On a generalized Trif's mapping in Banach modules over a $C^*$-algebra

ON A GENERALIZED TRIF’S MAPPING IN

BANACH MODULES OVER A C ^∗ -ALGEBRA

Chun-Gil Park

¹

and Themistocles M. Rassias

Abstract. Let X and Y be vector spaces. It is shown that a mapping f : X → Y satisfies the functional equation

(‡)

mn

mn−2

C

k−2

f

µ x

1

+ · · · + x

mn

mn

¶

+ m

mn−2

C

k−1

X

n i=1

f

µ x

mi−m+1

+ · · · + x

mi

m

¶

= k X

1≤i₁<···<i_k≤mn

f

µ x

i₁

+ · · · + x

i_k

k

¶

if and only if the mapping f : X → Y is additive, and we prove the Cauchy-Rassias stability of the functional equation (‡) in Ba- nach modules over a unital C

^∗

-algebra. Let A and B be unital C

^∗

-algebras or Lie JC

^∗

-algebras. As an application, we show that every almost homomorphism h : A → B of A into B is a homomor- phism when h(2

^d

uy) = h(2

^d

u)h(y) or h(2

^d

u ◦ y) = h(2

^d

u) ◦ h(y) for all unitaries u ∈ A, all y ∈ A, and d = 0, 1, 2, . . . , and that every almost linear almost multiplicative mapping h : A → B is a homomorphism when h(2x) = 2h(x) for all x ∈ A.

Moreover, we prove the Cauchy-Rassias stability of homomor- phisms in C

^∗

-algebras or in Lie JC

^∗

-algebras, and of Lie JC

^∗

- algebra derivations in Lie JC

^∗

-algebras.

1. Introduction

In 1940, S. M. Ulam [20] raised the following question: Under what conditions does there exist an additive mapping near an approximately

Received December 15, 2004.

2000 Mathematics Subject Classification: Primary 39B52, 46L05, 47B48, 17A36.

Key words and phrases: Trif’s functional equation, Cauchy-Rassias stability, C

^∗

- algebra homomorphism, Lie JC

^∗

-algebra homomorphism, Lie JC

^∗

-algebra deriva- tion.

1

Supported by Korea Research Foundation Grant KRF-2005-070-C00009.

additive mapping?

Let X and Y be Banach spaces with norms ||·|| and k·k, respectively.

Hyers [3] showed that if ² > 0 and f : X → Y such that kf (x + y) − f (x) − f (y)k ≤ ²

for all x, y ∈ X, then there exists a unique additive mapping T : X → Y such that

kf (x) − T (x)k ≤ ² for all x ∈ X.

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 ² ≥ 0 and p ∈ [0, 1) such that

(*) kf (x + y) − f (x) − f (y)k ≤ ²(||x|| ^p + ||y|| ^p )

for all x, y ∈ X. Th. M. Rassias [12] showed that there exists a unique R-linear mapping T : X → Y such that

kf (x) − T (x)k ≤ 2²

2 − 2 ^p ||x|| ^p

for all x ∈ X. The inequality (∗) that was introduced for the first time by Th. M. Rassias [12] we call Cauchy-Rassias inequality and the stability of the functional equation Cauchy-Rassias stability. This inequality has provided a lot of influence in the development of what is now known as Hyers–Ulam–Rassias stability of functional equations. Beginning around the year 1980 the topic of approximate homomorphisms, or the stability of the equation of homomorphism, was taken up by a number of math- ematicians (cf. [4], [10], [14]–[18]). Th. M. Rassias [13] during the 27

^th

International Symposium on Functional Equations asked the question whether such a theorem can also be proved for p ≥ 1. Z. Gajda [1] fol- lowing the same approach as in Th. M. Rassias [12], gave an affirmative solution to this question for p > 1.

G˘avruta [2] 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

e

ϕ(x, y) = X ∞ j=0

2 ^−j ϕ(2 ^j x, 2 ^j y) < ∞

for all x, y ∈ G. Suppose that f : G → Y is a mapping satisfying kf (x + y) − f (x) − f (y)k ≤ ϕ(x, y)

for all x, y ∈ G. Then there exists a unique additive mapping T : G → Y such that

kf (x) − T (x)k ≤ 1

2 ϕ(x, x) e

for all x ∈ G. C. Park [9] applied the G˘avruta’s result to linear functional equations in Banach modules over a C ^∗ -algebra.

Jun and Lee [5] proved the following: Denote by ϕ : X\{0}×X\{0} → [0, ∞) a function such that

e

ϕ(x, y) = X ∞ j=0

3 ^−j ϕ(3 ^j x, 3 ^j y) < ∞

for all x, y ∈ X \ {0}. Suppose that f : X → Y is a mapping satisfying k2f ( x + y

2 ) − f (x) − f (y)k ≤ ϕ(x, y)

for all x, y ∈ X \ {0}. Then there exists a unique additive mapping T : X → Y such that

kf (x) − f (0) − T (x)| ≤ 1

3 ( e ϕ(x, −x) + e ϕ(−x, 3x))

for all x ∈ X \ {0}. C. Park and W. Park [11] applied the Jun and Lee’s result to the Jensen’s equation in Banach modules over a C ^∗ -algebra.

Recently, T. Trif [19, Theorem 2.1] proved that, for vector spaces V and W , a mapping f : V → W with f (0) = 0 satisfies the functional equation

n _n−2 C _k−2 f

µ x

₁

+ · · · + x _n n

¶

+ _n−2 C _k−1 X n i=1

f (x _i )

= k X

1≤i1

<···<i

_k

≤n

f

µ x _i

₁

+ · · · + x _i

_k

k

¶ (T)

for all x

₁

, . . . , x _n ∈ V if and only if the mapping f : V → W satisfies

the additive Cauchy equation f (x + y) = f (x) + f (y) for all x, y ∈ V . He

proved the stability of the functional equation (T) (see [19, Theorems 3.1 and 3.2]).

Throughout this paper, assume that m, n, k are integers with 1 < m <

k < mn, and that s, q are integers with 1 ≤ s ≤ [ ⁿ

₂

] and 1 < 2q ≤ m, where [ ] denotes the Gauss symbol.

In this paper, we solve the following functional equation

(1.i)

mn mn−2 C k−2 f

µ x

1

+ · · · + x mn

mn

¶

+ m _mn−2 C _k−1 X n i=1

f

µ x _mi−m+1 + · · · + x _mi m

¶

= k X

1≤i1

<···<i

k

≤mn

f

µ x _i

₁

+ · · · + x _i

_k

k

¶ .

We moreover prove the Cauchy-Rassias stability of the functional equa- tion (1.i) in Banach modules over a unital C ^∗ -algebra. The main purpose of this paper is to investigate homomorphisms between C ^∗ -algebras and between Lie JC ^∗ -algebras, and to prove their Cauchy-Rassias stability.

2. A generalized Trif ’s mapping

Throughout this section, assume that X and Y are linear spaces.

Lemma 2.1. A mapping f : X → Y with f (0) = 0 satisfies the func- tional equation (1.i) for all x

1

, . . . , x mn ∈ X if and only if the mapping f : X → Y satisfies the additive Cauchy equation f (x+y) = f (x)+f (y) for all x, y ∈ X.

Proof. Assume that a mapping f : X → Y with f (0) = 0 satisfies the functional equation (1.i).

Replacing x m and x m+1 by x m+1 and x m in (1.i), respectively, we get

(2.1)

mn _mn−2 C _k−2 f

µ x

₁

+ · · · + x _mn mn

¶

+ m _mn−2 C _k−1 f

µ x

₁

+ · · · + x _m−1 + x _m+1 m

¶

+ m _mn−2 C _k−1 f

µ x m + x m+2 + · · · + x

2m

m

¶

+ m _mn−2 C _k−1 X n i=3

f

µ x _mi−m+1 + · · · + x _mi m

¶

= k X

1≤i₁

<···<i

_k

≤mn

f

µ x _i

₁

+ · · · + x _i

_k

k

¶

for all x

₁

, . . . , x _mn ∈ X. It follows from (1.i) and (2.1) that

(2.2) f

µ x

₁

+ · · · + x _m−1 + x _m m

¶ + f

µ x _m+1 + x _m+2 + · · · + x

_2m

m

¶

= f

µ x

1

+ · · · + x m−1 + x m+1

m

¶ + f

µ x m + x m+2 + · · · + x

2m

m

¶

for all x

1

, . . . , x

2m

∈ X. Letting x m−1 = x m = x, x m+1 = x m+2 = y and x

₁

= · · · = x _m−2 = x _m+3 = · · · = x

_2m

= 0 in (2.2), we get

(2.3) f

µ 2x m

¶ + f

µ 2y m

¶

= 2f

µ x + y m

¶

for all x, y ∈ X. Putting y = 0 in (2.3), we obtain f

µ 2x m

¶

= 2f µ x

m

¶

for all x ∈ X. So

(2.4) 2f

µ x m

¶ + 2f

µ y m

¶

= 2f

µ x + y m

¶

for all x, y ∈ X. Replacing x and y by mx and my in (2.4), respectively, we get

2f (x) + 2f (y) = 2f (x + y)

for all x, y ∈ X. Thus the mapping f : X → Y satisfies the additive Cauchy equation f (x + y) = f (x) + f (y) for all x, y ∈ X.

The converse is obvious. ¤

3. Cauchy-Rassias stability of the generalized Trif ’s mapping 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 with norms || · || and k · k, respectively.

Given a mapping f : X → Y , we set D _u f (x

₁

, . . . , x _mn ) := mn _mn−2 C _k−2 f

µ ux

₁

+ · · · + ux _mn mn

¶

+ m _mn−2 C _k−1 X n i=1

f

µ ux _mi−m+1 + · · · + ux _mi m

¶

− k X

1≤i₁

<···<i

_k

≤mn

uf

µ x i

₁

+ · · · + x i

_k

k

¶

for all u ∈ U(A) and all x

₁

, . . . , x _mn ∈ X.

Theorem 3.1. Let f : X → Y be a mapping satisfying f (0) = 0 for which there is a function ϕ : X ^mn → [0, ∞) such that

e

ϕ(x

₁

, . . . , x _mn ) :=

X ∞ j=0

1

2 ^j ϕ(2 ^j x

₁

, . . . , 2 ^j x _mn ) < ∞, (3.i)

kD _u f (x

₁

, . . . , x _mn )k ≤ ϕ(x

₁

, . . . , x _mn ) (3.ii)

for all u ∈ U(A) and all x

1

, . . . , x mn ∈ X. Then there exists a unique A-linear generalized Trif’s mapping T : X → Y such that

(3.iii)

kf (x) − T (x)k

≤ 1

2ms _mn−2 C _k−1 ϕ( 0, . . . , 0 e

| {z }

m−2q times

, mx

q , . . . , mx

| {z q }

q times

, 0, . . . , 0

| {z }

q times

,

mx

q , . . . , mx

| {z q }

q times

, 0, . . . , 0

| {z }

m−q times

, . . . , 0, . . . , 0

| {z }

m−2q times

, mx

q , . . . , mx

| {z q }

q times

, 0, . . . , 0

| {z }

q times

,

mx

q , . . . , mx

| {z q }

q times

, 0, . . . , 0

| {z }

m−q times

, 0, . . . , 0

| {z }

mn−2ms times

)

+ 1

2ms _mn−2 C _k−1 ϕ( 0, . . . , 0 e

| {z }

m−2q times

, mx

q , . . . , mx

| {z q }

q times

, mx

q , . . . , mx

| {z q }

q times

,

0, . . . , 0

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, . . . , 0, . . . , 0

| {z }

m−2q times

, mx

q , . . . , mx

| {z q }

q times

, mx

q , . . . , mx

| {z q }

q times

,

0, . . . , 0

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, 0, . . . , 0

| {z }

mn−2ms times

)

for all x ∈ X.

Proof. Let u = 1 ∈ U(A). Putting x im−2q+1 = · · · = x im−q = x, x _im−q+1 = · · · = x _im = 0, x _im+1 = · · · = x _im+q = x for i = 1, 3, . . . , 2s − 1, and x _j = 0 for other indices j in (3.ii), we have

(3.1)

° °

° °mn ^mn−2 C k−2 f µ 2qsx

mn

¶

+ 2ms mn−2 C k−1 f µ qx

m

¶

− k X

1≤i₁

<···<i

_k

≤mn

f

µ x _i

₁

+ · · · + x _i

_k

k

¶° °

° °

≤ ϕ( 0, . . . , 0

| {z }

m−2q times

, x, . . . , x

| {z }

q times

, 0, . . . , 0

| {z }

q times

, x, . . . , x

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, . . . , 0, . . . , 0

| {z }

m−2q times

, x, . . . , x

| {z }

q times

, 0, . . . , 0

| {z }

q times

, x, . . . , x

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, 0, . . . , 0

| {z }

mn−2ms times

)

for all x ∈ X. Putting x _im−2q+1 = · · · = x _im−q = x, x _im−q+1 = · · · = x _im = x for i = 1, 3, . . . , 2s − 1, and x _j = 0 for other indices j in (3.ii), we have

(3.2)

° °

° °mn ^mn−2 C _k−2 f µ 2qsx

mn

¶

+ ms _mn−2 C _k−1 f µ 2qx

m

¶

− k X

1≤i1

<···<i

k

≤mn

f

µ x _i

₁

+ · · · + x _i

_k

k

¶° °

° °

≤ ϕ( 0, . . . , 0

| {z }

m−2q times

, x, . . . , x

| {z }

q times

, x, . . . , x

| {z }

q times

, 0, . . . , 0

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, . . . , 0, . . . , 0

| {z }

m−2q times

, x, . . . , x

| {z }

q times

, x, . . . , x

| {z }

q times

, 0, . . . , 0

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, 0, . . . , 0

| {z }

mn−2ms times

)

for all x ∈ X. It follows from (3.1) and (3.2) that

° °

° °2ms ^mn−2 C k−1 f µ qx

m

¶

− ms mn−2 C k−1 f µ 2qx

m

¶° °

° °

≤ ϕ( 0, . . . , 0

| {z }

m−2q times

, x, . . . , x

| {z }

q times

, 0, . . . , 0

| {z }

q times

, x, . . . , x

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, . . . , 0, . . . , 0

| {z }

m−2q times

, x, . . . , x

| {z }

q times

, 0, . . . , 0

| {z }

q times

, x, . . . , x

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, 0, . . . , 0

| {z }

mn−2ms times

)

+ ϕ( 0, . . . , 0

| {z }

m−2q times

, x, . . . , x

| {z }

q times

, x, . . . , x

| {z }

q times

, 0, . . . , 0

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, . . . , 0, . . . , 0

| {z }

m−2q times

, x, . . . , x

| {z }

q times

, x, . . . , x

| {z }

q times

, 0, . . . , 0

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, 0, . . . , 0

| {z }

mn−2ms times

)

for all x ∈ X. So

(3.‡)

° °

° °f (x) − 1 2 f (2x)

° °

° °

≤ 1

2ms _mn−2 C _k−1 ϕ( 0, . . . , 0

| {z }

m−2q times

, mx

q , . . . , mx

| {z q }

q times

, 0, . . . , 0

| {z }

q times

,

mx

q , . . . , mx

| {z q }

q times

, 0, . . . , 0

| {z }

m−q times

, . . . , 0, . . . , 0

| {z }

m−2q times

, mx

q , . . . , mx

| {z q }

q times

,

0, . . . , 0

| {z }

q times

, mx

q , . . . , mx

| {z q }

q times

, 0, . . . , 0

| {z }

m−q times

, 0, . . . , 0

| {z }

mn−2ms times

)

+ 1

2ms _mn−2 C _k−1 ϕ( 0, . . . , 0

| {z }

m−2q times

, mx

q , . . . , mx

| {z q }

q times

, mx

q , . . . , mx

| {z q }

q times

,

0, . . . , 0

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, . . . , 0, . . . , 0

| {z }

m−2q times

, mx

q , . . . , mx

| {z q }

q times

,

mx

q , . . . , mx

| {z q }

q times

, 0, . . . , 0

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, 0, . . . , 0

| {z }

mn−2ms times

)

for all x ∈ X. Hence

(3.3)

° °

° ° 1

2 ^d f (2 ^d x) − 1

2 ^d+1 f (2 ^d+1 x)

° °

° °

= 1 2 ^d

° °

° °f (2 ^d x) − 1

2 f (2 · 2 ^d x)

° °

° °

≤ 1

2 ^d+1 ms _mn−2 C _k−1 ϕ( 0, . . . , 0

| {z }

m−2q times

, 2 ^d mx

q , . . . , 2 ^d mx

| {z q }

q times

, 0, . . . , 0

| {z }

q times

,

2 ^d mx

q , . . . , 2 ^d mx

| {z q }

q times

, 0, . . . , 0

| {z }

m−q times

, . . . , 0, . . . , 0

| {z }

m−2q times

, 2 ^d mx

q , . . . , 2 ^d mx

| {z q }

q times

,

0, . . . , 0

| {z }

q times

, 2 ^d mx

q , . . . , 2 ^d mx

| {z q }

q times

, 0, . . . , 0

| {z }

m−q times

, 0, . . . , 0

| {z }

mn−2ms times

)

+ 1

2 ^d+1 ms _mn−2 C _k−1 ϕ( 0, . . . , 0

| {z }

m−2q times

, 2 ^d mx

q , . . . , 2 ^d mx

| {z q }

q times

,

2 ^d mx

q , . . . , 2 ^d mx

| {z q }

q times

, 0, . . . , 0

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, . . . , 0, . . . , 0

| {z }

m−2q times

, 2 ^d mx

q , . . . , 2 ^d mx

| {z q }

q times

,

2 ^d mx

q , . . . , 2 ^d mx

| {z q }

q times

, 0, . . . , 0

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, 0, . . . , 0

| {z }

mn−2ms times

)

for all x ∈ X and all positive integers d. By (3.3), we have

(3.4)

° °

° ° 1

2 ^l f (2 ^l x) − 1 2 ^d f

µ 2 ^d x

¶° °

° °

≤

d−1 X

j=l

1

2 ^j+1 ms _mn−2 C _k−1 ϕ( 0, . . . , 0

| {z }

m−2q times

, 2 ^j mx

q , . . . , 2 ^j mx

| {z q }

q times

,

0, . . . , 0

| {z }

q times

, 2 ^j mx

q , . . . , 2 ^j mx

| {z q }

q times

,

0, . . . , 0

| {z }

m−q times

, . . . , 0, . . . , 0

| {z }

m−2q times

, 2 ^j x

q , . . . , 2 ^j mx

| {z q }

q times

,

0, . . . , 0

| {z }

q times

, 2 ^j mx

q , . . . , 2 ^j mx

| {z q }

q times

, 0, . . . , 0

| {z }

m−q times

, 0, . . . , 0

| {z }

mn−2ms times

)

+

d−1 X

j=l

1

2 ^j+1 ms _mn−2 C _k−1 ϕ( 0, . . . , 0

| {z }

m−2q times

, 2 ^j mx

q , . . . , 2 ^j mx

| {z q }

q times

,

2 ^j mx

q , . . . , 2 ^j mx

| {z q }

q times

, 0, . . . , 0

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, . . . , 0, . . . , 0

| {z }

m−2q times

, 2 ^j mx

q , . . . , 2 ^j mx

| {z q }

q times

,

2 ^j mx

q , . . . , 2 ^j mx

| {z q }

q times

, 0, . . . , 0

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, 0, . . . , 0

| {z }

mn−2ms times

)

for all x ∈ X and all positive integers l and d with l < d. This shows that the sequence {

₂¹_d

f (2 ^d x)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {

₂¹_d

f (2 ^d x)} converges for all x ∈ X. So we can define a mapping T : X → Y by

T (x) := lim

d→∞

1

2 ^d f (2 ^d x) for all x ∈ X. Also, we get

kD

1

T (x

1

, . . . , x mn )k = lim

d→∞

1

2 ^d kD

1

f (2 ^d x

1

, . . . , 2 ^d x mn )k

≤ lim

d→∞

1

2 ^d ϕ(2 ^d x

₁

, . . . , 2 ^d x _mn )

= 0

for all x

₁

, . . . , x _mn ∈ X. Thus T is a generalized Trif’s mapping. By Lemma 2.1, T is additive. Putting l = 0 and letting d → ∞ in (3.4), we get (3.iii).

Now, let L : X → Y be another generalized Trif’s mapping satisfying (3.iii). Then we have

kT (x) − L(x)k

= 1

2 ^d kT (2 ^d x) − L(2 ^d x)k

≤ 1

2 ^d (kT (2 ^d x) − f (2 ^d x)k + kL(2 ^d x) − f (2 ^d x)k)

≤ 2

2 ^d+1 ms _mn−2 C _k−1 ϕ( 0, . . . , 0 e

| {z }

m−2q times

, 2 ^d mx

q , . . . , 2 ^d mx

| {z q }

q times

, 0, . . . , 0

| {z }

q times

,

2 ^d mx

q , . . . , 2 ^d mx

| {z q }

q times

, 0, . . . , 0

| {z }

m−q times

, . . . , 0, . . . , 0

| {z }

m−2q times

, 2 ^d mx

q , . . . , 2 ^d mx

| {z q }

q times

,

0, . . . , 0

| {z }

q times

, 2 ^d mx

q , . . . , 2 ^d mx

| {z q }

q times

, 0, . . . , 0

| {z }

m−q times

, 0, . . . , 0

| {z }

mn−2ms times

)

+ 2

2 ^d+1 ms _mn−2 C _k−1 ϕ( 0, . . . , 0 e

| {z }

m−2q times

, 2 ^d mx

q , . . . , 2 ^d mx

| {z q }

q times

,

2 ^d mx

q , . . . , 2 ^d mx

| {z q }

q times

, 0, . . . , 0

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, . . . , 0, . . . , 0

| {z }

m−2q times

, 2 ^d mx

q , . . . , 2 ^d mx

| {z q }

q times

,

2 ^d mx

q , . . . , 2 ^d mx

| {z q }

q times

, 0, . . . , 0

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, 0, . . . , 0

| {z }

mn−2ms times

),

which tends to zero as d → ∞ for all x ∈ X. So we can conclude that T (x) = L(x) for all x ∈ X. This proves the uniqueness of T .

By the assumption, for each u ∈ U(A), we get kD u T (x, 0, . . . , 0

| {z }

mn−1 times

)k = lim

d→∞

1

2 ^d kD u f (2 ^d x, 0, . . . , 0

| {z }

mn−1 times

)k

≤ lim

d→∞

1

2 ^d ϕ(2 ^d x, 0, . . . , 0

| {z }

mn−1 times

) = 0

for all x ∈ X. So mn mn−2 C k−2 T ( ux

mn ) + m mn−2 C k−1 T ( ux

m ) = k mn−1 C k−1 uT ( x k ) for all u ∈ U(A) and all x ∈ X. Since _mn−2 C _k−2 + _mn−2 C _k−1 =

mn−1 C k−1 and T is additive,

(3.5) T (ux) = uT (x)

for all u ∈ U(A) and all x ∈ X.

Now let a ∈ A (a 6= 0) and M an integer greater than 4|a|. Then

| _M ^a | <

¹₄

< 1 −

²₃

=

¹₃

. By [6, Theorem 1], there exist three elements u

₁

, u

₂

, u

₃

∈ U(A) such that 3 _M ^a = u

₁

+ u

₂

+ u

₃

. So by (3.5)

T (ax) = T ( M 3 · 3 a

M x) = M · T ( 1 3 · 3 a

M x)

= M 3 T (3 a

M x)

= M

3 T (u

1

x + u

2

x + u

3

x)

= M

3 (T (u

1

x) + T (u

2

x) + T (u

3

x))

= M

3 (u

1

+ u

2

+ u

3

)T (x)

= M 3 · 3 a

M T (x) = aT (x) for all a ∈ A and all x ∈ X. Hence

T (ax + by) = T (ax) + T (by) = aT (x) + bT (y)

for all a, b ∈ A(a, b 6= 0) and all x, y ∈ X. And T (0x) = 0 = 0T (x) for all x ∈ X. So the generalized Trif’s mapping T : A → B is an A-linear

mapping, as desired. ¤

Corollary 3.2. Let θ and p < 1 be positive real numbers. Let f : X → Y be a mapping satisfying f (0) = 0 such that

kD _u f (x

₁

, . . . , x _mn )k ≤ θ X mn j=1

||x _j || ^p

for all u ∈ U(A) and all x

1

, . . . , x mn ∈ X. Then there exists a unique A-linear generalized Trif’s mapping T : X → Y such that

kf (x) − T (x)k ≤ 4m ^p−1 q

^1−p

θ

(2 − 2 ^p ) _mn−2 C _k−1 ||x|| ^p for all x ∈ X.

Proof. Define ϕ(x

₁

, . . . , x _mn ) = θ P _mn

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

3.1. ¤

Theorem 3.3. Let f : X → Y be a mapping satisfying f (0) = 0 for which there is a function ϕ : X ^mn → [0, ∞) such that

e

ϕ(x

₁

, . . . , x _mn ) :=

X ∞ j=1

2 ^j ϕ µ x

1

2 ^j , . . . , x mn

2 ^j

¶

< ∞, (3.iv)

kD _u f (x

₁

, . . . , x _mn )k ≤ ϕ(x

₁

, . . . , x _mn )

(3.v)

for all u ∈ U(A) and all x

1

, . . . , x mn ∈ X. Then there exists a unique A-linear generalized Trif’s mapping T : X → Y such that

(3.vi)

kf (x) − T (x)k

≤ 1

2ms mn−2 C k−1 ϕ( 0, . . . , 0 e

| {z }

m−2q times

, mx

q , . . . , mx

| {z q }

q times

, 0, . . . , 0

| {z }

q times

,

mx

q , . . . , mx

| {z q }

q times

, 0, . . . , 0

| {z }

m−q times

, . . . , 0, . . . , 0

| {z }

m−2q times

, mx

q , . . . , mx

| {z q }

q times

, 0, . . . , 0

| {z }

q times

,

mx

q , . . . , mx

| {z q }

q times

, 0, . . . , 0

| {z }

m−q times

, 0, . . . , 0

| {z }

mn−2ms times

)

+ 1

2ms mn−2 C k−1 ϕ( 0, . . . , 0 e

| {z }

m−2q times

, mx

q , . . . , mx

| {z q }

q times

, mx

q , . . . , mx

| {z q }

q times

,

0, . . . , 0

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, . . . , 0, . . . , 0

| {z }

m−2q times

, mx

q , . . . , mx

| {z q }

q times

, mx

q , . . . , mx

| {z q }

q times

,

0, . . . , 0

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, 0, . . . , 0

| {z }

mn−2ms times

)

for all x ∈ X.

Proof. Replacing x by ^x

₂

in (3.‡), we have

kf (x) − 2f ( x 2 )k

≤ 1

ms mn−2 C k−1 ϕ( 0, . . . , 0

| {z }

m−2q times

, mx

2q , . . . , mx

| {z 2q }

q times

, 0, . . . , 0

| {z }

q times

,

mx

2q , . . . , mx

| {z 2q }

q times

, 0, . . . , 0

| {z }

m−q times

, . . . , 0, . . . , 0

| {z }

m−2q times

, mx

2q , . . . , mx

| {z 2q }

q times

, 0, . . . , 0

| {z }

q times

,

mx

2q , . . . , mx

| {z 2q }

q times

, 0, . . . , 0

| {z }

m−q times

, 0, . . . , 0

| {z }

mn−2ms times

)

+ 1

ms _mn−2 C _k−1 ϕ( 0, . . . , 0

| {z }

m−2q times

, mx

2q , . . . , mx

| {z 2q }

q times

, mx

2q , . . . , mx

| {z 2q }

q times

,

0, . . . , 0

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, . . . , 0, . . . , 0

| {z }

m−2q times

, mx

2q , . . . , mx

| {z 2q }

q times

, mx

2q , . . . , mx

| {z 2q }

q times

,

0, . . . , 0

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, 0, . . . , 0

| {z }

mn−2ms times

)

for all x ∈ X. So (3.6) ° °

° °2 ^d f µ x

2 ^d

¶

− 2 ^d+1 f µ x

2 ^d+1

¶° °

° °

= 2 ^d

° °

° °f µ x

2 ^d

¶

− 2f µ x

2 · 2 ^d

¶° °

° °

≤ 2 ^d

ms _mn−2 C _k−1 ϕ( 0, . . . , 0

| {z }

m−2q times

, mx

2 ^d+1 q , . . . , mx 2 ^d+1 q

| {z }

q times

, 0, . . . , 0

| {z }

q times

,

mx

2 ^d+1 q , . . . , mx 2 ^d+1 q

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, . . . , 0, . . . , 0

| {z }

m−2q times

, mx

2 ^d+1 q , . . . , mx 2 ^d+1 q

| {z }

q times

,

0, . . . , 0

| {z }

q times

, mx

2 ^d+1 q , . . . , mx 2 ^d+1 q

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, 0, . . . , 0

| {z }

mn−2ms times

)

+ 2 ^d

ms mn−2 C k−1

ϕ( 0, . . . , 0

| {z }

m−2q times

, mx

2 ^d+1 q , . . . , mx 2 ^d+1 q

| {z }

q times

,

mx

2 ^d+1 q , . . . , mx 2 ^d+1 q

| {z }

q times

,

0, . . . , 0

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, . . . , 0, . . . , 0

| {z }

m−2q times

, mx

2 ^d+1 q , . . . , mx 2 ^d+1 q

| {z }

q times

,

mx

2 ^d+1 q , . . . , mx 2 ^d+1 q

| {z }

q times

, 0, . . . , 0

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, 0, . . . , 0

| {z }

mn−2ms times

)

for all x ∈ X and all positive integers d. By (3.6), we have (3.7) °

° °

°2 ^l f ( x

2 ^l ) − 2 ^d f ( x 2 ^d )

° °

° °

≤

d−1 X

j=l

2 ^j

ms _mn−2 C _k−1 ϕ( 0, . . . , 0

| {z }

m−2q times

, mx

2 ^j+1 q , . . . , mx 2 ^j+1 q

| {z }

q times

, 0, . . . , 0

| {z }

q times

,

mx

2 ^j+1 q , . . . , mx 2 ^j+1 q

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, . . . , 0, . . . , 0

| {z }

m−2q times

, mx

2 ^j+1 q , . . . , mx 2 ^j+1 q

| {z }

q times

,

0, . . . , 0

| {z }

q times

, mx

2 ^j+1 q , . . . , mx 2 ^j+1 q

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, 0, . . . , 0

| {z }

mn−2ms times

)

+

d−1 X

j=l

2 ^j

ms _mn−2 C _k−1 ϕ( 0, . . . , 0

| {z }

m−2q times

, mx

2 ^j+1 q , . . . , mx 2 ^j+1 q

| {z }

q times

,

mx

2 ^j+1 q , . . . , mx 2 ^j+1 q

| {z }

q times

, 0, . . . , 0

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, . . . , 0, . . . , 0

| {z }

m−2q times

,

mx

2 ^j+1 q , . . . , mx 2 ^j+1 q

| {z }

q times

, mx

2 ^j+1 q , . . . , mx 2 ^j+1 q

| {z }

q times

, 0, . . . , 0

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

,

0, . . . , 0

| {z }

mn−2ms times

)

for all x ∈ X and all positive integers l and d with l < d. This shows that the sequence {2 ^d f (

₂

^x

_d

)} is a Cauchy sequence for all x ∈ X. Since Y is complete, the sequence {2 ^d f (

₂

^x

d

)} converges for all x ∈ X. So we can define a mapping T : X → Y by

T (x) := lim

d→∞ 2 ^d f µ x

2 ^d

¶

for all x ∈ X. Also, we get

kD

₁

T (x

₁

, . . . , x _mn )k = lim

d→∞ 2 ^d

° °

° °D

¹

f µ x

₁

2 ^d , . . . , x _mn 2 ^d

¶° °

° °

≤ lim

d→∞ 2 ^d ϕ µ x

1

2 ^d , . . . , x mn

2 ^d

¶

= 0

for all x

1

, . . . , x mn ∈ X. Thus T is a generalized Trif’s mapping. By Lemma 2.1, T is additive. Putting l = 0 and letting d → ∞ in (3.7), we get (3.vi).

The rest of the proof is similar to the proof of Theorem 3.1. ¤ Corollary 3.4. Let θ and p > 1 be positive real numbers. Let f : X → Y be a mapping satisfying f (0) = 0 such that

kD u f (x

1

, . . . , x mn )k ≤ θ X mn j=1

||x j || ^p

for all u ∈ U(A) and all x

1

, . . . , x mn ∈ X. Then there exists a unique A-linear generalized Trif’s mapping T : X → Y such that

kf (x) − T (x)k ≤ 4m ^p−1 q

^1−p

θ

(2 ^p − 2) _mn−2 C _k−1 ||x|| ^p for all x ∈ X.

Proof. Define ϕ(x

₁

, . . . , x _mn ) = θ P _mn

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

3.3. ¤

4. Isomorphisms between unital C ^∗ -algebras

Throughout this section, assume that A is a unital C ^∗ -algebra with norm || · ||, unit e and unitary group U(A), and that B is a unital C ^∗ - algebra with norm k · k.

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

Theorem 4.1. Let h : A → B be a bijective mapping satisfying h(0) = 0 and h(2 ^d uy) = h(2 ^d u)h(y) for all u ∈ U(A), all y ∈ A, and d = 0, 1, 2, . . . , for which there is a function ϕ : A ^mn → [0, ∞) satisfying (3.i) for all x

₁

, . . . , x _mn ∈ A such that

° °

° °mn ^mn−2 C k−2 h

µ µx

1

+ · · · + µx mn

mn

¶

+ m _mn−2 C _k−1 X n i=1

h

µ µx mi−m+1 + · · · + µx mi

m

¶

− k X

1≤i1

<···<i

k

≤mn

µh

µ x _i

₁

+ · · · + x _i

_k

k

¶° °

° °

≤ ϕ µ

x

₁

, . . . , x _mn

¶ , (4.i)

° °

° °h(2 ^d u ^∗ ) − h(2 ^d u) ^∗

° °

° ° ≤ ϕ(2 ^d u, . . . , 2 ^d u

| {z }

mn times

) (4.ii)

for all u ∈ U(A), all x

₁

, . . . , x _mn ∈ A, all µ ∈ T

¹

:= {λ ∈ C | |λ| = 1}

and d = 0, 1, 2, . . . . Assume that (4.iii) lim _d→∞ ^h(2

_2d^d

^e) is invertible. Then the bijective mapping h : A → B is a C ^∗ -algebra isomorphism.

Proof. We can consider a C ^∗ -algebra as a Banach module over a unital C ^∗ -algebra C. So by Theorem 3.1, there exists a unique C-linear mapping H : A → B such that

(4.iv)

kh(x) − H(x)k

≤ 1

2ms _mn−2 C _k−1 ϕ( 0, . . . , 0 e

| {z }

m−2q times

, mx

q , . . . , mx

| {z q }

q times

, 0, . . . , 0

| {z }

q times

,

mx

q , . . . , mx

| {z q }

q times

, 0, . . . , 0

| {z }

m−q times

, . . . , 0, . . . , 0

| {z }

m−2q times

, mx

q , . . . , mx

| {z q }

q times

, 0, . . . , 0

| {z }

q times

,

mx

q , . . . , mx

| {z q }

q times

, 0, . . . , 0

| {z }

m−q times

, 0, . . . , 0

| {z }

mn−2ms times

)

+ 1

2ms _mn−2 C _k−1 ϕ( 0, . . . , 0 e

| {z }

m−2q times

, mx

q , . . . , mx

| {z q }

q times

, mx

q , . . . , mx

| {z q }

q times

,

0, . . . , 0

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, . . . , 0, . . . , 0

| {z }

m−2q times

, mx

q , . . . , mx

| {z q }

q times

, mx

q , . . . , mx

| {z q }

q times

,

0, . . . , 0

| {z }

q times

, 0, . . . , 0

| {z }

m−q times

, 0, . . . , 0

| {z }

mn−2ms times

)

for all x ∈ A. The mapping H : A → B is given by

(4.1) H(x) = lim

d→∞

1

2 ^d h(2 ^d x)

for all x ∈ A.

By (3.i) and (4.ii), we get

H(u ^∗ ) = lim

d→∞

h(2 ^d u ^∗ )

2 ^d = lim

d→∞

h(2 ^d u) ^∗ 2 ^d =

µ

d→∞ lim

h(2 ^d u) 2 ^d

¶ _∗

= 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 [7, Theorem 4.1.7]), i.e., x = P _d

j=1 λ _j u _j (λ _j ∈ C, u _j ∈ U(A)),

H(x ^∗ ) = H µ X _d

j=1

λ _j u ^∗ _j

¶

= X d j=1

λ _j H(u ^∗ _j ) = X d j=1

λ _j H(u _j ) ^∗

= µ X _d

j=1

λ _j H(u _j )

¶ _∗

= H µ X _d

j=1

λ _j u _j

¶ _∗

= H(x) ^∗ for all x ∈ A.

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

(4.2) H(uy) = lim

d→∞

1

2 ^d h(2 ^d uy) = lim

d→∞

1

2 ^d h(2 ^d u)h(y) = H(u)h(y) for all u ∈ U(A) and all y ∈ A. By the additivity of H and (4.2),

2 ^d H(uy) = H(2 ^d uy) = H(u(2 ^d y)) = H(u)h(2 ^d y) for all u ∈ U(A) and all y ∈ A. Hence

(4.3) H(uy) = 1

2 ^d H(u)h(2 ^d y) = H(u) 1

2 ^d h(2 ^d y)

for all u ∈ U(A) and all y ∈ A. Taking the limit in (4.3) as d → ∞, we obtain

(4.4) H(uy) = H(u)H(y)