Homomorphisms between $C^*$-algebras associated with the Trif functional equation and linear derivations on $C^*$-algebras

HOMOMORPHISMS BETWEEN C ^∗ -ALGEBRAS ASSOCIATED WITH THE

TRIF FUNCTIONAL EQUATION AND LINEAR DERIVATIONS ON C ^∗ -ALGEBRAS

Chun-Gil Park ^‡ and Jinchuan Hou ^∗

Abstract. It is shown that every almost linear mapping h : A → B of a unital C

-algebra A to a unital C

-algebra B is a homo- morphism under some condition on multiplication, and that every almost linear continuous mapping h : A → B of a unital C

-algebra A of real rank zero to a unital C

-algebra B is a homomorphism under some condition on multiplication.

Furthermore, we are going to prove the generalized Hyers-Ulam- Rassias stability of ∗-homomorphisms between unital C

-algebras, and of C-linear ∗-derivations on unital C

-algebras.

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(||x|| ^p + ||y|| ^p )

for all x, y ∈ X. Rassias [7] showed that there exists a unique R-linear mapping T : X → Y such that

f(x) − T (x)
≤ 2θ

2 − 2 ^p ||x|| ^p

Received March 10, 2003.

2000 Mathematics Subject Classification: Primary 47B48, 39B52, 46L05.

Key words and phrases: homomorphism, C

-algebra of real rank zero, linear derivation, stability.

‡

Supported by grant No. R05-2003-000-10006-0 from the Basic Research Program of the Korea Science & Engineering Foundation.

∗

Supported by NNSF of China and NSF of Shanxi Province.

for all x ∈ X. G˘avruta [2] generalized the Rassias’ result.

Recently, Trif [8] proved the following: let q := ^l(d−1) _d−l , r := − _d−l ^l . Denote by ϕ : X ^d → [0, ∞) a function such that

ϕ(x 1 , . . . , x d ) =

 ∞ j=0

q ^−j ϕ(q ^j x 1 , . . . , q ^j x d ) < ∞

for all x 1 , . . . , x d ∈ X. Suppose that f : X → Y is a mapping satisfying

d d−2 C l−2 f ( x 1 + · · · + x d

d ) + _d−2 C l−1

 d j=1

f (x j )

− l 

1≤j

<···<j

≤d

f ( x j

+ · · · + x j

l )

≤ ϕ(x 1 , . . . , x d )

for all x 1 , . . . , x d ∈ X. Then there exists a unique additive mapping T : X → Y such that

f(x) − f(0) − T (x)
≤ 1

l · d−1 C l−1 ϕ(qx, rx, . . . , rx
  

d−1 times

)

for all x ∈ X. And Park [6] applied the Trif’s result to the Trif functional equation in Banach modules over a C ^∗ -algebra.

B. E. Johnson [3, Theorem 7.2] also investigated almost algebra ∗- homomorphisms between Banach ∗-algebras : Suppose that U and B are Banach ∗-algebras which satisfy the conditions of [3, Theorem 3.1]. Then for each positive  and K there is a positive δ such that if T ∈ L(U, B) with
T
< K,
T ^∨
< δ and
T (x ^∗ ) ^∗ − T (x)
≤ δ
x
(x ∈ U) then there is a ∗-homomorphism T ^ : U → B with
T −T ^
< . Here L(U, B) is the space of bounded linear maps from U into B, and T ^∨ (x, y) = T (xy) − T (x)T (y) (x, y ∈ U). See [3] for details.

Throughout this paper, let A be a unital C ^∗ -algebra with norm || · ||

and unit e, and B a unital C ^∗ -algebra with norm
·
. Let U(A) be the set of unitary elements in A, A sa = {x ∈ A | x = x ^∗ }, and I 1 ( A sa ) = {v ∈ A sa | ||v|| = 1, v is invertible}. Let q = ^l(d−1) _d−l and r = − _d−l ^l for integers l, d with 2 ≤ l ≤ d − 1.

In this paper, we prove that every almost linear mapping h : A → B is

a homomorphism when h(q ⁿ uy) = h(q ⁿ u)h(y) holds for all u ∈ U(A), all

y ∈ A, and all n = 0, 1, 2, . . . , and that for a unital C ^∗ -algebra A of real rank zero (see [1]), every almost linear continuous mapping h : A → B is a homomorphism when h(q ⁿ uy) = h(q ⁿ u)h(y) holds for all u ∈ I 1 ( A sa ), all y ∈ A, and all n = 0, 1, 2, . . . .

Furthermore, we are going to prove the generalized Hyers-Ulam-Rassi- as stability of ∗-homomorphisms between unital C ^∗ -algebras, and of C- linear ∗-derivations on unital C ^∗ -algebras.

2. ∗-homomorphisms between unital C ^∗ -algebras

We are going to investigate ∗-homomorphisms between unital C ^∗ - algebras.

Theorem 1. Let h : A → B be a mapping satisfying h(0) = 0 and h(q ⁿ uy) = h(q ⁿ u)h(y) for all u ∈ U(A), all y ∈ A, and all n = 0, 1, 2, . . . , for which there exists a function ϕ : A ^d → [0, ∞) such that

ϕ(x 1 , . . . , x d ) :=

 ∞ j=0

q ^−j ϕ(q ^j x 1 , . . . , q ^j x d ) < ∞, (i)

d · d−2 C l−2 h( µx 1 + · · · + µx d

d ) + _d−2 C l−1

 d j=1

µh(x j )

− l 

1≤j

<···<j

≤d

µh( x j

+ · · · + x j

l )
≤ ϕ(x 1 , . . . , x d ), (ii)

h(q ⁿ u ^∗ ) − h(q ⁿ u) ^∗
≤ ϕ(q  ⁿ u, . . . , q  ⁿ u 

d times

) (iii)

for all µ ∈ T ¹ := {λ ∈ C | |λ| = 1}, all u ∈ U(A), all n = 0, 1, 2, . . . , and all x 1 , . . . , x d ∈ A. Assume that (iv) lim n→∞ h(q

e)

q

is invertible.

Then the mapping h : A → B is a ∗-homomorphism.

Proof. Put µ = 1 ∈ T ¹ . It follows from the Trif theorem [8, Theorem 3.1] that there exists a unique additive mapping Θ : A → B such that

( †)
h(x) − Θ(x)
≤ 1

l · d−1 C l−1 ϕ(qx, rx, . . . , rx
  

d−1 times

)

for all x ∈ A. The additive mapping Θ : A → B is given by Θ(x) = lim

n→∞

1

q ⁿ h(q ⁿ x)

for all x ∈ A.

Put x 1 = · · · = x d = x in (ii). For each µ ∈ T ¹ ,

d · d−2 C l−2 (h(µx) − µh(x))
≤ ϕ(x, . . . , x   

d times

)

for all x ∈ A. So

q ⁻ⁿ
d · d−2 C l−2 (h(µq ⁿ x) − µh(q ⁿ x))
≤ q ⁻ⁿ ϕ(q  ⁿ x, . . . , q  ⁿ x 

d times

)

for all x ∈ A. By (i),

q ⁻ⁿ
d · d−2 C l−2 (h(µq ⁿ x) − µh(q ⁿ x))
→ 0 as n → ∞ for all µ ∈ T ¹ and all x ∈ A. Thus

q ⁻ⁿ
h(µq ⁿ x) − µh(q ⁿ x)
→ 0 as n → ∞ for all µ ∈ T ¹ and all x ∈ A. Hence (1) Θ(µx) = lim

n→∞

h(q ⁿ µx)

q ⁿ = lim

n→∞

µh(q ⁿ x)

q ⁿ = µΘ(x) for all µ ∈ T ¹ and all x ∈ A.

Now let λ ∈ C (λ = 0) and M an integer greater than 4|λ|. Then

| _M ^λ | < ¹ ₄ < 1 − ² ₃ = ¹ ₃ . By [4, Theorem 1], there exist three elements µ 1 , µ 2 , µ 3 ∈ T ¹ such that 3 _M ^λ = µ 1 + µ 2 + µ 3 . So by (1)

Θ(λx) = Θ( M 3 · 3 λ

M x) = M · Θ( 1 3 · 3 λ

M x) = M 3 Θ(3 λ

M x)

= M

3 Θ(µ 1 x + µ 2 x + µ 3 x) = M

3 (Θ(µ 1 x) + Θ(µ 2 x) + Θ(µ 3 x))

= M

3 (µ 1 + µ 2 + µ 3 )Θ(x) = M 3 · 3 λ

M Θ(x) = λΘ(x) for all x ∈ A. Hence

Θ(ζx + ηy) = Θ(ζx) + Θ(ηy) = ζΘ(x) + ηΘ(y)

for all ζ, η ∈ C(ζ, η = 0) and all x, y ∈ A. And Θ(0x) = 0 = 0Θ(x)

for all x ∈ A. So the unique additive mapping Θ : A → B is a C-linear

mapping.

By (i) and (iii), we get Θ(u ^∗ ) = lim

n→∞

h(q ⁿ u ^∗ )

q ⁿ = lim

n→∞

h(q ⁿ u) ^∗

q ⁿ = ( lim

n→∞

h(q ⁿ u)

q ⁿ ) ^∗ = Θ(u) ^∗ for all u ∈ U(A). Since Θ is C-linear and each x ∈ A is a finite linear combination of unitary elements (see [5, Theorem 4.1.7]), i.e., x =  _m

j=1 λ j u j (λ j ∈ C, u j ∈ U(A)), Θ(x ^∗ ) = Θ(

 m j=1

λ j u ^∗ _j ) =

 m j=1

λ j Θ(u ^∗ _j ) =

 m j=1

λ j Θ(u j ) ^∗

= (

 m j=1

λ j Θ(u j )) ^∗ = Θ(

 m j=1

λ j u j ) ^∗ = Θ(x) ^∗ for all x ∈ A.

Since h(q ⁿ uy) = h(q ⁿ u)h(y) for all u ∈ U(A), all y ∈ A, and all n = 0, 1, 2, . . . ,

(2) Θ(uy) = lim

n→∞

1

q ⁿ h(q ⁿ uy) = lim

n→∞

1

q ⁿ h(q ⁿ u)h(y) = Θ(u)h(y) for all u ∈ U(A) and all y ∈ A. By the additivity of Θ and (2),

q ⁿ Θ(uy) = Θ(q ⁿ uy) = Θ(u(q ⁿ y)) = Θ(u)h(q ⁿ y) for all u ∈ U(A) and all y ∈ A. Hence

(3) Θ(uy) = 1

q ⁿ Θ(u)h(q ⁿ y) = Θ(u) 1

q ⁿ h(q ⁿ y)

for all u ∈ U(A) and all y ∈ A. Taking the limit in (3) as n → ∞, we obtain

(4) Θ(uy) = Θ(u)Θ(y)

for all u ∈ U(A) and all y ∈ A. Since Θ is C-linear and each x ∈ A is a finite linear combination of unitary elements, i.e., x =  _m

j=1 λ j u j (λ j ∈ C, u j ∈ U(A)), it follows from (4) that

Θ(xy) = Θ(

 m j=1

λ j u j y) =

 m j=1

λ j Θ(u j y)

=

 m j=1

λ j Θ(u j )Θ(y) = Θ(

 m j=1

λ j u j )Θ(y)

= Θ(x)Θ(y)

for all x, y ∈ A.

By (2) and (4),

Θ(e)Θ(y) = Θ(ey) = Θ(e)h(y) for all y ∈ A. Since lim n→∞ h(q

e)

q

= Θ(e) is invertible, Θ(y) = h(y)

for all y ∈ A.

Therefore, the mapping h : A → B is a ∗-homomorphism, as desired.

Corollary 2. Let h : A → B be a mapping satisfying h(0) = 0 and h(q ⁿ uy) = h(q ⁿ u)h(y) for all u ∈ U(A), all y ∈ A, and all n = 0, 1, 2, . . . , for which there exist constants θ ≥ 0 and p ∈ [0, 1) such that

d · d−2 C l−2 h( µx ₁ + · · · + µx d

d ) + _d−2 C l−1

 d j=1

µh(x j )

− l 

1≤j

<···<j

≤d

µh( x j

+ · · · + x j

l )
≤ θ(

 d j=1

||x j || ^p ),

h(q ⁿ u ^∗ ) − h(q ⁿ u) ^∗
≤ dq ^np θ

for all µ ∈ T ¹ , all u ∈ U(A), all n = 0, 1, 2, . . . , and all x 1 , . . . , x d ∈ A.

Assume that lim _n→∞ ^h(q _q

^e) is invertible. Then the mapping h : A → B is a ∗-homomorphism.

Proof. Define ϕ(x 1 , . . . , x d ) = θ(  _d

j=1 ||x j || ^p ), and apply Theorem

1.

Theorem 3. Let h : A → B be a mapping satisfying h(0) = 0 and h(q ⁿ uy) = h(q ⁿ u)h(y) for all u ∈ U(A), all y ∈ A, and all n = 0, 1, 2, . . . , for which there exists a function ϕ : A ^d → [0, ∞) satisfying (i), (iii), and (iv) such that

d · d−2 C l−2 h( µx 1 + · · · + µx d

d ) + _d−2 C l−1

 d j=1

µh(x j )

− l 

1≤j

<···<j

≤d

µh( x j

+ · · · + x j

l )

≤ ϕ(x 1 , . . . , x d ) (v)

for µ = 1, i, and all x ₁ , . . . , x d ∈ A. If h(tx) is continuous in t ∈ R for

each fixed x ∈ A, then the mapping h : A → B is a ∗-homomorphism.

Proof. Put µ = 1 in (v). By the same reasoning as the proof of Theorem 1, there exists a unique additive mapping Θ : A → B satisfying the inequality ( †). By the same reasoning as the proof of [7, Theorem], the additive mapping Θ : A → B is R-linear.

Put µ = i in (v). By the same method as the proof of Theorem 1, one can obtain that

Θ(ix) = lim

n→∞

h(q ⁿ ix)

q ⁿ = lim

n→∞

ih(q ⁿ x)

q ⁿ = iΘ(x) for all x ∈ A.

For each element λ ∈ C, λ = s + it, where s, t ∈ R. So Θ(λx) = Θ(sx + itx) = sΘ(x) + tΘ(ix)

= sΘ(x) + itΘ(x) = (s + it)Θ(x)

= λΘ(x) for all λ ∈ C and all x ∈ A. So

Θ(ζx + ηy) = Θ(ζx) + Θ(ηy) = ζΘ(x) + ηΘ(y)

for all ζ, η ∈ C, and all x, y ∈ A. Hence the additive mapping Θ : A → B is C-linear.

The rest of the proof is the same as the proof of Theorem 1.
From now on, assume that A is a unital C ^∗ -algebra of real rank zero, where “real rank zero” means that the set of invertible self-adjoint elements is dense in the set of self-adjoint elements (see [1]).

Now we are going to investigate continuous ∗-homomorphisms be- tween unital C ^∗ -algebras.

Theorem 4. Let h : A → B be a continuous mapping satisfying h(0) = 0 and h(q ⁿ uy) = h(q ⁿ u)h(y) for all u ∈ I ₁ ( A sa ), all y ∈ A, and all n = 0, 1, 2, . . . , for which there exists a function ϕ : A ^d → [0, ∞) satisfying (i), (ii), (iii), and (iv). Then the mapping h : A → B is a

∗-homomorphism.

Proof. By the same reasoning as the proof of Theorem 1, there exists a unique C-linear involution Θ : A → B satisfying the inequality (†).

Since h(q ⁿ uy) = h(q ⁿ u)h(y) for all u ∈ I 1 ( A sa ), all y ∈ A, and all n = 0, 1, 2, . . . ,

(5) Θ(uy) = lim

n→∞

1

q ⁿ h(q ⁿ uy) = lim

n→∞

1

q ⁿ h(q ⁿ u)h(y) = Θ(u)h(y)

for all u ∈ I 1 ( A sa ) and all y ∈ A. By the additivity of Θ and (5), q ⁿ Θ(uy) = Θ(q ⁿ uy) = Θ(u(q ⁿ y)) = Θ(u)h(q ⁿ y) for all u ∈ I 1 ( A sa ) and all y ∈ A. Hence

(6) Θ(uy) = 1

q ⁿ Θ(u)h(q ⁿ y) = Θ(u) 1

q ⁿ h(q ⁿ y)

for all u ∈ I 1 ( A sa ) and all y ∈ A. Taking the limit in (6) as n → ∞, we obtain

(7) Θ(uy) = Θ(u)Θ(y)

for all u ∈ I 1 ( A sa ) and all y ∈ A.

By (5) and (7),

Θ(e)Θ(y) = Θ(ey) = Θ(e)h(y) for all y ∈ A. Since lim n→∞ h(q

e)

q

= Θ(e) is invertible, Θ(y) = h(y)

for all y ∈ A. So Θ : A → B is continuous. But by the assumption that A has real rank zero, it is easy to show that I 1 ( A sa ) is dense in {x ∈ A sa | ||x|| = 1}. So for each w ∈ {z ∈ A sa | ||z|| = 1}, there is a sequence {κ j } such that κ j → w as j → ∞ and κ j ∈ I 1 ( A sa ). Since Θ : A → B is continuous, it follows from (7) that

Θ(wy) = Θ( lim

j→∞ κ j y) = lim

j→∞ Θ(κ j y)

= lim

j→∞ Θ(κ j )Θ(y) = Θ( lim

j→∞ κ j )Θ(y)

= Θ(w)Θ(y) (8)

for all w ∈ {z ∈ A sa | ||z|| = 1} and all y ∈ A.

For each x ∈ A, x = ^x+x ₂

+ i ^x−x _2i

, where x 1 := ^x+x ₂

and x 2 := ^x−x _2i

are self-adjoint.

First, consider the case that x 1 = 0, x 2 = 0. Since Θ : A → B is C-linear, it follows from (8) that

Θ(xy) = Θ(x 1 y + ix 2 y) = Θ(||x 1 || x 1

||x 1 || y + i||x 2 || x 2

||x 2 || y)

= ||x 1 ||Θ( x 1

||x 1 || y) + i||x 2 ||Θ( x 2

||x 2 || y)

= ||x 1 ||Θ( x 1

||x 1 || )Θ(y) + i||x 2 ||Θ( x 2

||x 2 || )Θ(y)

= {Θ(||x 1 || x 1

||x 1 || ) + iΘ(||x 2 || x 2

||x 2 || ) }Θ(y) = Θ(x 1 + ix 2 )Θ(y)

= Θ(x)Θ(y) for all y ∈ A.

Next, consider the case that x ₁ = 0, x ₂ = 0. Since Θ : A → B is C-linear, it follows from (8) that

Θ(xy) = Θ(x 1 y) = Θ(||x 1 || x 1

||x 1 || y) = ||x 1 ||Θ( x 1

||x 1 || y)

= ||x 1 ||Θ( x 1

||x 1 || )Θ(y) = Θ(||x 1 || x 1

||x 1 || )Θ(y) = Θ(x 1 )Θ(y)

= Θ(x)Θ(y) for all y ∈ A.

Finally, consider the case that x 1 = 0, x 2 = 0. Since Θ : A → B is C-linear, it follows from (8) that

Θ(xy) = Θ(ix 2 y) = Θ(i||x 2 || x 2

||x 2 || y) = i||x 2 ||Θ( x 2

||x 2 || y)

= i||x 2 ||Θ( x 2

||x 2 || )Θ(y) = Θ(i||x 2 || x 2

||x 2 || )Θ(y) = Θ(ix 2 )Θ(y)

= Θ(x)Θ(y) for all y ∈ A. Hence

Θ(xy) = Θ(x)Θ(y) for all x, y ∈ A.

Therefore, the mapping h : A → B is a ∗-homomorphism, as desired.

Corollary 5. Let h : A → B be a continuous mapping satisfying h(0) = 0 and h(q ⁿ uy) = h(q ⁿ u)h(y) for all u ∈ I 1 ( A sa ), all y ∈ A, and all n = 0, 1, 2, . . . , for which there exist constants θ ≥ 0 and p ∈ [0, 1) such that

d · d−2 C l−2 h( µx 1 + · · · + µx d

d ) + _d−2 C l−1

 d j=1

µh(x j )

− l 

1≤j

<···<j

≤d

µh( x j

+ · · · + x j

l )
≤ θ(

 d j=1

||x j || ^p ),

h(q ⁿ u ^∗ ) − h(q ⁿ u) ^∗
≤ dq ^np θ

for all µ ∈ T ¹ , all u ∈ I 1 ( A sa ), all n = 0, 1, 2, . . . , and all x 1 , . . . , x d ∈ A.

Assume that lim _n→∞ ^h(q _q

^e) is invertible. Then the mapping h : A → B is a ∗-homomorphism.

Proof. Define ϕ(x 1 , . . . , x d ) = θ(  _d

j=1 ||x j || ^p ), and apply Theorem

4.

Theorem 6. Let h : A → B be a continuous mapping satisfying h(0) = 0 and h(q ⁿ uy) = h(q ⁿ u)h(y) for all u ∈ I 1 ( A sa ), all y ∈ A, and all n = 0, 1, 2, . . . , for which there exists a function ϕ : A ^d → [0, ∞) satisfying (i), (iii), (iv), and (v). Then the mapping h : A → B is a

∗-homomorphism.

Proof. By the same reasoning as the proof of Theorem 3, there exists a unique C-linear mapping Θ : A → B satisfying the inequality (†).

The rest of the proof is the same as the proofs of Theorems 1 and 4.

3. Stability of ∗-homomorphisms between unital C ^∗ -algebras We are going to show the generalized Hyers-Ulam-Rassias stability of

∗-homomorphisms between unital C ^∗ -algebras.

Theorem 7. Let h : A → B be a mapping with h(0) = 0 for which

there exists a function ϕ : A ^d+2 → [0, ∞) such that

ϕ(x 1 , . . . , x d , z, w) :=

 ∞ j=0

q ^−j ϕ(q ^j x 1 , . . . , q ^j x d , q ^j z, q ^j w) < ∞, (vi)

d · d−2 C l−2 h( µx 1 + · · · + µx d

d + zw

d d−2 C l−2

) + _d−2 C l−1

 d j=1

µh(x j )

− l 

1≤j

<···<j

≤d

µh( x j

+ · · · + x j

l ) − h(z)h(w)

(vii)

≤ ϕ(x 1 , . . . , x d , z, w),

h(q ⁿ u ^∗ ) − h(q ⁿ u) ^∗
≤ ϕ(q  ⁿ u, . . . , q  ⁿ u 

d times

, 0, 0) (viii)

for all µ ∈ T ¹ , all u ∈ U(A), all n = 0, 1, 2, . . . , and all x 1 , . . . , x d , z, w ∈ A. Then there exists a unique ∗-homomorphism Θ : A → B such that

(ix)
h(x) − Θ(x)
≤ 1

l · d−1 C l−1 ϕ(qx, rx, . . . , rx
  

d−1 times

, 0, 0)

for all x ∈ A.

Proof. Put z = w = 0 and µ = 1 ∈ T ¹ in (vii). By the same reasoning as the proof of Theorem 1, there exists a unique C-linear involution Θ : A → B satisfying the inequality (ix). The C-linear mapping Θ : A → B is given by

(9) Θ(x) = lim

n→∞

1

q ⁿ h(q ⁿ x) for all x ∈ A.

It follows from (9) that

(10) Θ(x) = lim

n→∞

h(q ²ⁿ x) q ²ⁿ

for all x ∈ A. Let x 1 = · · · = x d = 0 in (vii). Then we get

d · d−2 C l−2 h( zw d · d−2 C l−2

) − h(z)h(w)
≤ ϕ(0, . . . , 0   

d times

, z, w)

for all z, w ∈ A. Since 1

q ²ⁿ ϕ(0, . . . , 0   

d times

, q ⁿ z, q ⁿ w) ≤ 1

q ⁿ ϕ(0, . . . , 0   

d times

, q ⁿ z, q ⁿ w),

1

q ²ⁿ
d · d−2 C l−2 h( 1

d · d−2 C l−2 q ⁿ z · q ⁿ w) − h(q ⁿ z)h(q ⁿ w)

≤ 1

q ²ⁿ ϕ(0, . . . , 0   

d times

, q ⁿ z, q ⁿ w) ≤ 1

q ⁿ ϕ(0, . . . , 0   

d times

, q ⁿ z, q ⁿ w) (11)

for all z, w ∈ A. By (vi), (10), and (11), d · d−2 C l−2 Θ( zw

d · d−2 C l−2

) = lim

n→∞

d · d−2 C l−2 h( _d·

¹ _C

q ²ⁿ zw) q ²ⁿ

= lim

n→∞

d · d−2 C l−2 h( _d·

¹ _C

q ⁿ z · q ⁿ w) q ⁿ · q ⁿ

= lim

n→∞ ( h(q ⁿ z)

q ⁿ · h(q ⁿ w) q ⁿ )

= lim

n→∞

h(q ⁿ z) q ⁿ · lim

n→∞

h(q ⁿ w) q ⁿ

= Θ(z)Θ(w) for all z, w ∈ A. But since Θ is C-linear,

Θ(zw) = d · d−2 C l−2 Θ( zw d · d−2 C l−2

) = Θ(z)Θ(w)

for all z, w ∈ A. Hence the C-linear mapping Θ : A → B is a ∗- homomorphism satisfying the inequality (ix), as desired.
Corollary 8. Let h : A → B be a mapping with h(0) = 0 for which there exist constants θ ≥ 0 and p ∈ [0, 1) such that

d · d−2 C l−2 h( µx 1 + · · · + µx d

d + zw

d · d−2 C l−2

) + _d−2 C l−1

 d j=1

µh(x j )

− l 

1≤j

<···<j

≤d

µh( x j

+ · · · + x j

l ) − h(z)h(w)

≤ θ(

 d j=1

||x j || ^p + ||z|| ^p + ||w|| ^p ),

h(q ⁿ u ^∗ ) − h(q ⁿ u) ^∗
≤ dq ^np θ

for all µ ∈ T ¹ , all u ∈ U(A), all n = 0, 1, 2, . . . , and all x 1 , . . . , x d , z, w ∈ A. Then there exists a unique ∗-homomorphism Θ : A → B such that

h(x) − Θ(x)
≤ q ^1−p (q ^p + (d − 1)r ^p )θ l · d−1 C l−1 (q ^1−p − 1) ||x|| ^p for all x ∈ A.

Proof. Define ϕ(x 1 , . . . , x d , z, w) = θ(  _d

j=1 ||x j || ^p + ||z|| ^p + ||w|| ^p ),

and apply Theorem 7.

Theorem 9. Let h : A → B be a mapping with h(0) = 0 for which there exists a function ϕ : A ^d+2 → [0, ∞) satisfying (vi) and (viii) such that

d · d−2 C l−2 h( µx ₁ + · · · + µx d

d + zw

d · d−2 C l−2

) + _d−2 C l−1

 d j=1

µh(x j )

− l 

1≤j

<···<j

≤d

µh( x j

+ · · · + x j

l ) − h(z)h(w)

≤ ϕ(x 1 , . . . , x d , z, w)

for µ = 1, i, and all x 1 , . . . , x d , z, w ∈ A. If h(tx) is continuous in t ∈ R for each fixed x ∈ A, then there exists a unique ∗-homomorphism Θ : A → B satisfying the inequality (ix).

Proof. By the same reasoning as the proof of Theorem 3, there exists a unique C-linear mapping Θ : A → B satisfying the inequality (ix).

The rest of the proof is the same as the proofs of Theorems 1 and 7.

4. Stability of linear ∗-derivations on unital C ^∗ -algebras We are going to show the generalized Hyers-Ulam-Rassias stability of linear ∗-derivations on unital C ^∗ -algebras.

Theorem 10. Let h : A → A be a mapping with h(0) = 0 for which

there exists a function ϕ : A ^d+2 → [0, ∞) satisfying (vi) and (viii) such

that

d · d−2 C l−2 h( µx 1 + · · · + µx d

d + zw

d · d−2 C l−2

)

+ _d−2 C l−1

 d j=1

µh(x j )

− l 

1≤j

<···<j

≤d

µh( x j

+ · · · + x j

l ) − zh(w) − wh(z)

≤ ϕ(x 1 , . . . , x d , z, w) (x)

for all µ ∈ T ¹ and all x ₁ , . . . , x d , z, w ∈ A. Then there exists a unique C-linear ∗-derivation D : A → A such that

(xi)
h(x) − D(x)
≤ 1

l · d−1 C l−1

ϕ(qx, rx, . . . , rx
  

d−1 times

, 0, 0)

for all x ∈ A.

Proof. Put z = w = 0 and µ = 1 ∈ T ¹ in (x). By the same reasoning as the proof of Theorem 1, there exists a unique C-linear involution D : A → A satisfying the inequality (xi). The C-linear mapping D : A → A is given by

(12) D(x) = lim

n→∞

1

q ⁿ h(q ⁿ x) for all x ∈ A.

It follows from (12) that

(13) D(x) = lim

n→∞

h(q ²ⁿ x) q ²ⁿ

for all x ∈ A. Let x 1 = · · · = x d = 0 in (x). Then we get

d · d−2 C l−2 h( zw d · d−2 C l−2

) − zh(w) − wh(z)
≤ ϕ(0, . . . , 0   

d times

, z, w)

for all z, w ∈ A. Since 1

q ²ⁿ ϕ(0, . . . , 0   

d times

, q ⁿ z, q ⁿ w) ≤ 1

q ⁿ ϕ(0, . . . , 0   

d times

, q ⁿ z, q ⁿ w),

1

q ²ⁿ
d · d−2 C l−2 h( 1

d · d−2 C l−2 q ⁿ z · q ⁿ w)

− q ⁿ zh(q ⁿ w) − q ⁿ wh(q ⁿ z)

≤ 1

q ²ⁿ ϕ(0, . . . , 0   

d times

, q ⁿ z, q ⁿ w) ≤ 1

q ⁿ ϕ(0, . . . , 0   

d times

, q ⁿ z, q ⁿ w) (14)

for all z, w ∈ A.

By (x), (13), and (14),

d · d−2 C l−2 D( zw d · d−2 C l−2

) = lim

n→∞

d · d−2 C l−2 h( _d·

¹ _C

q ²ⁿ zw) q ²ⁿ

= lim

n→∞

d · d−2 C l−2 h( _d· ¹

d−2

C

q ⁿ z · q ⁿ w) q ⁿ · q ⁿ

= lim

n→∞ ( q ⁿ z

q ⁿ · h(q ⁿ w)

q ⁿ + q ⁿ w

q ⁿ · h(q ⁿ z) q ⁿ )

= zD(w) + wD(z) for all z, w ∈ A. But since D is C-linear,

D(zw) = d · d−2 C l−2 D( zw

d · d−2 C l−2 ) = zD(w) + wD(z)

for all z, w ∈ A. Hence the C-linear mapping D : A → A is a ∗-derivation satisfying the inequality (xi), as desired.
Corollary 11. Let h : A → A be a mapping with h(0) = 0 for which there exist constants θ ≥ 0 and p ∈ [0, 1) such that

d · d−2 C l−2 h( µx 1 + · · · + µx d

d + zw

d · d−2 C l−2

) + _d−2 C l−1

 d j=1

µh(x j )

− l 

1≤j

<···<j

≤d

µh( x j

+ · · · + x j

l ) − zh(w) − wh(z)

≤ θ(

 d j=1

x j
^p +
z
^p +
w
^p ),

h(q ⁿ u ^∗ ) − h(q ⁿ u) ^∗
≤ dq ^np θ

for all µ ∈ T ¹ , all u ∈ U(A), all n = 0, 1, 2, . . . , and all x 1 , . . . , x d , z, w ∈ A. Then there exists a unique C-linear ∗-derivation D : A → A such that

h(x) − D(x)
≤ q ^1−p (q ^p + (d − 1)r ^p )θ l · d−1 C l−1 (q ^1−p − 1)
x
^p for all x ∈ A.

Proof. Define ϕ(x 1 , . . . , x d , z, w) = θ(  _d

j=1
x j
^p +
z
^p +
w
^p ), and

apply Theorem 10.

Theorem 12. Let h : A → A be a mapping with h(0) = 0 for which there exists a function ϕ : A ^d+2 → [0, ∞) satisfying (vi) and (viii) such that

d · d−2 C l−2 h( µx 1 + · · · + µx d

d + zw

d · d−2 C l−2

) + _d−2 C l−1

 d j=1

µh(x j )

− l 

1≤j

<···<j

≤d

µh( x j

+ · · · + x j

l ) − zh(w) − wh(z)

≤ ϕ(x 1 , . . . , x d , z, w)

for µ = 1, i, and all x 1 , . . . , x d , z, w ∈ A. If h(tx) is continuous in t ∈ R for each fixed x ∈ A, then there exists a unique C-linear ∗-derivation D : A → A satisfying the inequality (xi).

Proof. By the same reasoning as the proof of Theorem 3, there exists a unique C-linear mapping D : A → A satisfying the inequality (xi).

The rest of the proof is the same as the proofs of Theorems 1 and

10.

References

[1] L. Brown and G. Pedersen, C

-algebras of real rank zero, J. Funct. Anal. 99 (1991), 131–149.

[2] P. G˘ avruta, A generalization of the Hyers-Ulam-Rassias stability of approxi- mately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436.

[3] B. E. Johnson, Approximately multiplicative maps between Banach algebras, J.

London Math. Soc. 37 (1988), no. 2, 294–316.

[4] R. V. Kadison and G. Pedersen, Means and convex combinations of unitary operators, Math. Scand. 57 (1985), 249–266.

[5] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator

Algebras, Elementary Theory, Academic Press, New York, 1983.

[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] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc.

Amer. Math. Soc. 72 (1978), 297–300.

[8] T. Trif, On the stability of a functional equation deriving from an inequality of Popoviciu for convex functions, J. Math. Anal. Appl. 272 (2002), 604–616.

Chun-Gil Park

Department of Mathematics Chungnam National University Daejeon 305-764, Korea

E-mail: [email protected] Jinchuan Hou

Department of Mathematics Shanxi Teachers University Linfen, 041004, P. R. China and

Department of Mathematics Shanxi University

Taiyuan, P. R. China

E-mail: [email protected]