈ E )

for all minimal projections e ∈ A . Thus hT (x), S(x)i = γ|x|2

(x ∈ E ).

Now, by the polarization identity, we get hT (x), S(y)i = γhx, yi

(x, y ∈ E ). 

Corollary 2.6. Let A be a standard C ∗ -algebra and let {E , h., .i1 } be an inner product A -module. Suppose that h., .i2 is a second A -valued inner products on E . If hx, yi1 = 0 implies hx, yi2 = 0 for any x, y ∈ E , then there exists a positive constant γ ∈ C such that hx, yi2 = γhx, yi1 for any x, y ∈ E . Proof. Take E = F as A -modules and set T = S = id : {E , h., .i1 } → {E , h., .i2 }.  Applying Theorem 2.5 the assertion follows.

ORTHOGONALITY PRESERVING PROPERTY FOR PAIRS OF OPERATORS

7

Remark 2.7. According to [14, Example 3.15], the assumption of A -linearity, even in the case T = S, is necessary in Theorem 2.5. In the following result, we employ some ideas of [11] to consider local maps between inner product A -modules, i.e. linear maps T : E → F such that ax = 0 for any x ∈ E , a ∈ A forces aT (x) = 0 in F . We are interested in maps which preserve orthogonality. Theorem 2.8. Let A be a standard C ∗ -algebra on a Hilbert space H and let E , F be two inner product A -modules. Suppose that T, S : E −→ F are two nonzero local mappings such that x ⊥ y =⇒ T (x) ⊥ S(y)

(x, y ∈ E ).

Then there exists γ ∈ C such that hT (x), S(y)i = γhx, yi

(x, y ∈ E ).

Proof. Let (ei )i∈I and (fj )j∈J be approximate units for A and K(H ), respectively. Suppose that p ∈ K(H ) is a projection. For x ∈ E we have p(1 − p)ei (1 − p)x = 0 and (1 − p)pei px = 0. Since T is local, we obtain

p(1 − p)ei T ((1 − p)x) = 0 and (1 − p)pei T (px) = 0. From lim pei p = p and lim(1 − p)ei (1 − p) = 1 − p, we derive i i

p T (x) = p T ((1 − p)x) + T (px) = pT ((1 − p)x) − (1 − p)T (px) + T (px)

= lim pei p T ((1 − p)x)) − lim(1 − p)ei (1 − p) T (px) + T (px) i i

= lim pei p T ((1 − p)x) − lim(1 − p)ei (1 − p) T (xp) + T (px) i

i

= T (px). Thus, T (ax) = aT (x) for all finite rank operators a ∈ K(H ). Now, for any x ∈ K(H ) · E , there exist c ∈ K(H ) and z ∈ E such that x = cz. Consider the linear mapping Te : K(H ) · E −→ K(H ) · F defined by Te(x) := cT (z). Therefore, Te(ax) = Te(acz) = acT (z) = aTe(x) for all x ∈ E and all a ∈ K(H ). Since (fj )j∈J is an approximate unit for K(H )






it follows from fj T (x) − Te(x) = T (fj cz) −cT (z) = fj cT (z) −cT (z) that

8

M. FRANK, M. S. MOSLEHIAN, A. ZAMANI

e lim fj T (x) = Te(x) for all x ∈ E ·K(H ) and all j ∈ J. Similarly, lim fk S(y) = S(y) j

k

for all y ∈ E · K(H ) and all k ∈ J. Therefore, if x, y ∈ K(H ) · E with hx, yi = 0, then hT (x), S(y)i = 0, which implies hfj T (x), fk S(y)i = 0 for all j, k ∈ J. Thus e hTe(x), S(y)i = 0. Hence for K(H )-linear mappings Te, Se we have e x ⊥ y =⇒ Te(x) ⊥ S(y)

(x, y ∈ K(H ) · E ).

e So, by Theorem 2.5, there exists γ ∈ C such that hTe(x), S(x)i = γ|x|2 for all x ∈ K(H ) · E . Thus e j x)i = γ|fj x|2 = fj γ|x|2 fj fj hT (x), S(x)ifj = hTe(fj x), S(f

for all x ∈ E and all j ∈ J. Hence, hT (x), S(x)i = γ|x|2 for all x ∈ E and by the polarization identity, we obtain hT (x), S(y)i = γhx, yi

(x, y ∈ E ). 

Combining Proposition 2.4 and Theorem 2.8 we get the next result. In fact, this result is a generalization of [9, Theorem 3.1] and [21, Theorem 4.10]. The result generalizes [11, Cor. 3.2]. Theorem 2.9. Let A be a standard C ∗ -algebra and let E , F be two inner product A -modules. Suppose that T, S : E → F are two nonzero local mappings. The following statements are mutually equivalent: (i) x ⊥ y =⇒ T (x) ⊥ S(y) for all x, y ∈ E . (ii) There exists γ ∈ C such that hT (x), S(x)i = γ|x|2 for all x ∈ E . (iii) There exists γ ∈ C such that hT (x), S(y)i = γhx, yi for all x, y ∈ E . Corollary 2.10. Let A be a standard C ∗ -algebra and let E be an inner product A -module. Suppose that T : E → E is a nonzero local mapping such that x ⊥ y =⇒ T (x) ⊥ y

(x, y ∈ E ).

Then there exists γ ∈ C such that T (x) = γx for all x ∈ E . Proof. Applying Theorem 2.9 to T and S = id we obtain, with some γ ∈ C, hT (x), yi = γhx, yi for all x, y ∈ E . Hence, hT (x) − γx, yi = 0 for all x, y ∈ E . Thus T (x) = γx for all x ∈ E .  Corollary 2.11. Let A be a standard C ∗ -algebra and let E and F be two inner product A -modules. Let T0 , S0 : E → F be two nonzero local mappings such that x ⊥ y =⇒ T0 (x) ⊥ S0 (y)

(x, y ∈ E ).

ORTHOGONALITY PRESERVING PROPERTY FOR PAIRS OF OPERATORS

9

Suppose, the mappings T, S : E → F are sufficiently close to T0 and S0 , respectively, namely that for θ1 , θ2 ∈ [0, 1) and for all x, y ∈ E kT (x) − T0 (x)k ≤ θ1 kT (x)k

and

kS(y) − S0 (y)k ≤ θ2 kS(y)k.

Then x ⊥ y =⇒ T (x) ⊥ε S(y)

(x, y ∈ E ),

where ε = θ1 θ2 + θ1 (θ2 + 1) + (θ1 + 1)θ2 . Proof. From the assumption, we obtain kT0 (x)k ≤ (θ1 + 1)kT (x)k

and

kS0 (y)k ≤ (θ2 + 1)kS(y)k

(x, y ∈ E ). (2.3)

Also, by Theorem 2.9, there exists γ0 ∈ C such that hT0 (x), S0 (y)i = γ0 hx, yi

(x, y ∈ E ).

(2.4)

Now let x, y ∈ E and x ⊥ y. By (2.3) and (2.4), we get khT (x), S(y)ik = khT (x), S(y)i − hT0 (x), S0 (y)ik

= hT (x) − T0 (x), S(y) − S0 (y)i + hT (x) − T0 (x), S0 (y)i

+ hT0 (x), S(y) − S0 (y)i

≤ kT (x) − T0 (x)k kS(y) − S0 (y)k + kT (x) − T0 (x)k kS0 (y)k 

+ kT0 (x)k kS(y) − S0 (y)k

 ≤ θ1 θ2 + θ1 (θ2 + 1) + (θ1 + 1)θ2 kT (x)k kS(y)k = εkT (x)k kS(y)k. Thus khT (x), S(y)ik ≤ εkT (x)k kS(y)k and hence T (x) ⊥ε S(y).



3. C ∗ -linear orthogonality preserving mappings We want to show the properties of pairs of bounded C ∗ -linear mappings {T, S} for which orthogonality of two elements x, y of the domain Hilbert C ∗ -module implies the orthogonality of their respective images T (x), S(y). To get reasonable results we have either to suppose or we derive C ∗ -linearity of the maps. The proof of the key equality relies on the theory of bidual von Neumann algebras of the C ∗ -algebra of coefficients, on the existence of predual Banach spaces for von Neumann algebras and for selfdual Hilbert C ∗ -modules over them. We need a construction by W. L. Paschke ([15]): for any Hilbert A -module E over any C ∗ -algebra A one can extend E canonically to a Hilbert A ∗∗ -module

10

M. FRANK, M. S. MOSLEHIAN, A. ZAMANI

E # over the bidual Banach space and von Neumann algebra A ∗∗ of A [15, Theorem 3.2, Proposition 3.8, §4]. For this aim the A ∗∗ -valued pre-inner product can be defined by the formula [a ⊗ x, b ⊗ y] = a∗ hx, yib, for elementary tensors of A ∗∗ ⊗ E , where a, b ∈ A ∗∗ , x, y ∈ E . The quotient module of A ∗∗ ⊗ E by the set of all isotropic vectors is denoted by E # . It can be canonically completed to a self-dual Hilbert A ∗∗ -module G which is isometrically algebraically isomorphic to the A ∗∗ -dual A ∗∗ -module of E # . G is a dual Banach space itself, cf. [15, Theorem 3.2, Proposition 3.8, §4]. Every A -linear bounded map T : E → E can be continued to a unique A ∗∗ -linear map T : E # → E # preserving the operator norm and obeying the canonical embedding π ′ (E ) of E into E # . Similarly, T can be further extended to the self-dual Hilbert A ∗∗ module G . The extension is such that the isometrically algebraically embedded copy π ′ (E ) of E in G is a w ∗ -dense A -submodule of G , and that A -valued inner product values of elements of E embedded in G are preserved with respect to the A ∗∗ -valued inner product on G and to the canonical isometric embedding π of A into its bidual Banach space A ∗∗ . Any bounded A -linear operator T on E extends to a unique bounded A ∗∗ -linear operator on G preserving the operator norm, cf. [15, Proposition 3.6, Corollary 3.7, §4]. The extension of bounded A linear operators from E to G is continuous with respect to the w ∗ -topology on G . For topological characterizations of self-duality of Hilbert C ∗ -modules over W ∗ algebras we refer to [15], [7, Theorem 3.2] and to [17, 18]: a Hilbert C ∗ -module K over a W ∗ -algebra B is self-dual, if and only if its unit ball is complete with respect to the topology induced by the semi-norms {|f (hx, .i)| : x ∈ K, f ∈ B ∗ , kxk ≤ 1, kf k ≤ 1}, if and only if its unit ball is complete with respect to the topology induced by the semi-norms {f (h., .i)1/2 : f ∈ B ∗ , kf k ≤ 1}. The first topology coincides with the w ∗ -topology on K in that case. Note, that in the construction above E is always w ∗ -dense in G , as well as for any subset of E the respective construction is w ∗ -dense in its biorthogonal complement with respect to G . However, starting with a subset of G its biorthogonal complement with respect to G might not have a w ∗ -dense intersection with the embedding of E into G , cf. [16, Proposition 3.11.9]. Theorem 3.1. Let A be a C ∗ algebra and let E ,F be two full Hilbert A -modules. Suppose that T, S : E → F are two nonzero bounded A -linear mappings such that x ⊥ y =⇒ T (x) ⊥ S(y) (x, y ∈ E ).

ORTHOGONALITY PRESERVING PROPERTY FOR PAIRS OF OPERATORS

11

Then there exists an element γ of the center Z(M(A )) of the multiplier algebra M(A ) of A such that hT (x), S(y)i = γhx, yi

(x, y ∈ E ).

Proof. First, we make use of the existing canonical non-degenerate isometric ∗representation π of a C ∗ -algebra A in its bidual Banach space and von Neumann algebra A ∗∗ of A , as well as of its extension π ′ : E → E # → G , π ′ : F → F # → H and of the unique w ∗ -continuous A ∗∗ -linear bounded operator extensions T, S : G → H . In other words, we extend the set {A , E , F , T, S} to the set {A ∗∗ , G , H , T, S}. The Hilbert A ∗∗ -modules G , H are self-dual and admit a predual Banach space, hence, a w ∗ -topology. The extended operators T, S are w ∗ -continuous, A ∗∗ -linear and bounded by the same constants as the original operators T, S. The properties of the modules and operators can be derived from the results in [15, 7]. Secondly, we have to demonstrate that for any pair of elements x ⊥ y with x, y ∈ G the property T (x) ⊥ S(y) still holds for the extended operators. In the sequel, we leave out the denotation π ′ and identify E with its image π ′ (E ) ⊆ G . Since the subsets E ∩ {x}⊥ and E ∩ {y}⊥ are w ∗ -dense in their biorthogonal completions inside G by construction (cf. [7]), there exists two nets {yα }α∈I ⊂ E ∩ {x}⊥ and {xβ }β∈J ⊂ E ∩ {y}⊥ such that the w ∗ -limit of {yα}α∈I is y and the w ∗ -limit of {xβ }β∈J is x in G . Since x ⊥ y by supposition, yα ⊥ xβ for any α ∈ I, β ∈ J. Consequently, T (xβ ) ⊥ S(yα) for any α ∈ I, β ∈ J by supposition. Since both the extended operators T, S are w ∗ -continuous and the operation of taking the orthogonal complement in H commutes with taking w ∗ -limits, we arrive at T (x) ⊥ S(y). Next, we want to consider only discrete W ∗ -algebras, i.e. W ∗ -algebras for which the supremum of all minimal projections contained in them equals their identity. (We prefer to use the word discrete instead of atomic.) To connect to the general C ∗ -case we make use of a theorem by Ch. A. Akemann stating that the ∗-homomorphism of a C ∗ -algebra A into the discrete part of its bidual von Neumann algebra A ∗∗ which arises as the composition of the canonical embedding π of A into A ∗∗ followed by the projection to the discrete part of A ∗∗ is an injective ∗-homomorphism ρ, [1, p. 278]. This injective ∗-homomorphism ρ is implemented by a central projection p ∈ Z(A ∗∗ ) in such a way that A ∗∗ multiplied by p gives the discrete part of A ∗∗ . Applying this approach to our situation we reduce the problem further by investigating the set {A ∗∗ p, G p, H p, T p, Sp}

12

M. FRANK, M. S. MOSLEHIAN, A. ZAMANI

instead of the set {A ∗∗ , G , H , T, S}, where we rely on the injectivity of the algebraic embeddings A → A ∗∗ p, E → G p and F → H p. The latter two maps are injective since hx, xi = 6 0 forces hxp, xpi = hx, xip 6= 0. Obviously, the bounded ∗∗ pA -linear operators T p and Sp have again the property that xp ⊥ yp forces T (x)p ⊥ S(y)p for any such x, y ∈ pG . Since the von Neumann algebra pA ∗∗ is discrete its identity p can be represented as the w ∗ -sum of a maximal set of pairwise orthogonal atomic projections P {qα : α ∈ I} of the center Z(pA ∗∗ ) of pA ∗∗ . Note, that w ∗ - α∈I qα = p. Take a single atomic projection qα ∈ Z(pA ∗∗ ) of this collection and consider the part {A ∗∗ pqα , E pqα , F pqα , T pqα , Spqα } of the problem. Since A ∗∗ pqα is a discrete (type I) W ∗ -factor (finite- or infinite-dimensional), the equality hT (x), S(y)i = λqα hx, yi = hx, yiλqα holds for a specific for T, S, qα complex number λqα , cf. Theorem 2.5. Now, we can follow the decomposition process in reverse direction. Note, that the multiplier algebra of A p is ∗-isometrically embedded in A ∗∗ p. Since P kλα k ≤ kSk kT k for any α ∈ I, the sum α∈I λα qα is w ∗ -convergent in Z(A ∗∗ p). P Moreover, since λα commutes with hx, yipqα for any α, the sum α∈I λα qα commutes with hx, yip. What is more, since hG , G ip is dense in A ∗∗ p and the prodP P uct α∈I λα qα hx, yi = α∈I hx, yiλαqα belongs to A ∗∗ p for any x, y ∈ G , the P element α∈I λα qα is in pZ(M(A )). We arrive at the equality hT (x), S(y)ip = P hx, yi α∈I λα qα for any x, y ∈ G . The remaining step is to pull back this equality to the initial context along the two injective ∗-homomorphisms used.  Corollary 3.2. Let A be a C ∗ -algebra and E be a full Hilbert A -module. Let T : E → E be a bounded A -linear map such that x ⊥ y implies T (x) ⊥ y for any suitable pair x, y ∈ E . The there exists an element γ ∈ Z(M(A )) such that T (x) = xγ for any x ∈ E . The proof is the same as for Corollary 2.10 changing the origin of γ and the theorem referred to. The next theorem generalizes [8, Theorem 3] and [11, Thm. 3.2], and it gives a partial solution of [8, Problem 1]. Theorem 3.3. Let A be a C ∗ -algebra and E be a full Hilbert A -module. Let T : E → E be a bounded linear map and let S : E → E be an invertible linear map with bounded inverse operator. Suppose, γhx, yi = hT (x), S(y)i for some element γ ∈ Z(M(A )). Then S, T are bounded, A -linear, adjointable, invertible, and ST ∗ (x) = xγ and S ∗ T (x) = xγ for any x ∈ E , i.e. the pairs of operators {S, T ∗ }

ORTHOGONALITY PRESERVING PROPERTY FOR PAIRS OF OPERATORS

13

and {S ∗ , T } commute, and also S = γ(T ∗ )−1 and T = γ(S ∗ )−1 . Furthermore, the set of all such pairs of operators can be described as n o {T, S} : T = κγU, S = κ −1 U, U − unitary, κ ∈ Z(M(A )) − invertible . In the special situation of T = S the element γ is positive and T = unitary A -linear operator U on E .

√

γU for some

Proof. Since S is boundedly invertible, we derive the equality γhx, S −1 (z)i = hT (x), zi for any x, z ∈ E . By the boundedness of S −1 and T , the operator T is adjointable and T ∗ = γS −1 . Since adjointable linear module maps on Hilbert C ∗ modules are A -linear and bounded, both T and S have to be A -linear, bounded, invertible with bounded inverses and adjointable. Since γS −1 = T ∗ and T ∗ is invertible along with T , γ has to be invertible, too. So T T −1 = γ −1 T S ∗ = id, T −1 T = γ −1 S ∗ T = id. We arrive at ST ∗ = γ · id and S ∗ T = γ · id. Taking the adjoint of one of these equalities we obtain the commutation results. If, in particular, T = S in our initial equality we derive T T ∗ = γ · id and √ T ∗ T = γ · id. Consequently, γ is positive and T = γU for some A -linear unitary operator U on E . This shows the last assertion. Considering the equality hT (x), S(y)i = γhx, yi for x, y ∈ E , along with a certain pair of bounded C ∗ -linear operators {T, S} also all pairs {κT, κ −1 S : κ ∈ Z(M(A )) − invertible} are solutions of it. Recall, that T = γ(S ∗ )−1 and replace T in the equality under consideration, what gives hγ(S ∗ )−1 (x), S(y)i = γh(S ∗ )−1 (x), S(y)i = γhx, yi. Since γ is invertible and belongs to Z(M(A )) we can cut it. Therefore, h(S ∗ )−1 (x), S(y)i = hx, yi. We can transform the latter equality in two ways: firstly, switching from (S ∗ )−1 to its adjoint operator S −1 , and secondly, replacing x by S ∗ (z), where z ∈ E is arbitrary. We get the two equalities hx, S(y)i = hS −1 (x), yi and hz, S(y)i = hS ∗ (z), yi that are valid for any x, y, z ∈ E . Comparing the right sides and replacing the symbol z by the symbol x again, we obtain S −1 = S ∗ , i.e. the operator S has to be unitary on E . Also, T = γS. Replacing now S in the initial equality we get the equality hT (x), T (y)iγ −1 = γhx, yi. This transforms to the equality hT (x), T (y)i = |γ|2 hx, yi. But, this has been solved: T = |γ|U, where U is a unitary operator on E . For a better representation of the result, we correct T by a certain unitary factor. Let γ = u|γ| be the polar decomposition of γ. Then set T = |γ|U = |γ|u∗uU = γ(uU) = γS. So S = uU is the compatible choice.

14

M. FRANK, M. S. MOSLEHIAN, A. ZAMANI

Just recall the role of factors from Z(M(A )) and the set of solutions is complete.  Remark 3.4. The situation treated in the previous theorem can be reconsidered taking bounded C ∗ -linear operators S admitting a bounded generalized inverse operator in the context of Hilbert C ∗ -modules and bounded C ∗ -linear operators T. Corollary 3.5. Let A be a C ∗ -algebra and let h., .i2 be another A -valued inner product on a full Hilbert A -module {E , h., .i1 } inducing an equivalent norm to the given one. Suppose, hx, yi1 = 0 implies hx, yi2 = 0 for suitable x, y ∈ E . Then there exists an invertible positive element γ ∈ Z(M(A )) such that hx, yi1 = γhx, yi2 for any x, y ∈ E . Proof. Set F = E as Hilbert A -modules and add the alternative A -valued inner product h., .i2 . Set T = S = id and note, that both these operators are bounded if considered as operators from E to F . Then Theorem 3.1 and Corollary 3.2 give hx, yi1 = γhx, yi2 for any x, y ∈ E .  4. Additional results Recall that a C ∗ -algebra A has real rank zero if every selfadjoint element in A can be approximated in norm by invertible selfadjoint elements. Note that if A has real rank zero, then the ∗-algebra generated by all the idempotents in A is dense in A (see, e.g., [3]). The result extends [11, Thm. 2.3]. Theorem 4.1. Let A be a C ∗ -algebra of real rank zero with identity e and E and F be Hilbert A -modules. Suppose that A -linear mappings T, S : E −→ F satisfy the condition x ⊥ y =⇒ T (x) ⊥ S(y)

(x, y ∈ E ).

Suppose that there is z ∈ E with hz, zi being invertible and hT (z), S(z)i is selfadjoint. Then, there exits a selfadjoint element γ in the center Z(A ) of A such that hT (x), S(y)i = γhx, yi

(x, y ∈ E ).

Proof. By replacing z with |z|−1 z, we assume hz, zi = e. For any symmetry u ∈ A , we have hz + uz, z − uzi = |z|2 − u|z|2 + |z|2 u∗ − u|z|2 u∗ = 0

ORTHOGONALITY PRESERVING PROPERTY FOR PAIRS OF OPERATORS

15

whence, z + uz ⊥ z − uz. Hence our assumption yields T (z + uz) ⊥ S(z − uz), or equivalently hT (z), S(z)i − uhT (z), S(z)i + hT (z), S(z)iu∗ − uhT (z), S(z)iu∗ = 0. Now, let γ := hT (z), S(z)i. So, the above equality becomes γ −uγ +γu−uγu = 0. Since γ is selfadjoint, hence by taking adjoint γ − γu + uγ − uγu = 0. Thus γ = uγu. As A is generated by projections, and thus also by symmetries we

 get γ ∈ Z(A ). On the other hand, for any x ∈ E we have z, x − hx, ziz = hz, xi − |z|2 hz, xi = 0. Hence z ⊥ x − hx, ziz

and

x − hx, ziz ⊥ z.

(4.1)

and


T x − hx, ziz ⊥ S(z).

(4.2)

So, our assumption yields T (z) ⊥ S x − hx, ziz




Furthermore, from (4.1) it follows that D E x − hx, ziz + x − hx, ziz z, x − hx, ziz − x − hx, ziz z 2

 = x − hx, ziz − x − hx, ziz, z x − hx, ziz

 + x − hx, ziz z, x − hx, ziz − x − hx, ziz hz, zi x − hx, ziz 2 2 = x − hx, ziz − x − hx, ziz = 0. x − hx, ziz z is orthogonal to x − hx, ziz − x − hx, ziz z and Then x − hx, ziz +     hence T x−hx, ziz+ x−hx, ziz z is orthogonal to S x−hx, ziz− x−hx, ziz z . Thus, from (4.2) it follows that D    E 0 = T x − hx, ziz + x − hx, ziz z , S x − hx, ziz − x − hx, ziz z D E

E D
= T x − hx, ziz , S x − hx, ziz − T x − hx, ziz , S(z) x − hx, ziz D
E + x − hx, ziz T (z), S x − hx, ziz − x − hx, ziz hT (z), S(z)i x − hx, ziz D

E = T x − hx, ziz , S x − hx, ziz − x − hx, ziz γ x − hx, ziz . D

E Then T x − hx, ziz , S x − hx, ziz = x − hx, ziz γ x − hx, ziz . Since γ ∈ Z(A ), by (4.1) we obtain D 2

E T x − hx, ziz , S x − hx, ziz = γ x − hx, ziz D E D E = γ x − hx, ziz, x − γ x − hx, ziz, hx, ziz = γ|x|2 − γ|hx, zi|2 .

(4.3)

16

M. FRANK, M. S. MOSLEHIAN, A. ZAMANI

From (4.2) and (4.3) we get D E

hT (x), S(x)i = T x − hx, ziz + hx, ziT (z), S x − hx, ziz + hx, ziS(z) D D
E

E = T x − hx, ziz , S x − hx, ziz + hx, zi T (z), S x − hx, ziz D E
+ T x − hx, ziz , S(z) hz, xi + hx, zihT (z), S(z)ihz, xi = γ|x|2 − γ|hx, zi|2 + γ|hx, zi|2 = γ|x|2 . Hence hT (x), S(x)i = γ|x|2

(x ∈ E ).

(4.4)

Now, by (4.4) and the polarization identity, we obtain hT (x), S(y)i = γhx, yi

(x, y ∈ E ). 

Remark 4.2. Notice that orthogonality preserving functions may be very nonlinear and discontinuous; see [4, Example 2]. Now let E be a Hilbert K(H )-module and let F be a Hilbert A -module. Let g, h : E −→ F be additive functions such that x ⊥ y =⇒ g(x) ⊥ h(y)

(x, y ∈ E ).

Suppose that function f : E −→ A defined by f (x) := hg(x), h(x)i is continuous. Fix x, y ∈ E such that x ⊥ y. Hence hx, yi = hy, xi = 0. Therefore hg(x), h(y)i = hg(y), h(x)i = 0. Then f (x + y) = hg(x + y), h(x + y)i = hg(x), h(x)i + hg(x), h(y)i + hg(y), h(x)i + hg(y), h(y)i = hg(x), h(x)i + hg(y), h(y)i = f (x) + f (y). Thus f is orthogonally additive. By [10, Theorem 4.4 (ii)], there are a unique continuous additive function A : E −→ A and a unique mapping Φ : hE , E i −→ A such that f (x) = A(x) + Φ(hx, xi)

(x ∈ E ).

References 1. Ch. A. Akemann, The general Stone-Weierstrass problem for C ∗ -algebras, J. Funct. Anal. 4 (1969), 277–294. 2. Lj. Arambaˇsi´c and R. Raji´c, A strong version of the Birkhoff–James orthogonality in Hilbert C ∗ -modules, Ann. Funct. Anal. 5 (2014), no. 1, 109–120.

ORTHOGONALITY PRESERVING PROPERTY FOR PAIRS OF OPERATORS

17

3. L. G. Brown and G. K. Pedersen, Hilbert C ∗ -algebras of real rank zero, J. Funct. Anal. 99 (1991), 131–149. 4. J. Chmieli´ nski, Linear mappings approximately preserving orthogonality, J. Math. Anal. Appl. 304 (2005), 158–169. 5. J. Chmieli´ nski, Orthogonality equation with two unknown functions, Aequationes Math. 90 (2016), 11–23. 6. J. Chmieli´ nski, R. Lukasik and P. W´ojcik, On the stability of the orthogonality equation and the orthogonality-preserving property with two unknown functions, Banach J. Math. Anal. 10 (2016), no. 4, 828–847. 7. M. Frank, Self-duality and C ∗ -reflexivity of Hilbert C ∗ -modules, Z. Anal. Anwend. 9 (1990), 165–176. 8. M. Frank, A. S. Mishchenko and A. A. Pavlov, Orthogonality-preserving, C ∗ -conformal and conformal module mappings on Hilbert C ∗ -modules, J. Funct. Anal. 260 (2011), 327–339. 9. D. Iliˇsevi´c and A. Turnˇsek, Approximately orthogonality preserving mappings on C ∗ modules, J. Math. Anal. Appl. 341 (2008), 298–308. 10. D. Iliˇsevi´c, A. Turnˇsek and D. Yang, Orthogonally additive mappings on Hilbert modules, Studia Math. 221 (2014), 209–229. 11. C.-W. Leung, C.-K. Ng and N.-C. Wong, Linear orthogonality preservers of Hilbert C ∗ modules over C ∗ -algebras with real rank zero, Proc. Amer. Math. Soc. 140 (2012), no. 9, 3151–3160. 12. C.-W. Leung, C.-K. Ng and N.-C. Wong, Linear orthogonality preservers of Hilbert C ∗ modules, J. Operator Theory 71 (2014), no. 2, 571–584. 13. V. M. Manuilov and E. V. Troitsky, Hilbert C ∗ -modules, In: Translations of Mathematical Monographs. 226, American Mathematical Society, Providence, RI, 2005. 14. M. S. Moslehian and A. Zamani, Mappings preserving approximate orthogonality in Hilbert C ∗ -modules, Math Scand. (to appear). 15. W. L. Paschke, Inner product modules over B ∗ -algebras, Trans. Amer. Math. Soc. 182 (1973), 443–468. 16. G. K. Pedersen, C ∗ -algebras and their automorphism groups, Academic Press, London–New York–San Francisco, 1979. 17. J. Schweizer, Interplay between noncommutative topology and operators on C ∗ -algebras, Habilitation Thesis, Mathematische Fakult¨ at der Universit¨at T¨ ubingen, T¨ ubingen, Germany, 1996. 18. J. Schweizer, Hilbert C ∗ -modules with a predual, J. Operator Theory 48 (2002), 621–632. 19. A. Turnˇsek, On mappings approximately preserving orthogonality, J. Math. Anal. Appl. 336 (1) (2007), 625–631. 20. P. W´ ojcik, On certain basis connected with operator and its applications, J. Math. Anal. Appl. 423 (2) (2015), 1320–1329. 21. A. Zamani, M. S. Moslehian and M. Frank, Angle preserving mappings, Z. Anal. Anwend. 34 (2015), 485–500. 22. Y. Zhang, Y. Chen, D. Hadwin and L. Kong, AOP mappings and the distance to the scalar multiples of isometries, J. Math. Anal. Appl. 431 (2) (2015), 1275–1284.

18

M. FRANK, M. S. MOSLEHIAN, A. ZAMANI 1

¨r Technik, Wirtschaft und Kultur (HTWK) Leipzig, Fakulta ¨t Hochschule fu IMN, PF 301166, 04251 Leipzig, Germany E-mail address: [email protected] 2

Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O.

Box 1159, Mashhad 91775, Iran E-mail address: [email protected], [email protected] 3

Department of Mathematics, Farhangian University, Iran

E-mail address: [email protected]