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Generalized additive mapping in Banach modules and isomorphisms between C∗ -algebras
J. Math. Anal. Appl. 314 (2006) 150–161 www.elsevier.com/locate/jmaa
Generalized additive mapping in Banach modules and isomorphisms between C ∗ -algebras ✩ Choonkil Baak a , Deok-Hoon Boo a,∗ , Themistocles M. Rassias b a Department of Mathematics, Chungnam National University, Daejeon 305-764, Republic of Korea b Department of Mathematics, National Technical University of Athens, Zografou Campus,
15780 Athens, Greece Received 25 January 2005 Available online 17 May 2005 Submitted by R. Curto
Abstract Let X, Y be vector spaces. It is shown that if an odd mapping f : X → Y satisfies the functional equation d (−1)ι(j ) x d x d j j =1 j j =1 + = (d−1 Cl − d−1 Cl−1 + 1) rf f (xj ) rf r r ι(j )=0,1 d j =1 ι(j )=l
j =1
(1)
then the odd mapping f : X → Y is additive, and we prove the Hyers–Ulam stability of the functional equation (1) in Banach modules over a unital C ∗ -algebra. As an application, we show that every almost linear bijection h : A → B of a unital C ∗ -algebra A onto a unital C ∗ -algebra B is a C ∗ -algebra isomorphism when n n 2 2 h n uy = h n u h(y) r r for all unitaries u ∈ A, all y ∈ A, and n = 0, 1, 2, . . . .
2005 Elsevier Inc. All rights reserved. ✩
Supported by Korea Research Foundation Grant KRF-2004-041-C00023.
* Corresponding author. Fax: (8242) 823 1856.
E-mail addresses: [email protected] (C. Baak), [email protected] (D.-H. Boo), [email protected] (Th.M. Rassias). 0022-247X/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2005.03.099
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Keywords: Banach module over C ∗ -algebra; Functional equation in d variables; Hyers–Ulam stability; C ∗ -algebra isomorphism
1. Introduction Let X and Y be Banach spaces with norms · and · , respectively. Consider f : X → Y to be a mapping such that f (tx) is continuous in t ∈ R for each fixed x ∈ X. Assume that there exist constants θ 0 and p ∈ [0, 1) such that f (x + y) − f (x) − f (y) θ xp + yp for all x, y ∈ X. Th.M. Rassias [15] showed that there exists a unique R-linear mapping T : X → Y such that f (x) − T (x) 2θ xp 2 − 2p for all x ∈ X. P. G˘avruta [1] generalized the Rassias’ result: Let G be an abelian group and Y a Banach space. Denote by ϕ : G × G → [0, ∞) a function such that ϕ(x, ˜ y) =
∞ 1 j ϕ 2 x, 2j y < ∞ j 2 j =0
for all x, y ∈ G. Suppose that f : G → Y is a mapping satisfying f (x + y) − f (x) − f (y) ϕ(x, y) for all x, y ∈ G. Then there exists a unique additive mapping T : G → Y such that f (x) − T (x) 1 ϕ(x, ˜ x) 2 for all x ∈ G. C. Park [4] applied the G˘avruta’s result to linear functional equations in Banach modules over a C ∗ -algebra. Several functional equations have been investigated in [5–22]. Throughout this paper, assume that r is a positive rational number and that d and l are integers with 1 < l < d/2. In this paper, we solve the following functional equation d d ι(j ) x j j =1 xj j =1 (−1) + rf rf r r ι(j )=0,1 d j =1 ι(j )=l
= (d−1 Cl − d−1 Cl−1 + 1)
d
f (xj ).
(1.1)
j =1
We moreover prove the Hyers–Ulam stability of the functional equation (1.1) in Banach modules over a unital C ∗ -algebra. These results are applied to investigate C ∗ -algebra isomorphisms between unital C ∗ -algebras.
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2. An odd functional equation in d variables Throughout this section, assume that X and Y are real linear spaces. Lemma 2.1. An odd mapping f : X → Y satisfies (1.1) for all x1 , x2 , . . . , xd ∈ X if and only if f is Cauchy additive. Proof. Assume that f : X → Y satisfies (1.1) for all x1 , x2 , . . . , xd ∈ X. Note that f (0) = 0 and f (−x) = −f (x) for all x ∈ X since f is an odd mapping. Putting x1 = x, x2 = y and x3 = · · · = xd = 0 in (1.1), we get x+y = (d−1 Cl − d−1 Cl−1 + 1) f (x) + f (y) (2.1) (d−2 Cl − d−2 Cl−2 + 1)rf r for all x, y ∈ X. Since d−2 Cl − d−2 Cl−2 + 1 = d−1 Cl − d−1 Cl−1 + 1, x+y = f (x) + f (y) rf r for all x, y ∈ X. Letting y = 0 in (2.1), we get rf (x/r) = f (x) for all x ∈ X. Hence x +y f (x + y) = rf = f (x) + f (y) r for all x, y ∈ X. Thus f is Cauchy additive. The converse is obviously true. 2 In the proof of Lemma 2.1, we prove the following. Corollary 2.2. An odd mapping f : X → Y satisfies rf ((x + y)/r) = f (x) + f (y) for all x, y ∈ X if and only if f is Cauchy additive. 3. Stability of an odd functional equation in Banach modules over a C ∗ -algebra Throughout this section, assume that A is a unital C ∗ -algebra with norm | · | and unitary group U (A), and that X and Y are left Banach modules over a unital C ∗ -algebra A with norms · and · , respectively. Given a mapping f : X → Y , we set d d ι(j ) ux j j =1 uxj j =1 (−1) + rf Du f (x1 , . . . , xd ) := rf r r ι(j )=0,1 d j =1 ι(j )=l
− (d−1 Cl − d−1 Cl−1 + 1)
d j =1
for all u ∈ U (A) and all x1 , . . . , xd ∈ X.
uf (xj )
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Theorem 3.1. Let r = 2. Let f : X → Y be an odd mapping for which there is a function ϕ : X d → [0, ∞) such that ∞ j j r 2 2j (3.1) ϕ(x ˜ 1 , . . . , xd ) := ϕ j x1 , . . . , j xd < ∞, 2j r r j =0 Du f (x1 , . . . , xd ) ϕ(x1 , . . . , xd ) (3.2) for all u ∈ U (A) and all x1 , . . . , xd ∈ X. Then there exists a unique A-linear generalized additive mapping L : X → Y such that f (x) − L(x)
1 2(d−2 Cl − d−2 Cl−2 + 1)
ϕ(x, ˜ x, 0, . . . , 0 )
(3.3)
d−2 times
for all x ∈ X. Proof. Note that f (0) = 0 and f (−x) = −f (x) for all x ∈ X since f is an odd mapping. Let u = 1 ∈ U (A). Putting x1 = x2 = x and x3 = · · · = xd = 0 in (3.2), we have 1 rf 2 x − 2f (x) ϕ(x, x, 0, . . . , 0 ) r d−2 Cl − d−2 Cl−2 + 1 d−2 times
for all x ∈ X. Letting t := d−2 Cl − d−2 Cl−2 + 1, we get f (x) − r f 2 x 1 ϕ(x, x, 0, . . . , 0 ) 2 r 2t d−2 times
for all x ∈ X. Hence n n n n+1 2n+1 n r rn f 2 x −r = f 2 x − r f 2 · 2 x f x 2n n n n n n+1 n+1 r 2 r 2 r r 2 r rn 2n 2n n+1 ϕ n x, n x, 0, . . . , 0 r r 2 t
(3.4)
d−2 times
for all x ∈ X and all positive integers n. By (3.4), we have m m k n−1 n n r 2 rk 2k f 2 x − r f 2 x ϕ x, x, 0, . . . , 0 2m rm 2n rn r k r k 2k+1 t k=m
(3.5)
d−2 times
for all x ∈ X and all positive integers m and n with m < n. This shows that the sen n quence { 2r n f ( 2r n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence n n { 2r n f ( 2r n x)} converges for all x ∈ X. So we can define a mapping L : X → Y by n 2 rn L(x) := lim n f n x n→∞ 2 r for all x ∈ X. Since f (−x) = −f (x) for all x ∈ X, we have L(−x) = −L(x) for all x ∈ X. Also, we get
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n n n D1 L(x1 , . . . , xd ) = lim r D1 f 2 x1 , . . . , 2 xd n→∞ 2n rn rn n n n r 2 2 lim n ϕ n x1 , . . . , n xd = 0 n→∞ 2 r r for all x1 , . . . , xd ∈ X. So L is a generalized additive mapping. Putting m = 0 and letting n → ∞ in (3.5), we get (3.3). Now, let L : X → Y be another additive mapping satisfying (3.3). By Lemma 2.1, L and L are additive. So we have n n n L(x) − L (x) = r L 2 x − L 2 x n n n 2 r r n n n n n 2 2 r 2 2 + L L n x − f x x − f x 2 rn rn rn rn n 2r n 2 2n n+1 ϕ˜ n x, n x, 0, . . . , 0 , r r 2 t d−2 times
which tends to zero as n → ∞ for all x ∈ X. So we can conclude that L(x) = L (x) for all x ∈ X. This proves the uniqueness of L. By the assumption, for each u ∈ U (A), we get n n Du L(x, 0, . . . , 0 ) = lim r Du f 2 x, 0, . . . , 0 n n n→∞ 2 r d−1 times d−1 times n rn 2 lim n ϕ n x, 0, . . . , 0 = 0 n→∞ 2 r d−1 times
for all x ∈ X. So
ux (d−1 Cl − d−1 Cl−1 + 1)rL = (d−1 Cl − d−1 Cl−1 + 1)uL(x) r for all u ∈ U (A) and all x ∈ X. Since L is additive, ux = uL(x) (3.6) L(ux) = rL r for all u ∈ U (A) and all x ∈ X. Now let a ∈ A (a = 0) and M an integer greater than 4|a|. Then a 1 < < 1 − 2 = 1. M 4 3 3 By [2], there exist three elements u1 , u2 , u3 ∈ U (A) such that 3a/M = u1 + u2 + u3 . So by (3.6) M a 1 a M a L(ax) = L ·3 x =M ·L ·3 x = L 3 x 3 M 3 M 3 M M M L(u1 x) + L(u2 x) + L(u3 x) = L(u1 x + u2 x + u3 x) = 3 3 a M M · 3 L(x) = aL(x) = (u1 + u2 + u3 )L(x) = 3 3 M
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for all a ∈ A and all x ∈ X. Hence L(ax + by) = L(ax) + L(by) = aL(x) + bL(y) for all a, b ∈ A (a, b = 0) and all x, y ∈ X. And L(0x) = 0 = 0L(x) for all x ∈ X. So the unique generalized additive mapping L : X → Y is an A-linear mapping. 2 Corollary 3.2. Let r > 2, and let θ and p > 1 be positive real numbers. Or let r < 2, and let θ and p < 1 be positive real numbers. Let f : X → Y be an odd mapping such that d Du f (x1 , . . . , xd ) θ xj p j =1
for all u ∈ U (A) and all x1 , . . . , xd ∈ X. Then there exists a unique A-linear generalized additive mapping L : X → Y such that f (x) − L(x)
r p−1 θ (r p−1
− 2p−1 )(d−2 Cl
− d−2 Cl−2 + 1)
xp
for all x ∈ X. Proof. Define ϕ(x1 , . . . , xd ) = θ
d
j =1 xj
p,
and apply Theorem 3.1.
2
Theorem 3.3. Let r = 2. Let f : X → Y be an odd mapping for which there is a function ϕ : X d → [0, ∞) satisfying (3.2) such that ∞ j j 2 r rj ϕ(x ˜ 1 , . . . , xd ) := ϕ j x1 , . . . , j xd < ∞ rj 2 2 j =1
for all u ∈ U (A) and all x1 , . . . , xd ∈ X. Then there exists a unique A-linear generalized additive mapping L : X → Y such that 1 f (x) − L(x) ϕ(x, ˜ x, 0, . . . , 0 ) 2(d−2 Cl − d−2 Cl−2 + 1) d−2 times
for all x ∈ X. Proof. Note that f (0) = 0 and f (−x) = −f (x) for all x ∈ X since f is an odd mapping. Let u = 1 ∈ U (A). Putting x1 = x2 = x and x3 = · · · = xd = 0 in (3.2), we have 1 rf 2 x − 2f (x) ϕ(x, x, 0, . . . , 0 ) r d−2 Cl − d−2 Cl−2 + 1 d−2 times
for all x ∈ X. Letting t := d−2 Cl − d−2 Cl−2 + 1, we get f (x) − 2 f r x 1 ϕ r x, r x, 0, . . . , 0 r 2 rt 2 2 d−2 times
for all x ∈ X. The rest of the proof is similar to the proof of Theorem 3.1.
2
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Corollary 3.4. Let r < 2, and let θ and p > 1 be positive real numbers. Or let r > 2, and let θ and p < 1 be positive real numbers. Let f : X → Y be an odd mapping such that d Du f (x1 , . . . , xd ) θ xj p j =1
for all u ∈ U (A) and all x1 , . . . , xd ∈ X. Then there exists a unique A-linear generalized additive mapping L : X → Y such that f (x) − L(x)
r p−1 θ xp (2p−1 − r p−1 )(d−2 Cl − d−2 Cl−2 + 1)
for all x ∈ X. Proof. Define ϕ(x1 , . . . , xd ) = θ
d
j =1 xj
p,
and apply Theorem 3.3.
2
Now we investigate the Hyers–Ulam stability of linear mappings for the case d = 2. Theorem 3.5. Let r = 2. Let f : X → Y be an odd mapping for which there is a function ϕ : X 2 → [0, ∞) such that ∞ j j r 2 2j ϕ(x, ˜ y) := ϕ x, y < ∞, 2j rj rj j =0 rf ux + uy − uf (x) − uf (y) ϕ(x, y) (3.7) r for all u ∈ U (A) and all x, y ∈ X. Then there exists a unique A-linear mapping L : X → Y such that f (x) − L(x) 1 ϕ(x, ˜ x) 2 for all x ∈ X. Proof. Let u = 1 ∈ U (A). Putting x = y in (3.7), we have rf 2 x − 2f (x) ϕ(x, x) r for all x ∈ X. So f (x) − r f 2 x 1 ϕ(x, x) 2 r 2 for all x ∈ X. The rest of the proof is the same as in the proof of Theorem 3.1.
2
Corollary 3.6. Let r > 2, and let θ and p > 1 be positive real numbers. Or let r < 2, and let θ and p < 1 be positive real numbers. Let f : X → Y be an odd mapping such that rf ux + uy − uf (x) − uf (y) θ xp + yp r
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for all u ∈ U (A) and all x, y ∈ X. Then there exists a unique A-linear mapping L : X → Y such that r p−1 θ f (x) − L(x) xp r p−1 − 2p−1 for all x ∈ X. Proof. Define ϕ(x, y) = θ (xp + yp ), and apply Theorem 3.5.
2
Theorem 3.7. Let r = 2. Let f : X → Y be an odd mapping for which there is a function ϕ : X 2 → [0, ∞) satisfying (3.7) such that ∞ j j 2 r rj ϕ(x, ˜ y) := ϕ x, y <∞ rj 2j 2j j =1
for all u ∈ U (A) and all x, y ∈ X. Then there exists a unique A-linear mapping L : X → Y such that f (x) − L(x) 1 ϕ(x, ˜ x) 2 for all x ∈ X. Proof. Let u = 1 ∈ U (A). Putting x = y in (3.7), we have rf 2 x − 2f (x) ϕ(x, x) r for all x ∈ X. So f (x) − 2 f r x 1 ϕ r x, r x r 2 r 2 2 for all x ∈ X. The rest of the proof is similar to the proof of Theorem 3.1.
2
Corollary 3.8. Let r < 2, and let θ and p > 1 be positive real numbers. Or let r > 2, and let θ and p < 1 be positive real numbers. Let f : X → Y be an odd mapping such that rf ux + uy − uf (x) − uf (y) θ xp + yp r for all u ∈ U (A) and all x, y ∈ X. Then there exists a unique A-linear mapping L : X → Y such that r p−1 θ f (x) − L(x) xp p−1 2 − r p−1 for all x ∈ X. Proof. Define ϕ(x, y) = θ (xp + yp ), and apply Theorem 3.7.
2
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4. Isomorphisms between unital C ∗ -algebras Throughout this section, assume that A is a unital C ∗ -algebra with norm · and unit e, and that B is a unital C ∗ -algebra with norm · . Let U (A) be the set of unitary elements in A. We are going to investigate C ∗ -algebra isomorphisms between unital C ∗ -algebras. n
Theorem 4.1. Let r = 2. Let h : A → B be an odd bijective mapping satisfying h( 2r n uy) = n h( 2r n u)h(y) for all u ∈ U (A), all y ∈ A, and n = 0, 1, 2, . . . , for which there exists a function ϕ : Ad → [0, ∞) such that ∞ j j r 2 2j < ∞, ϕ x , . . . , x 1 d 2j rj rj
(4.1)
j =0
Dµ h(x1 , . . . , xd ) ϕ(x1 , . . . , xd ), n n n ∗ n 2 ∗ 2 h ϕ 2 u, . . . , 2 u − h u u rn n n rn r r
(4.2)
d times
for all µ ∈ S 1 := {λ ∈ C | |λ| = 1}, all u ∈ U (A), n = 0, 1, 2, . . . , and all x1 , . . . , xd ∈ A. n n Assume that limn→∞ 2r n h( 2r n e) is invertible. Then the odd bijective mapping h : A → B is a C ∗ -algebra isomorphism. Proof. Consider the C ∗ -algebras A and B as left Banach modules over the unital C ∗ algebra C. By Theorem 3.1, there exists a unique C-linear generalized additive mapping H : A → B such that h(x) − H (x)
1 2(d−2 Cl − d−2 Cl−2 + 1)
ϕ(x, ˜ x, 0, . . . , 0 )
(4.3)
d−2 times
for all x ∈ A. The generalized additive mapping H : A → B is given by n rn 2 H (x) = lim n h n x n→∞ 2 r for all x ∈ A. By (4.1) and (4.2), we get n n ∗ n ∗ rn rn rn 2 ∗ 2 2 ∗ H (u ) = lim n h n u = lim n h n u = lim n h n u n→∞ 2 n→∞ n→∞ r 2 r 2 r = H (u)∗ for all u ∈ U (A). Since H is C-linear and each x ∈ A is a finite linear combination of unitary elements (see [3]), i.e., x = m j =1 λj uj (λj ∈ C, uj ∈ U (A)),
C. Baak et al. / J. Math. Anal. Appl. 314 (2006) 150–161
H (x ∗ ) = H
=H
m
λj u∗j =
j =1 m
∗
m
m ∗ ∗ λj H uj = λj H uj =
j =1
j =1
159 m
∗ λj H (uj )
j =1
= H (x)∗
λj uj
j =1
for all x ∈ A. n n Since h( 2r n uy) = h( 2r n u)h(y) for all u ∈ U (A), all y ∈ A, and all n = 0, 1, 2, . . . ,
n 2n rn rn 2 (4.4) H (uy) = lim n h n uy = lim n h n u h(y) = H (u)h(y) n→∞ 2 n→∞ 2 r r for all u ∈ U (A) and all y ∈ A. By the additivity of H and (4.4), n n n 2 2 2 2n H (uy) = H uy = H u y = H (u)h y n n n r r r rn for all u ∈ U (A) and all y ∈ A. Hence n n rn 2 rn 2 H (uy) = n H (u)h n y = H (u) n h n y 2 r 2 r
(4.5)
for all u ∈ U (A) and all y ∈ A. Taking the limit in (4.5) as n → ∞, we obtain H (uy) = H (u)H (y)
(4.6)
for all u ∈ U (A) and all y ∈ A. Since H is C-linear and each x ∈ A is a finite linear combination of unitary elements, i.e., x = m j =1 λj uj (λj ∈ C, uj ∈ U (A)), it follows from (4.6) that
m m m λj uj y = λj H (uj y) = λj H (uj )H (y) H (xy) = H
=H
j =1 m
j =1
j =1
λj uj H (y) = H (x)H (y)
j =1
for all x, y ∈ A. By (4.4) and (4.6), H (e)H (y) = H (ey) = H (e)h(y) for all y ∈ A. Since limn→∞
2n rn 2n h( r n e) = H (e)
is invertible,
H (y) = h(y) for all y ∈ A. Therefore, the odd bijective mapping h : A → B is a C ∗ -algebra isomorphism.
2
Corollary 4.2. Let r > 2, and let θ and p > 1 be positive real numbers. Or let r < 2, and let θ and p < 1 be positive real numbers. Let h : A → B be an odd bijective mapping
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n
satisfying h( 2r n uy) = h( 2r n u)h(y) for all u ∈ U (A), all y ∈ A, and all n = 0, 1, 2, . . . , such that d Dµ h(x1 , . . . , xd ) θ xj p , j =1
n n ∗ pn 2 ∗ 2 h d2 θ − h u u rn rn r pn for all µ ∈ S 1 , all u ∈ U (A), n = 0, 1, 2, . . . , and all x1 , . . . , xd ∈ A. Assume that n n limn→∞ 2r n h( 2r n e) is invertible. Then the odd bijective mapping h : A → B is a C ∗ -algebra isomorphism. Proof. Define ϕ(x1 , . . . , xd ) = θ
d
j =1 xj
p,
and apply Theorem 4.1.
2 n
Theorem 4.3. Let r = 2. Let h : A → B be an odd bijective mapping satisfying h( 2r n uy) = n h( 2r n u)h(y) for all u ∈ U (A), all y ∈ A, and n = 0, 1, 2, . . . , for which there exists a function ϕ : Ad → [0, ∞) satisfying (4.1) and (4.2) such that Dµ h(x1 , . . . , xd ) ϕ(x1 , . . . , xd ) (4.7) n
n
for µ = 1, i, and all x1 , . . . , xd ∈ A. Assume that limn→∞ 2r n h( 2r n e) is invertible. If h(tx) is continuous in t ∈ R for each fixed x ∈ A, then the odd bijective mapping h : A → B is a C ∗ -algebra isomorphism. Proof. Put µ = 1 in (4.7). By the same reasoning as in the proof of Theorem 3.1, there exists a unique generalized additive mapping H : A → B satisfying (4.3). By the same reasoning as in the proof of [15], the generalized additive mapping H : A → B is R-linear. Put µ = i in (4.7). By the same method as in the proof of Theorem 3.1, one can obtain that n n rn ir n 2 2 H (ix) = lim n h n ix = lim n h n x = iH (x) n→∞ 2 n→∞ 2 r r for all x ∈ A. For each element λ ∈ C, λ = s + it, where s, t ∈ R. So H (λx) = H (sx + itx) = sH (x) + tH (ix) = sH (x) + itH (x) = (s + it)H (x) = λH (x) for all λ ∈ C and all x ∈ A. So H (ζ x + ηy) = H (ζ x) + H (ηy) = ζ H (x) + ηH (y) for all ζ, η ∈ C, and all x, y ∈ A. Hence the generalized additive mapping H : A → B is C-linear. The rest of the proof is the same as in the proof of Theorem 3.1. 2
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Acknowledgment The authors thank the referee for a number of valuable suggestions regarding a previous version of this paper.
References [1] P. G˘avruta, A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994) 431–436. [2] R.V. Kadison, G. Pedersen, Means and convex combinations of unitary operators, Math. Scand. 57 (1985) 249–266. [3] R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, Academic Press, New York, 1983. [4] C. Park, On the stability of the linear mapping in Banach modules, J. Math. Anal. Appl. 275 (2002) 711–720. [5] C. Park, Linear functional equations in Banach modules over a C ∗ -algebra, Acta Appl. Math. 77 (2003) 125–161. [6] C. Park, Modified Trif’s functional equations in Banach modules over a C ∗ -algebra and approximate algebra homomorphisms, J. Math. Anal. Appl. 278 (2003) 93–108. [7] C. Park, Universal Jensen’s equations in Banach modules over a C ∗ -algebra and its unitary group, Acta Math. Sinica 20 (2004) 1047–1056. [8] C. Park, Lie ∗-homomorphisms between Lie C ∗ -algebras and Lie ∗-derivations on Lie C ∗ -algebras, J. Math. Anal. Appl. 293 (2004) 419–434. [9] C. Park, Homomorphisms between Lie J C ∗ -algebras and Cauchy–Rassias stability of Lie J C ∗ -algebra derivations, J. Lie Theory 15 (2005) 393–414. [10] C. Park, Generalized Hyers–Ulam–Rassias stability of n-sesquilinear-quadratic mappings on Banach modules over C ∗ -algebras, J. Comput. Appl. Math., in press. [11] C. Park, Approximate homomorphisms on J B ∗ -triples, J. Math. Anal. Appl. 306 (2005) 375–381. [12] C. Park, Homomorphisms between Poisson J C ∗ -algebras, Bull. Braz. Math. Soc., in press. [13] C. Park, J. Hou, Homomorphisms between C ∗ -algebras associated with the Trif functional equation and linear derivations on C ∗ -algebras, J. Korean Math. Soc. 41 (2004) 461–477. [14] C. Park, J. Hou, S. Oh, Homomorphisms between J C ∗ -algebras and between Lie C ∗ -algebras, Acta Math. Sinica, in press. [15] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300. [16] Th.M. Rassias, On the stability of the quadratic functional equation and its applications, Studia Univ. Babes– Bolyai XLIII (1998) 89–124. [17] Th.M. Rassias, The problem of S.M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246 (2000) 352–378. [18] Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000) 264–284. [19] Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000) 23–130. [20] Th.M. Rassias, P. Šemrl, On the Hyers–Ulam stability of linear mappings, J. Math. Anal. Appl. 173 (1993) 325–338. [21] Th.M. Rassias, K. Shibata, Variational problem of some quadratic functionals in complex analysis, J. Math. Anal. Appl. 228 (1998) 234–253. [22] S.M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1960.
Generalized additive mapping in Banach modules and isomorphisms between C ∗ -algebras ✩ Choonkil Baak a , Deok-Hoon Boo a,∗ , Themistocles M. Rassias b a Department of Mathematics, Chungnam National University, Daejeon 305-764, Republic of Korea b Department of Mathematics, National Technical University of Athens, Zografou Campus,
15780 Athens, Greece Received 25 January 2005 Available online 17 May 2005 Submitted by R. Curto
Abstract Let X, Y be vector spaces. It is shown that if an odd mapping f : X → Y satisfies the functional equation d (−1)ι(j ) x d x d j j =1 j j =1 + = (d−1 Cl − d−1 Cl−1 + 1) rf f (xj ) rf r r ι(j )=0,1 d j =1 ι(j )=l
j =1
(1)
then the odd mapping f : X → Y is additive, and we prove the Hyers–Ulam stability of the functional equation (1) in Banach modules over a unital C ∗ -algebra. As an application, we show that every almost linear bijection h : A → B of a unital C ∗ -algebra A onto a unital C ∗ -algebra B is a C ∗ -algebra isomorphism when n n 2 2 h n uy = h n u h(y) r r for all unitaries u ∈ A, all y ∈ A, and n = 0, 1, 2, . . . .
2005 Elsevier Inc. All rights reserved. ✩
Supported by Korea Research Foundation Grant KRF-2004-041-C00023.
* Corresponding author. Fax: (8242) 823 1856.
E-mail addresses: [email protected] (C. Baak), [email protected] (D.-H. Boo), [email protected] (Th.M. Rassias). 0022-247X/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2005.03.099
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Keywords: Banach module over C ∗ -algebra; Functional equation in d variables; Hyers–Ulam stability; C ∗ -algebra isomorphism
1. Introduction Let X and Y be Banach spaces with norms · and · , respectively. Consider f : X → Y to be a mapping such that f (tx) is continuous in t ∈ R for each fixed x ∈ X. Assume that there exist constants θ 0 and p ∈ [0, 1) such that f (x + y) − f (x) − f (y) θ xp + yp for all x, y ∈ X. Th.M. Rassias [15] showed that there exists a unique R-linear mapping T : X → Y such that f (x) − T (x) 2θ xp 2 − 2p for all x ∈ X. P. G˘avruta [1] generalized the Rassias’ result: Let G be an abelian group and Y a Banach space. Denote by ϕ : G × G → [0, ∞) a function such that ϕ(x, ˜ y) =
∞ 1 j ϕ 2 x, 2j y < ∞ j 2 j =0
for all x, y ∈ G. Suppose that f : G → Y is a mapping satisfying f (x + y) − f (x) − f (y) ϕ(x, y) for all x, y ∈ G. Then there exists a unique additive mapping T : G → Y such that f (x) − T (x) 1 ϕ(x, ˜ x) 2 for all x ∈ G. C. Park [4] applied the G˘avruta’s result to linear functional equations in Banach modules over a C ∗ -algebra. Several functional equations have been investigated in [5–22]. Throughout this paper, assume that r is a positive rational number and that d and l are integers with 1 < l < d/2. In this paper, we solve the following functional equation d d ι(j ) x j j =1 xj j =1 (−1) + rf rf r r ι(j )=0,1 d j =1 ι(j )=l
= (d−1 Cl − d−1 Cl−1 + 1)
d
f (xj ).
(1.1)
j =1
We moreover prove the Hyers–Ulam stability of the functional equation (1.1) in Banach modules over a unital C ∗ -algebra. These results are applied to investigate C ∗ -algebra isomorphisms between unital C ∗ -algebras.
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2. An odd functional equation in d variables Throughout this section, assume that X and Y are real linear spaces. Lemma 2.1. An odd mapping f : X → Y satisfies (1.1) for all x1 , x2 , . . . , xd ∈ X if and only if f is Cauchy additive. Proof. Assume that f : X → Y satisfies (1.1) for all x1 , x2 , . . . , xd ∈ X. Note that f (0) = 0 and f (−x) = −f (x) for all x ∈ X since f is an odd mapping. Putting x1 = x, x2 = y and x3 = · · · = xd = 0 in (1.1), we get x+y = (d−1 Cl − d−1 Cl−1 + 1) f (x) + f (y) (2.1) (d−2 Cl − d−2 Cl−2 + 1)rf r for all x, y ∈ X. Since d−2 Cl − d−2 Cl−2 + 1 = d−1 Cl − d−1 Cl−1 + 1, x+y = f (x) + f (y) rf r for all x, y ∈ X. Letting y = 0 in (2.1), we get rf (x/r) = f (x) for all x ∈ X. Hence x +y f (x + y) = rf = f (x) + f (y) r for all x, y ∈ X. Thus f is Cauchy additive. The converse is obviously true. 2 In the proof of Lemma 2.1, we prove the following. Corollary 2.2. An odd mapping f : X → Y satisfies rf ((x + y)/r) = f (x) + f (y) for all x, y ∈ X if and only if f is Cauchy additive. 3. Stability of an odd functional equation in Banach modules over a C ∗ -algebra Throughout this section, assume that A is a unital C ∗ -algebra with norm | · | and unitary group U (A), and that X and Y are left Banach modules over a unital C ∗ -algebra A with norms · and · , respectively. Given a mapping f : X → Y , we set d d ι(j ) ux j j =1 uxj j =1 (−1) + rf Du f (x1 , . . . , xd ) := rf r r ι(j )=0,1 d j =1 ι(j )=l
− (d−1 Cl − d−1 Cl−1 + 1)
d j =1
for all u ∈ U (A) and all x1 , . . . , xd ∈ X.
uf (xj )
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Theorem 3.1. Let r = 2. Let f : X → Y be an odd mapping for which there is a function ϕ : X d → [0, ∞) such that ∞ j j r 2 2j (3.1) ϕ(x ˜ 1 , . . . , xd ) := ϕ j x1 , . . . , j xd < ∞, 2j r r j =0 Du f (x1 , . . . , xd ) ϕ(x1 , . . . , xd ) (3.2) for all u ∈ U (A) and all x1 , . . . , xd ∈ X. Then there exists a unique A-linear generalized additive mapping L : X → Y such that f (x) − L(x)
1 2(d−2 Cl − d−2 Cl−2 + 1)
ϕ(x, ˜ x, 0, . . . , 0 )
(3.3)
d−2 times
for all x ∈ X. Proof. Note that f (0) = 0 and f (−x) = −f (x) for all x ∈ X since f is an odd mapping. Let u = 1 ∈ U (A). Putting x1 = x2 = x and x3 = · · · = xd = 0 in (3.2), we have 1 rf 2 x − 2f (x) ϕ(x, x, 0, . . . , 0 ) r d−2 Cl − d−2 Cl−2 + 1 d−2 times
for all x ∈ X. Letting t := d−2 Cl − d−2 Cl−2 + 1, we get f (x) − r f 2 x 1 ϕ(x, x, 0, . . . , 0 ) 2 r 2t d−2 times
for all x ∈ X. Hence n n n n+1 2n+1 n r rn f 2 x −r = f 2 x − r f 2 · 2 x f x 2n n n n n n+1 n+1 r 2 r 2 r r 2 r rn 2n 2n n+1 ϕ n x, n x, 0, . . . , 0 r r 2 t
(3.4)
d−2 times
for all x ∈ X and all positive integers n. By (3.4), we have m m k n−1 n n r 2 rk 2k f 2 x − r f 2 x ϕ x, x, 0, . . . , 0 2m rm 2n rn r k r k 2k+1 t k=m
(3.5)
d−2 times
for all x ∈ X and all positive integers m and n with m < n. This shows that the sen n quence { 2r n f ( 2r n x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence n n { 2r n f ( 2r n x)} converges for all x ∈ X. So we can define a mapping L : X → Y by n 2 rn L(x) := lim n f n x n→∞ 2 r for all x ∈ X. Since f (−x) = −f (x) for all x ∈ X, we have L(−x) = −L(x) for all x ∈ X. Also, we get
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n n n D1 L(x1 , . . . , xd ) = lim r D1 f 2 x1 , . . . , 2 xd n→∞ 2n rn rn n n n r 2 2 lim n ϕ n x1 , . . . , n xd = 0 n→∞ 2 r r for all x1 , . . . , xd ∈ X. So L is a generalized additive mapping. Putting m = 0 and letting n → ∞ in (3.5), we get (3.3). Now, let L : X → Y be another additive mapping satisfying (3.3). By Lemma 2.1, L and L are additive. So we have n n n L(x) − L (x) = r L 2 x − L 2 x n n n 2 r r n n n n n 2 2 r 2 2 + L L n x − f x x − f x 2 rn rn rn rn n 2r n 2 2n n+1 ϕ˜ n x, n x, 0, . . . , 0 , r r 2 t d−2 times
which tends to zero as n → ∞ for all x ∈ X. So we can conclude that L(x) = L (x) for all x ∈ X. This proves the uniqueness of L. By the assumption, for each u ∈ U (A), we get n n Du L(x, 0, . . . , 0 ) = lim r Du f 2 x, 0, . . . , 0 n n n→∞ 2 r d−1 times d−1 times n rn 2 lim n ϕ n x, 0, . . . , 0 = 0 n→∞ 2 r d−1 times
for all x ∈ X. So
ux (d−1 Cl − d−1 Cl−1 + 1)rL = (d−1 Cl − d−1 Cl−1 + 1)uL(x) r for all u ∈ U (A) and all x ∈ X. Since L is additive, ux = uL(x) (3.6) L(ux) = rL r for all u ∈ U (A) and all x ∈ X. Now let a ∈ A (a = 0) and M an integer greater than 4|a|. Then a 1 < < 1 − 2 = 1. M 4 3 3 By [2], there exist three elements u1 , u2 , u3 ∈ U (A) such that 3a/M = u1 + u2 + u3 . So by (3.6) M a 1 a M a L(ax) = L ·3 x =M ·L ·3 x = L 3 x 3 M 3 M 3 M M M L(u1 x) + L(u2 x) + L(u3 x) = L(u1 x + u2 x + u3 x) = 3 3 a M M · 3 L(x) = aL(x) = (u1 + u2 + u3 )L(x) = 3 3 M
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for all a ∈ A and all x ∈ X. Hence L(ax + by) = L(ax) + L(by) = aL(x) + bL(y) for all a, b ∈ A (a, b = 0) and all x, y ∈ X. And L(0x) = 0 = 0L(x) for all x ∈ X. So the unique generalized additive mapping L : X → Y is an A-linear mapping. 2 Corollary 3.2. Let r > 2, and let θ and p > 1 be positive real numbers. Or let r < 2, and let θ and p < 1 be positive real numbers. Let f : X → Y be an odd mapping such that d Du f (x1 , . . . , xd ) θ xj p j =1
for all u ∈ U (A) and all x1 , . . . , xd ∈ X. Then there exists a unique A-linear generalized additive mapping L : X → Y such that f (x) − L(x)
r p−1 θ (r p−1
− 2p−1 )(d−2 Cl
− d−2 Cl−2 + 1)
xp
for all x ∈ X. Proof. Define ϕ(x1 , . . . , xd ) = θ
d
j =1 xj
p,
and apply Theorem 3.1.
2
Theorem 3.3. Let r = 2. Let f : X → Y be an odd mapping for which there is a function ϕ : X d → [0, ∞) satisfying (3.2) such that ∞ j j 2 r rj ϕ(x ˜ 1 , . . . , xd ) := ϕ j x1 , . . . , j xd < ∞ rj 2 2 j =1
for all u ∈ U (A) and all x1 , . . . , xd ∈ X. Then there exists a unique A-linear generalized additive mapping L : X → Y such that 1 f (x) − L(x) ϕ(x, ˜ x, 0, . . . , 0 ) 2(d−2 Cl − d−2 Cl−2 + 1) d−2 times
for all x ∈ X. Proof. Note that f (0) = 0 and f (−x) = −f (x) for all x ∈ X since f is an odd mapping. Let u = 1 ∈ U (A). Putting x1 = x2 = x and x3 = · · · = xd = 0 in (3.2), we have 1 rf 2 x − 2f (x) ϕ(x, x, 0, . . . , 0 ) r d−2 Cl − d−2 Cl−2 + 1 d−2 times
for all x ∈ X. Letting t := d−2 Cl − d−2 Cl−2 + 1, we get f (x) − 2 f r x 1 ϕ r x, r x, 0, . . . , 0 r 2 rt 2 2 d−2 times
for all x ∈ X. The rest of the proof is similar to the proof of Theorem 3.1.
2
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Corollary 3.4. Let r < 2, and let θ and p > 1 be positive real numbers. Or let r > 2, and let θ and p < 1 be positive real numbers. Let f : X → Y be an odd mapping such that d Du f (x1 , . . . , xd ) θ xj p j =1
for all u ∈ U (A) and all x1 , . . . , xd ∈ X. Then there exists a unique A-linear generalized additive mapping L : X → Y such that f (x) − L(x)
r p−1 θ xp (2p−1 − r p−1 )(d−2 Cl − d−2 Cl−2 + 1)
for all x ∈ X. Proof. Define ϕ(x1 , . . . , xd ) = θ
d
j =1 xj
p,
and apply Theorem 3.3.
2
Now we investigate the Hyers–Ulam stability of linear mappings for the case d = 2. Theorem 3.5. Let r = 2. Let f : X → Y be an odd mapping for which there is a function ϕ : X 2 → [0, ∞) such that ∞ j j r 2 2j ϕ(x, ˜ y) := ϕ x, y < ∞, 2j rj rj j =0 rf ux + uy − uf (x) − uf (y) ϕ(x, y) (3.7) r for all u ∈ U (A) and all x, y ∈ X. Then there exists a unique A-linear mapping L : X → Y such that f (x) − L(x) 1 ϕ(x, ˜ x) 2 for all x ∈ X. Proof. Let u = 1 ∈ U (A). Putting x = y in (3.7), we have rf 2 x − 2f (x) ϕ(x, x) r for all x ∈ X. So f (x) − r f 2 x 1 ϕ(x, x) 2 r 2 for all x ∈ X. The rest of the proof is the same as in the proof of Theorem 3.1.
2
Corollary 3.6. Let r > 2, and let θ and p > 1 be positive real numbers. Or let r < 2, and let θ and p < 1 be positive real numbers. Let f : X → Y be an odd mapping such that rf ux + uy − uf (x) − uf (y) θ xp + yp r
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157
for all u ∈ U (A) and all x, y ∈ X. Then there exists a unique A-linear mapping L : X → Y such that r p−1 θ f (x) − L(x) xp r p−1 − 2p−1 for all x ∈ X. Proof. Define ϕ(x, y) = θ (xp + yp ), and apply Theorem 3.5.
2
Theorem 3.7. Let r = 2. Let f : X → Y be an odd mapping for which there is a function ϕ : X 2 → [0, ∞) satisfying (3.7) such that ∞ j j 2 r rj ϕ(x, ˜ y) := ϕ x, y <∞ rj 2j 2j j =1
for all u ∈ U (A) and all x, y ∈ X. Then there exists a unique A-linear mapping L : X → Y such that f (x) − L(x) 1 ϕ(x, ˜ x) 2 for all x ∈ X. Proof. Let u = 1 ∈ U (A). Putting x = y in (3.7), we have rf 2 x − 2f (x) ϕ(x, x) r for all x ∈ X. So f (x) − 2 f r x 1 ϕ r x, r x r 2 r 2 2 for all x ∈ X. The rest of the proof is similar to the proof of Theorem 3.1.
2
Corollary 3.8. Let r < 2, and let θ and p > 1 be positive real numbers. Or let r > 2, and let θ and p < 1 be positive real numbers. Let f : X → Y be an odd mapping such that rf ux + uy − uf (x) − uf (y) θ xp + yp r for all u ∈ U (A) and all x, y ∈ X. Then there exists a unique A-linear mapping L : X → Y such that r p−1 θ f (x) − L(x) xp p−1 2 − r p−1 for all x ∈ X. Proof. Define ϕ(x, y) = θ (xp + yp ), and apply Theorem 3.7.
2
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4. Isomorphisms between unital C ∗ -algebras Throughout this section, assume that A is a unital C ∗ -algebra with norm · and unit e, and that B is a unital C ∗ -algebra with norm · . Let U (A) be the set of unitary elements in A. We are going to investigate C ∗ -algebra isomorphisms between unital C ∗ -algebras. n
Theorem 4.1. Let r = 2. Let h : A → B be an odd bijective mapping satisfying h( 2r n uy) = n h( 2r n u)h(y) for all u ∈ U (A), all y ∈ A, and n = 0, 1, 2, . . . , for which there exists a function ϕ : Ad → [0, ∞) such that ∞ j j r 2 2j < ∞, ϕ x , . . . , x 1 d 2j rj rj
(4.1)
j =0
Dµ h(x1 , . . . , xd ) ϕ(x1 , . . . , xd ), n n n ∗ n 2 ∗ 2 h ϕ 2 u, . . . , 2 u − h u u rn n n rn r r
(4.2)
d times
for all µ ∈ S 1 := {λ ∈ C | |λ| = 1}, all u ∈ U (A), n = 0, 1, 2, . . . , and all x1 , . . . , xd ∈ A. n n Assume that limn→∞ 2r n h( 2r n e) is invertible. Then the odd bijective mapping h : A → B is a C ∗ -algebra isomorphism. Proof. Consider the C ∗ -algebras A and B as left Banach modules over the unital C ∗ algebra C. By Theorem 3.1, there exists a unique C-linear generalized additive mapping H : A → B such that h(x) − H (x)
1 2(d−2 Cl − d−2 Cl−2 + 1)
ϕ(x, ˜ x, 0, . . . , 0 )
(4.3)
d−2 times
for all x ∈ A. The generalized additive mapping H : A → B is given by n rn 2 H (x) = lim n h n x n→∞ 2 r for all x ∈ A. By (4.1) and (4.2), we get n n ∗ n ∗ rn rn rn 2 ∗ 2 2 ∗ H (u ) = lim n h n u = lim n h n u = lim n h n u n→∞ 2 n→∞ n→∞ r 2 r 2 r = H (u)∗ for all u ∈ U (A). Since H is C-linear and each x ∈ A is a finite linear combination of unitary elements (see [3]), i.e., x = m j =1 λj uj (λj ∈ C, uj ∈ U (A)),
C. Baak et al. / J. Math. Anal. Appl. 314 (2006) 150–161
H (x ∗ ) = H
=H
m
λj u∗j =
j =1 m
∗
m
m ∗ ∗ λj H uj = λj H uj =
j =1
j =1
159 m
∗ λj H (uj )
j =1
= H (x)∗
λj uj
j =1
for all x ∈ A. n n Since h( 2r n uy) = h( 2r n u)h(y) for all u ∈ U (A), all y ∈ A, and all n = 0, 1, 2, . . . ,
n 2n rn rn 2 (4.4) H (uy) = lim n h n uy = lim n h n u h(y) = H (u)h(y) n→∞ 2 n→∞ 2 r r for all u ∈ U (A) and all y ∈ A. By the additivity of H and (4.4), n n n 2 2 2 2n H (uy) = H uy = H u y = H (u)h y n n n r r r rn for all u ∈ U (A) and all y ∈ A. Hence n n rn 2 rn 2 H (uy) = n H (u)h n y = H (u) n h n y 2 r 2 r
(4.5)
for all u ∈ U (A) and all y ∈ A. Taking the limit in (4.5) as n → ∞, we obtain H (uy) = H (u)H (y)
(4.6)
for all u ∈ U (A) and all y ∈ A. Since H is C-linear and each x ∈ A is a finite linear combination of unitary elements, i.e., x = m j =1 λj uj (λj ∈ C, uj ∈ U (A)), it follows from (4.6) that
m m m λj uj y = λj H (uj y) = λj H (uj )H (y) H (xy) = H
=H
j =1 m
j =1
j =1
λj uj H (y) = H (x)H (y)
j =1
for all x, y ∈ A. By (4.4) and (4.6), H (e)H (y) = H (ey) = H (e)h(y) for all y ∈ A. Since limn→∞
2n rn 2n h( r n e) = H (e)
is invertible,
H (y) = h(y) for all y ∈ A. Therefore, the odd bijective mapping h : A → B is a C ∗ -algebra isomorphism.
2
Corollary 4.2. Let r > 2, and let θ and p > 1 be positive real numbers. Or let r < 2, and let θ and p < 1 be positive real numbers. Let h : A → B be an odd bijective mapping
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n
satisfying h( 2r n uy) = h( 2r n u)h(y) for all u ∈ U (A), all y ∈ A, and all n = 0, 1, 2, . . . , such that d Dµ h(x1 , . . . , xd ) θ xj p , j =1
n n ∗ pn 2 ∗ 2 h d2 θ − h u u rn rn r pn for all µ ∈ S 1 , all u ∈ U (A), n = 0, 1, 2, . . . , and all x1 , . . . , xd ∈ A. Assume that n n limn→∞ 2r n h( 2r n e) is invertible. Then the odd bijective mapping h : A → B is a C ∗ -algebra isomorphism. Proof. Define ϕ(x1 , . . . , xd ) = θ
d
j =1 xj
p,
and apply Theorem 4.1.
2 n
Theorem 4.3. Let r = 2. Let h : A → B be an odd bijective mapping satisfying h( 2r n uy) = n h( 2r n u)h(y) for all u ∈ U (A), all y ∈ A, and n = 0, 1, 2, . . . , for which there exists a function ϕ : Ad → [0, ∞) satisfying (4.1) and (4.2) such that Dµ h(x1 , . . . , xd ) ϕ(x1 , . . . , xd ) (4.7) n
n
for µ = 1, i, and all x1 , . . . , xd ∈ A. Assume that limn→∞ 2r n h( 2r n e) is invertible. If h(tx) is continuous in t ∈ R for each fixed x ∈ A, then the odd bijective mapping h : A → B is a C ∗ -algebra isomorphism. Proof. Put µ = 1 in (4.7). By the same reasoning as in the proof of Theorem 3.1, there exists a unique generalized additive mapping H : A → B satisfying (4.3). By the same reasoning as in the proof of [15], the generalized additive mapping H : A → B is R-linear. Put µ = i in (4.7). By the same method as in the proof of Theorem 3.1, one can obtain that n n rn ir n 2 2 H (ix) = lim n h n ix = lim n h n x = iH (x) n→∞ 2 n→∞ 2 r r for all x ∈ A. For each element λ ∈ C, λ = s + it, where s, t ∈ R. So H (λx) = H (sx + itx) = sH (x) + tH (ix) = sH (x) + itH (x) = (s + it)H (x) = λH (x) for all λ ∈ C and all x ∈ A. So H (ζ x + ηy) = H (ζ x) + H (ηy) = ζ H (x) + ηH (y) for all ζ, η ∈ C, and all x, y ∈ A. Hence the generalized additive mapping H : A → B is C-linear. The rest of the proof is the same as in the proof of Theorem 3.1. 2
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Acknowledgment The authors thank the referee for a number of valuable suggestions regarding a previous version of this paper.
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