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Linear *-derivations on C*-algebras.

Abstract

It is shown that every almost linear almost *-derivation h : A [right arrow] A on a unital C*-algebra, JC*-algebra, or Lie C*-algebra A is a linear *-derivation when h(rx) = rh(x) (r > 1) for all x [member of] A.

We moreover prove the Cauchy-Rassias stability of linear *-derivations on unital C*-algebras, on unital JC*-algebras, or on unital Lie C*-algebras.

Keywords and Phrases: JC*-algebra, Lie C*-algebra, Linear *-derivation, Stability, Linear functional equation.

1. Introduction

Let X and Y be Banach spaces with norms ||dot|| and ||dot||, respectively. Consider f : X [right arrow] Y to be a mapping such that f(tx) is continuous in t [member of] R for each fixed x [member of] X. Assume that there exist constants [theta] [greater than or equal to] 0 and p [member of] [0, 1) such that

||f(x + y) - f(x) - f(y)|| [less than or equal to] [theta](||x||[.sup.p] + ||y||[.sup.p])

for all x, y [member of] X. Rassias [8] showed that there exists a unique R-linear mapping T : X [right arrow] Y such that

||f(x) - T(x)|| [less than or equal to] [2[theta]/[2 - [2.sup.p]]]||x||[.sup.p]

for all x [member of] X. Gavruta [1] generalized the Rassias' result: Let G be an abelian group and Y a Banach space. Denote by [phi]: G x G [right arrow] [0, [infinity]) a function such that

[~.[phi]](x, y) = [[infinity].summation over (j=0)] [2.sup.-j] [phi]([2.sup.j]x, [2.sup.j]y) < [infinity]

for all x, y [member of] G. Suppose that f : G [right arrow] Y is a mapping satisfying

||f(x + y) - f(x) - f(y)|| [less than or equal to] [phi](x, y)

for all x, y [member of] G. Then there exists a unique additive mapping T : G [right arrow] Y such that

||f(x) - T(x)|| [less than or equal to] [1/2][~.[phi]](x, x)

for all x [member of] G. Park [5] applied the Gavruta's result to linear functional equations in Banach modules over a C*-algebra.

Jun and Lee [2] proved the Cauchy-Rassias stability of Jensen's equation. C. Park and W. Park [7] applied the Jun and Lee's result to the Jensen's equation in Banach modules over a C*-algebra

Recently, Trif [9] proved the stability of a functional equation deriving from an inequality of Popoviciu for convex functions. And Park [6] applied the Trif's result to the Trif functional equation in Banach modules over a C*-algebra.

Throughout this paper, let A be a unital C*-algebra with norm ||dot||, and U(A) be the unitary group of A. Let l and d be integers with 2 [less than or equal to] l [less than or equal to] d - 1, and r a real number greater than 1.

In this paper, we prove that every almost linear almost *-derivation h : A [right arrow] A on a unital C*-algebra, JC*-algebra, or Lie C*-algebra A is a linear *-derivation when h(rx) = rh(x) (r > 1) for all x [member of] A, and prove the Cauchy-Rassias stability of linear *-derivations on unital C*-algebras, on unital JC*-algebras, or on unital Lie C*-algebras.

2. Linear *-Derivations on C*-Algebras

Throughout this section, assume that h(rx) = rh(x) for all x [member of] A.

We are going to investigate linear *-derivations on C*-algebras associated with the Cauchy functional equation.

Theorem 2.1 Let h : A [right arrow] A be a mapping for which there exists a function [phi] : [A.sup.2] [right arrow] [0, [infinity]) such that

[[infinity].summation over (j=0)] [r.sup.-j][phi]([r.sup.j]x, [r.sup.j]y) < [infinity], (2.i)

||h([mu]x + [mu]y) - [mu]h(x) - [mu]h(y)|| [less than or equal to] [phi](x, y), (2.ii)

||h([r.sup.n]u*) - h([r.sup.n]u)*|| [less than or equal to] [phi]([r.sup.n]u, [r.sup.n]u), (2.iii)

||h([r.sup.n]uy) - h([r.sup.n]u)y - [r.sup.n]uh(y)|| [less than or equal to] [phi]([r.sup.n]u, y) (2.iv)

for all [mu] [member of] [T.sup.1] [colon, equals] {[lambda] [member of] C | |[lambda]| = 1}, all u [member of] U(A), n = 0, 1,..., and all x, y [member of] A. Then the mapping h : A [right arrows] A is a C-linear *-derivation.

Proof. Since h(0) = rh(0), h(0) = 0. Put [mu] = 1 [member of] [T.sup.1] in (2.ii). By (2.ii) and the assumption that h(rx) = rh(x) for all x [member of] A,

||h(x + y) - h(x) - h(y)|| = [1/[r.sup.n]]||h([r.sup.n]x + [r.sup.n]y) - h([r.sup.n]x) - h([r.sup.n]y)||

[less than or equal to] [1/[r.sup.n]][phi]([r.sup.n]x, [r.sup.n]y)

which tends to zero as n [right arrow] [infinity] by (2.i). So

h(x + y) = h(x) + h(y) (2.1)

for all x, y [member of] A.

Put y = 0 in (2ii). By (2ii) and the assumption that h(rx) = rh(x) for all x [member of] A,

||h([mu]x) - [mu]h(x)|| = [1/[r.sup.n]]||h([r.sup.n][mu]x) - [mu]h([r.sup.n]x)|| [less than or equal to] [1/[r.sup.n]][phi]([r.sup.n]x, 0),

which tends to zero as n [right arrow] [infinity] by (2.i). So

h([mu]x) = [mu]h(x) (2.2)

for all [member of] A.

Now let [lambda] [member of] C ([lambda] [not equal to] 0) and M an integer greater than 4|[lambda]|. Then |[lambda]/M| < 1/4 < 1 - [2/3] = 1/3. By [3, Theorem 1], there exist three elements [[mu].sub.1], [[mu].sub.2], [[mu].sub.3] [member of] [T.sup.1] such that 3[[lambda]/M] = [[mu].sub.1] + [[mu].sub.2] + [[mu].sub.3]. Thus by (2.1) and (2.2)

h([lambda]x) = h([M/3] x 3[[lambda]/M]x) = M x h([1/3] x 3[[lambda]/M]x) =[M/3]h(3[[lambda]/M]x)

= [M/3]h([[mu].sub.1]x + [[mu].sub.2]x + [[mu].sub.3]x) = [M/3](h([[mu].sub.1]x) + h([[mu].sub.2]x) + h([[mu].sub.3]x))

= [M/3]([[mu].sub.1] + [[mu].sub.2] + [[mu].sub.3])h(x) = [M/3] x 3[[lambda]/M]h(x)

= [lambda]h(x)

for all x [member of] A. Hence

h([zeta]x + [eta]y) = h([zeta]x) + h([eta]y) = [zeta]h(x) + [eta]h(y)

for all [zeta], [eta] [member of] C([zeta], [eta] [not equal to] 0) and all x, y [member of] A. And h(0x) = 0 = 0h(x) for all x [member of] A. So the mapping h : A [right arrow] A is a C-linear mapping.

By (2.iii) and the assumption that h(rx) = rh(x) for all x [member of] A,

||h(u*) - h(u)*|| = [1/[r.sup.n]] ||h([r.sup.n]u*) - h([r.sup.n]u)*|| [less than or equal to] [1/[r.sup.n]][phi]([r.sup.n]u, [r.sup.n]u),

which tends to zero as n [right arrow] [infinity] by (2.i). So

h(u*) = h(u)*

for all u [member of] U(A). Since h : A [right arrow] A is C-linear and each x [member of] A is a finite linear combination of unitary elements (see [4, Theorem 4.1.7]), i.e., x [[summation].sub.j=1.sup.m] [[lambda].sub.j][u.sub.j] ([[lambda].sub.j] [member of] C, [u.sub.j] [member of] U(A)),

h(x*) = h([m.summation over (j=1)] [bar.[lambda].sub.j][u*.sub.j]) = [m.summation over (j=1)] [bar.[lambda].sub.j]h([u*.sub.j]) = [m.summation over (j=1)] [bar.[lambda].sub.j]h([u.sub.j])* = ([m.summation over (j=1)] [[lambda].sub.j]h([u.sub.j]))*

= h([m.summation over (j=1)] [[lambda].sub.j][u.sub.j])* = h(x)*

for all x [member of] A.

By (2.iv) and the assumption that h(rx) = rh(x) for all x [member of] A,

||h(uy) - h(u)y - uh(y)|| = [1/[r.sup.2n]]||h([r.sup.n]u x [r.sup.n]y) - h([r.sup.n]u)[r.sup.n]y - [r.sup.n]uh([r.sup.n]y)||

[less than or equal to] [1/[r.sup.2n]][phi]([r.sup.n]u, [r.sup.n]y) [less than or equal to] [1/[r.sup.n]][phi]([r.sup.n]u, [r.sup.n]y),

Which tends to zero as n [right arrow] [infinity] by (2.i), so

h(uy) = h(u)y + uh(y)

for all u [member of] U(A) and all y [member of] A. Since h : A [right arrow] A is C-linear and each x [member of] A is a finite linear combination of unitary elements, i.e., x [[summation].sub.j=1.sup.m] [[lambda].sub.j][u.sub.j] ([[lambda].sub.j] [member of] C, [u.sub.j] [member of] U(A)),

h(xy) = h([m.summation over (j=1)] [[lambda].sub.j][u.sub.j]y) = [m.summation over (j=1)] [[lambda].sub.j]h([u.sub.j]y) = [m.summation over (j=1)] [[lambda].sub.j](h([u.sub.j])y + [u.sub.j](y))

= h([m.summation over (j=1)] [[lambda].sub.j][u.sub.j])y + ([m.summation over (j=1)] [[lambda].sub.j][u.sub.j])h(y) = h(x)y + xh(y)

for all x, y [member of] A. Hence the mapping h : A [right arrow] A is a C-lincar *-derivation, as desired. [square]

Corollary 2.2. let h : A [right arrow] A be a mapping for which there exist constants [theta] [greater than or equal to] 0 and p [member of] [0, 1) such that

||h([mu]x + [mu]y) - [mu]h(x) -[mu]h(y)|| [less than or equal to] [theta](||x||[.sup.p] + ||y||[.sup.p]),

||h([r.sup.n]u*) - h([r.sup.n]u)*|| [less than or equal to] 2[r.sup.np][theta],

||h([r.sup.n]uy) - h([r.sup.n]u)y - [r.sup.n]uh(y)|| [less than or equal to] [theta]([r.sup.np] + ||y||[.sup.p])

for all [mu] [member of] [T.sup.1], all u [member of] U(A), n =0, 1,..., and all x, y [member of] A. Then the mapping h : A [right arrow] A is a C-linear *-derivalion.

Proof. Define [phi](x, y) = [theta](||x||[.sup.p] + ||y||[.sup.p]), and apply Theorem 2.1. [square]

Theorem 2.3. Let h: A [right arrow] A be a mapping for which there exists a function [phi] : [A.sup.2] [right arrow] [0, [infinity]) satisfying (2.i), (2.iii), and (2.iv) such that

||h([mu]x + [mu]y) - [mu]h(x) - [mu]h(y)|| [less than or equal to] [phi](x, y) (2.v)

for [mu] = 1, i, and all x, y [member of] A. If h(tx) is continuous in t [member of] R for each fixed x [member of] A, then the mapping h : A [right arrow] A is a C-linear *-derivation.

Proof. Put [mu] = 1 in (2.v). By the same reasoning as in the proof of Theorem 2.1, the mapping h : A [right arrow] A is additive. By the same reasoning as in the proof of [8, Theorem], the additive mapping h : A [right arrow] A is R-linear.

Put [mu] = i in (2.v). By the same method as in the proof of Theorem 2.1, one can obtain that

h(ix) = ih(x)

for all x [member of] A. For each element [lambda] [member of] C, [lambda] = s + it, where s, t [member of] R. So

h([lambda]x) = h(sx + itx) = sh(x) + th(ix) = sh(x) + ith(x) = (s + it)h(x)

= [lambda]h(x)

for all [lambda] [member of] C and all x [member of] A. So

h([zeta]x + [eta]y) = h([zeta]x) + h([eta]y) = [zeta]h(x) + [eta]h(y)

for all [zeta], [eta] [member of] C, and all x, y [member of] A. Hence the mapping h : A [right arrow] A is C-linear.

The rest of the proof is the same as in the proof of Theorem 2.1. [square]

Now we are going to investigate linear *-derivations on C*-algebras associated with the Jensen functional equation.

Theorem 2.4. Let h : A [right arrow] A be a mapping for which there exists a function [phi] : [A.sup.2] [right arrow] [0, [infinity]) satisfying (2.i), (2.iii), and (2.iv) such that

||2h([[mu]x| + [mu]y]/2) - [mu]h(x) - [mu]h(y)|| [less than or equal to] [phi](x, y), (2.vi)

for all [mu] [member of] [T.sup.1] and all x, y [member of] A. Then the mapping h : A [right arrow] A is a C-linear *-derivation.

Proof. Put [mu] = 1 [member of] [T.sup.1] in (2.vi). By (2.vi) and the assumption that h(rx) = rh(x) for all x [member of] A,

||2h([x + y]/2) - h(x) - h(y)|| = [1/[r.sup.n]]||2h([[r.sup.n]x + [r.sup.n]y]/2) - h([r.sup.n]x) - h ([r.sup.n]y)|| [less than or equal to] [1/[r.sup.n]][phi]([r.sup.n]x, [r.sup.n]y),

which tends to zero as n [right arrow] [infinity] by (2.i). So

2h([x + y]/2) = h(x) + h(y) (2.3)

for all x, y [member of] A. Let y = 0 in (2.3). Then 2h(x/2) = h(x) for all x [member of] A. Thus

h(x + y) = 2h([x + y]/2) = h(x) + h(y)

for all x, y [member of] A.

The rest of the proof is the same as in the proof of Theorem 2.1. [square]

One can obtain similar results to Corollary 2.2 and Theorem 2.3 for the Jensen functional equation.

We are going to investigate linear *-derivations on C*-algebras associated with the Trif functional equation.

Theorem 2.5. Let h : A [right arrow] A be a mapping for which there exists a function [phi] : [A.sup.d] [right arrow] [0, [infinity]) such that

[[infinity].summation over (j=0)] [r.sup.-j][phi]([r.sup.j][x.sub.1],..., [r.sup.j][x.sub.d]) < [infinity], (2.vii)

||d [.sub.d-2.C.sub.l-2]h([[mu][x.sub.1] + ... + [mu][x.sub.d]]/d) + [.sub.d-2.C.sub.l-1] [d.summation over (j=1)] [mu]h([x.sub.j]) -l [summation over (1[less than or equal to][j.sub.1] < ... < [j.sub.l][less than or equal to]d)] [mu]h([[x.sub.j.sub.1] + ... + [x.sub.j.sub.l]]/l)|| [less than or equal to] [phi]([x.sub.1],..., [x.sub.d]), (2.viii)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.ix)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all [mu] [member of] [T.sup.1], all u [member of] U(A), n = 0, 1,..., and all [x.sub.1],..., [x.sub.d] [member of] A. Then the mapping h : A [right arrow] A is a C-linear *-derivation.

Proof. The proof is similar to the proofs of Theorems 2.1 and 2.4. [square]

One can obtain similar results to Corollary 2.2 and Theorem 2.3 for the Trif functional equation.

3. Linear *-Derivations on JC*-Algebras and on Lie C*-Algebras

The original motivation to introduce the class of nonassociative algebras known as Jordan algebras came from quantum mechanics (see [?]). Let H be a complex Hilbert space, regarded as the "state space" of a quantum mechanical system. Let L(H) be the real vector space of all bounded self-adjoint linear operators on H, interpreted as the (bounded) observables of the system. In 1932, Jordan observed that L(H) is a (nonassociative) algebra via the anti-commutator product x [o] y [colon, equals] [[xy+yx]/2]. A commutative algebra X with product x [o] y is called a Jordan algebra if [x.sup.2] [o] (x [o] y) = x [o] ([x.sup.2] [o] y) holds.

A complex Jordan algebra C with product x [o] y and involution x [??] x* is called a JB*-algebra if C carries a Banach space norm ||dot|| satisfying ||x[o]y|| [less than or equal to] ||x||dot||y|| and ||{xx*x}|| = ||x||[.sup.3]. Here {xy*z} [colon, equals] x[o](y*[o]z)-y*[o](z[o]x)+z[o](x[o]y*) denotes the Jordan triple product of x, y, z [member of] C. A Jordan C*-subalgebra of a C*-algebra, endowed with the anticommutator product, is called a JC*-algebra.

Throughout this section, assume that h(rx) = rh(x) for all x [member of] A.

We are going to investigate linear *-derivations on JC*-algebras associated with the Cauchy functional equation.

Theorem 3.1. Let A be a unital JC*-algebra. Let h : A [right arrow] A be a mapping for which there exists a function [phi] : [A.sup.2] [right arrow] [0, [infinity]) satisfying (2.i), (2.ii), and (2.iii) such that

||h([r.sup.n]u [o] y) - h([r.sup.n]u) [o] y - [r.sup.n]u [o] h(y)|| [less than or equal to] [phi]([r.sup.n]u, y)

for all u [member of] U(A), n = 0, 1,..., and all y [member of] A. Then the mapping h : A [right arrow] A is a C-linear *-derivation.

Proof. The proof is similar to the proof of Theorem 2.1.

Corollary 3.2. Let A be a unital JC*-algebra. Let h : A [right arrow] A be a mapping for which there exist constants [theta] [greater than or equal to] 0 and p [member of] [0, 1) such that

||h([mu]x + [mu]y) - [mu]h(x) - [mu]h(y)|| [less than or equal to] [theta](||x||[.sup.p] + ||y||[.sup.p]),

||h([r.sup.n]u*) - h([r.sup.n]u)*|| [less than or equal to] 2[r.sup.np][theta],

||h([r.sup.n]u [o] y) - h([r.sup.n]u) [o] y - [r.sup.n]u [o] h(y)|| [less than or equal to] [theta]([r.sup.np] + ||y||[.sup.p])

for all [mu] [member of] [T.sup.1], all u [member of] U(A), n = 0, 1,..., and all x, y [member of] A. Then the mapping h : A [right arrow] A is a C-linear *-derivation.

Proof. Define [phi](x, y) = [theta](||x||[.sup.p] + ||y||[.sup.p]), and apply Theorem 3.1.

On JC*-algebras, one can obtain similar results to Theorems 2.3, 2.4, and 2.5.

A unital C*-algebra A, endowed with the Lie product [x, y] = xy - yx on A, is called a Lie C*-algebra. A C-linear mapping D on a Lie C*-algebra A is called a Lie derivation if D([x, y]) = [D(x), y] + [x, D(y)] holds for all x, y [member of] A.

Theorem 3.3. Let A be a unital Lie C*-algebra. Let h : A [right arrow] A be a mapping for which there exists a function [phi] : [A.sup.2] [right arrow] [0, [infinity]) satisfying (2.i), (2.ii), and (2.iii) such that

||h([[r.sup.n]u, y]) - [h([r.sup.n]u), y] - [[r.sup.n]u, h(y)]|| [less than or equal to] [phi]([r.sup.n]u, y)

for all u [member of] U(A), n = 0, 1,..., and all y [member of] A. Then the mapping h : A [right arrow] A is a C-linear *-derivation.

Proof. The proof is similar to the proof of Theorem 2.1. [square]

Corollary 3.4. Let A be a unital Lie C*-algebra. Let h : A [right arrow] A be a mapping for which there exist constants [theta] [greater than or equal to] 0 and p [member of] [0, 1) such that

||h([mu]x + [mu]y) - [mu]h(x) - [mu]h(y)|| [less than or equal to] [theta](||x||[.sup.p] + ||y||[.sup.p]),

||h([r.sup.n]u*) - h([r.sup.n]u)*|| [less than or equal to] 2[r.sup.np][theta],

||h([[r.sup.n]u, y]) - [h([r.sup.n]u), y] - [[r.sup.n]u, h(y)]|| [less than or equal to] [theta]([r.sup.np] + ||y||[.sup.p])

for all [mu] [member of] [T.sup.1], all u [member of] U(A), n = 0, 1,..., and all x, y [member of] A. Then the mapping h : A [right arrow] A is a C-linear *-derivation.

Proof. Define [phi](x, y) = [theta](||x||[.sup.p] + ||y||[.sup.p]), and apply Theorem 3.3. [square]

On Lie C*-algebras, one can obtain similar results to Theorems 2.3, 2.4, and 2.5.

4. Stability of Linear *-Derivations on C*-Algebras

We are going to show the Cauchy--Rassias stability of linear *-derivations on C*-algebras associated with the Cauchy functional equation.

Theorem 4.1. Let h : A [right arrow] A be a mapping with h(0) = 0 for which there exists a function [phi] : [A.sup.2] [right arrow] [0, [infinity]) such that

[~.[phi]](x, y) [colon, equals] [[infinity].summation over (j=0)] [2.sup.-j][phi]([2.sup.j]x, [2.sup.j]y) < [infinity], (4.i)

||h([mu]x + [mu]y) - [mu]h(x) - [mu]h(y)|| [less than or equal to] [phi](x, y), (4.ii)

||h([2.sup.n]u*) - h([2.sup.n]u)*|| [less than or equal to] [phi]([2.sup.n]u, [2.sup.n]u), (4.iii)

||h([2.sup.n]uy) - h([2.sup.n]u)y - [2.sup.n]uh(y)|| [less than or equal to] [phi]([2.sup.n]u, y) (4.iv)

for all [mu] [member of] [T.sup.1], all u [member of] U(A), n = 0, 1,..., and all x, y [member of] A. Then there exists a unique C-linear *-derivation D : A [right arrow] A such that

||h(x) - D(x)|| [less than or equal to] [1/2][~.[phi]](x, x) (4.v)

for all x [member of] A.

Proof. Put [mu] = 1 [member of] [T.sup.1] in (4.ii). It follows from Gavruta Theorem [?] that there exists a unique additive mapping D : A [right arrow] A satisfying the inequality (4.v). The additive mapping D : A [right arrow] A is given by

D(x) = [lim.[n[right arrow][infinity]]] [1/[2.sup.n]]h([2.sup.n]x) (4.1)

for all x [member of] A.

Put y = 0 in (4.ii). It follows from (4.ii) that

[1/[2.sup.n]]||h([2.sup.n][mu]x) - [mu]h([2.sup.n]x)|| [less than or equal to] [1/[2.sup.n]][phi]([2.sup.n]x, 0),

which tends to zero as n [right arrow] [infinity] by (4.i) for all [mu] [member of] [T.sup.1] and all x [member of] A. Hence

D([mu]x) = [lim.[n[right arrow][infinity]]] [[h([2.sup.n][mu]x)]/[2.sup.n]] = [lim.[n[right arrow][infinity]]] [[[mu]h([2.sup.n]x)]/[2.sup.n]] = [mu]D(x)

for all [mu] [member of] [T.sup.1] and all x [member of] A.

By the same method as in the proof of Theorem 2.1, one can show that the mapping D : A [right arrow] A is C-linear.

By (4.i) and (4.iii), we get

D(u*) = [lim.[n[right arrow][infinity]]] [[h([2.sup.n]u*)]/[2.sup.n]] = [lim.[n[right arrow][infinity]]] [[h([2.sup.n]u)*]/[2.sup.n]] = ([lim.[n[right arrow][infinity]]] [[h([2.sup.n]u)]/[2.sup.n]])* = D(u)*

for all u [member of] U(A). By the same method as the proof of Theorem 2.1, one can show that

D(x*) = D(x)*

for all x [member of] A.

It follows from (4.1) that

D(x) = [lim.[n[right arrow][infinity]]] [[h([2.sup.2n]x)]/[2.sup.2n]] (4.2)

for all x [member of] A. By (4.iv),

[1/[2.sup.2n]] ||h([2.sup.n]u x [2.sup.n]y) - h([2.sup.n]u)[2.sup.n]y - [2.sup.n]uh([2.sup.n]y)|| [less than or equal to] [1/[2.sup.2n]][phi]([2.sup.n]u, [2.sup.n]y)

[less than or equal to] [1/[2.sup.n]][phi]([2.sup.n]u, [2.sup.n]y) (4.3)

for all x, y [member of] A. By (4.i), (4.2), and (4.3),

D(uy) = [lim.[n[right arrow][infinity]]] [[h([2.sup.2n]uy)]/[2.sup.2n]] = [lim.[n[right arrow][infinity]]] [[h([2.sup.n]u x [2.sup.n]y)]/[2.sup.2n]] = [lim.[n[right arrow][infinity]]] ([1/[2.sup.n]]h([2.sup.n]u)y + u[1/[2.sup.n]]h([2.sup.n]y)) = D(u)y + uD(y)

for all u [member of] U(A) and all y [member of] A.

The rest of the proof is similar to the proof of Theorem 2.1. Hence the mapping D : A [right arrow] A is a C-linear *-derivation satisfying the inequality (4.v), as desired. [square]

Corollary 4.2. Let h : A [right arrow] A be a mapping with h(0) = 0 for which there exist constants [theta] [greater than or equal to] 0 and p [member of] [0, 1) such that

||h([mu]x + [mu]y) - [mu]h(x) - [mu]h(y)|| [less than or equal to] [theta](||x||[.sup.p] + ||y||[.sup.p]),

||h([2.sup.n]u*) - h([2.sup.n]u)*|| [less than or equal to] 2 x [2.sup.np][theta],

||h([2.sup.n]uy) - h([2.sup.n]u)y - [2.sup.n]uh(y)|| [less than or equal to] [theta]([2.sup.np] + ||y||[.sup.p])

for all [mu] [member of] [T.sup.1], all u [member of] U(A), n = 0, 1,..., and all x, y [member of] A. Then there exists a unique C-linear *-derivation D : A [right arrow] A such that

||h(x) - D(x)|| [less than or equal to] [2[theta]/[2 - [2.sup.p]]]||x||[.sup.p]

for all x [member of] A.

Proof. Define [phi](x, y) = [theta](||x||[.sup.p] + ||y||[.sup.p]), and apply Theorem 4.1. [square]

One can obtain a similar result to Theorem 2.3 for the Cauchy functional equation.

Now we are going to show the Cauchy-Rassias stability of linear *-derivations on C*-algebras associated with the Jensen functional equation.

Theorem 4.3. Let h : A [right arrow] A be a mapping with h(0) = 0 for which there exists a function [phi] : [A.sup.2] [right arrow] [0, [infinity]) such that

[~.[phi]](x, y) [colon, equals] [[infinity].summation over (j=0)] [3.sup.-j][phi]([3.sup.j]x, [3.sup.j]y) < [infinity], (4.vi)

||2h([[mu]x + [mu]y]/2) - [mu]h(x) - [mu]h(y)|| [less than or equal to] [phi](x, y), (4.vii)

||h([3.sup.n]u*) - h([3.sup.n]u)*|| [less than or equal to] [phi]([3.sup.n]u, [3.sup.n]u), (4.viii)

||h([3.sup.n]uy) - h([3.sup.n]u)y - [3.sup.n]uh(y)|| [less than or equal to] [phi]([3.sup.n]u, y) (4.ix)

for all [mu] [member of] [T.sup.1], all u [member of] U(A), n = 0, 1,..., and all x, y [member of] A \ {0}. Then there exists a unique C-linear *-derivation D : A [right arrow] A such that

||h(x) - D(x)|| [less than or equal to] [1/3]([~.[phi]](x, -x) + [~.[phi]](-x, 3x)) (4.x)

for all x [member of] A \ {0}.

Proof. Put [mu] = 1 [member of] [T.sup.1] in (4.vii). It follows from Jun and Lee Theorem [2, Theorem 1] that there exists a unique additive mapping D : A [right arrow] A satisfying the inequality (4.x). The additive mapping D : A [right arrow] A is given by

D(x) = [lim.[n[right arrow][infinity]]] [1/[3.sup.n]]h([3.sup.n]x)

for all x [member of] A.

The rest of the proof is similar to the proofs of Theorems 2.1 and 4.1. [square]

Corollary 4.4. Let h : A [right arrow] A be a mapping with h(0) = 0 for which there exist constants [theta] [greater than or equal to] 0 and p [member of] [0, 1) such that

||h([mu]x + [mu]y) - [mu]h(x) - [mu]h(y)|| [less than or equal to] [theta](||x||[.sup.p] + ||y||[.sup.p]),

||h([3.sup.n]u*) - h([3.sup.n]u)*|| [less than or equal to] 2 x [3.sup.np][theta],

||h([3.sup.n]uy) - h([3.sup.n]u)y - [3.sup.n]uh(y)|| [less than or equal to] [theta]([3.sup.np] + ||y||[.sup.p])

for all [mu] [member of] [T.sup.1], all u [member of] U(A), n = 0, 1,..., and all x, y [member of] A. Then there exists a unique C-linear *-derivation D : A [right arrow] A such that

||h(x) - D(x)|| [less than or equal to] [[3 + [3.sup.p]]/[3 - [3.sup.p]]][theta]||x||[.sup.p]

for all x [member of] A \ {0}.

Proof. Define [phi](x, y) = [theta](||x||[.sup.p] + ||y||[.sup.p]), and apply Theorem 4.3. [square]

One can obtain a similar result to Theorem 2.3 for the Jensen functional equation.

Now we are going to show the Cauchy-Rassias stability of linear *-derivations on C*-algebras associated with the Trif functional equation.

Theorem 4.5. Let q = [l(d-1)]/[d-l] and q' = -l/[d-l]. Let h : A [right arrow] A be a mapping with h(0) = 0 for which there exists a function [phi] : [A.sup.d] [right arrow] [0, [infinity]) such that

[~.[phi]]([x.sub.1],..., [x.sub.d]) [colon, equals] [[infinity].summation over (j=0)] [q.sup.-j][phi]([q.sup.j][x.sub.1],..., [q.sup.j][x.sub.d]) < [infinity], (4.xi)

||d[.sub.d-2.C.sub.l-2]h([[mu][x.sub.1] + ... + [mu][x.sub.d]]/d) + [.sub.d-2.C.sub.l-1] [d.summation over (j=1)] [mu]h([x.sub.j]) -l [summation over (1[less than or equal to][j.sub.1]< ... <[j.sub.l][less than or equal to]d)] [mu]h([[x.sub.j.sub.1] + ... + [x.sub.j.sub.l]]/l)|| [less than or equal to] [phi]([x.sub.1],..., [x.sub.d]), (4.xii)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], 4.xiii)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.xiv)

for all [mu] [member of] [T.sup.1], all u [member of] U(A), n = 0, 1,..., and all [x.sub.1],..., [x.sub.d] [member of] A. Then there exists a unique C-linear *-derivation D : A [right arrow] A such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.xv)

for all x [member of] A.

Proof. Put [mu] = 1 [member of] [T.sup.1] in (4.xii). It follows from Trif Theorem [9, Theorem 3.1] that there exists a unique additive mapping D : A [right arrow] A satisfying the inequality (4.xv). The additive mapping D : A [right arrow] A is given by

D(x) = [lim.[n[right arrow][infinity]]] [1/[q.sup.n]] h([q.sup.n]x)

for all x [member of] A.

The rest of the proof is similar to the proofs of Theorems 2.1 and 4.1. [square]

Corollary 4.6. Let q = [l(d-1)]/[d-l]. Let h : A [right arrow] A be a mapping with h(0) = 0 for which there exist constants [theta] [greater than or equal to] 0 and p [member of] [0, 1) such that

||d [.sub.d-2.C.sub.l-2]h([[mu][x.sub.1] + ... + [mu][x.sub.d]]/d) + [.sub.d-2.C.sub.l-1][d.summation over (j=1)] [mu]h([x.sub.j]) -l [summation over (1[less than or equal to][j.sub.1]< ... <[j.sub.l][less than or equal to]d)] [mu]h([[x.sub.j.sub.1] + ... + [x.sub.j.sub.l]]/l)|| [less than or equal to] [theta]([d.summation over (j=1)] ||[x.sub.j]||[.sup.p]),

||h([q.sup.n]u*) - h([q.sup.n]u)*|| [less than or equal to] d[q.sup.np][theta],

||h([q.sup.n]uy) - h([q.sup.n]u)y - [q.sup.n]uh(y)|| [less than or equal to] [theta]([q.sup.np] + (d - 1)||y||[.sup.p])

for all [mu] [member of] [T.sup.1], all u [member of] U(A), n = 0, 1,..., and all [x.sub.1],..., [x.sub.d] [member of] A. Then there exists a unique C-linear *-derivation D : A [right arrow] A such that

||h(x) - D(x)|| [less than or equal to] [[[q.sup.1-p]([q.sup.p] + (d - 1)[r.sup.p])[theta]]/[l [.sub.d-1.C.sub.l-1]([q.sup.1-p] - 1)]]||x||[.sup.p]

for all x [member of] A.

Proof. Define [phi]([x.sub.1],..., [x.sub.d]) = [theta]([[summation].sub.j=1.sup.d] ||[x.sub.j]||[.sup.p]), and apply Theorem 4.5. [square]

One can obtain a similar result to Theorem 2.3 for the Trif functional equation.

5. Stability of Linear *-Derivations on JC*-Algebras and On Lie C*-Algebras

We are going to show the Cauchy-Rassias stability of linear *-derivations on JC*-algebras associated with the Cauchy functional equation.

Theorem 5.1. Let A be a unital JC*-algebra. Let h : A [right arrow] A be a mapping with h(0) = 0 for which there exists a function [phi] : [A.sup.2] [right arrow] [0,1) satisfying (4.i), (4.ii), and (4.iii) such that

||h([2.sup.n]u [o] y) - h([2.sup.n]u) [o] y - [2.sup.n]u [o] h(y)|| [less than or equal to] [phi]([2.sup.n]u, y)

for all u [member of] U(A), n = 0, 1,..., and all y [member of] A. Then there exists a unique C-linear *-derivation D : A [right arrow] A satisfying the inequality (4.v).

Proof. The proof is similar to the proofs of Theorems 2.1 and 4.1. [square]

Corollary 5.2. Let A be a unital JC*-algebra. Let h : A [right arrow] A be a mapping with h(0) = 0 for which there exist constants [theta] [greater than or equal to] 0 and p [member of] [0, 1) such that

||h([mu]x + [mu]y) - [mu]h(x) - [mu]h(y)|| [less than or equal to] [theta](||x||[.sup.p] + ||y||[.sup.p]),

||h([2.sup.n]u*) - h([2.sup.n]u)*|| [less than or equal to] 2 x [2.sup.np][theta],

||h([2.sup.n]u [o] y) - h([2.sup.n]u) [o] y - [2.sup.n]u [o] h(y)|| [less than or equal to] [theta]([2.sup.np] + ||y||[.sup.p])

for all [mu] [member of] [T.sup.1], all u [member of] U(A), n = 0, 1,..., and all x, y [member of] A. Then there exists a unique C-linear *-derivation D : A [right arrow] A such that

||h(x) - D(x)|| [less than or equal to] [2[theta]/[2 - [2.sup.p]]]||x||[.sup.p]

for all x [member of] A.

Proof. Define [phi](x, y) = [theta](||x||[.sup.p] + ||y||[.sup.p]), and apply Theorem 5.1. [square]

On JC*-algebras, one can obtain similar results to Theorems 4.3 and 4.5, and Corollaries 4.4 and 4.6.

Now we are going to show the Cauchy-Rassias stability of linear *-derivations on Lie C*-algebras associated with the Cauchy functional equation.

Theorem 5.3. Let A be a unital Lie C*-algebra. Let h : A [right arrow] A be a mapping with h(0) = 0 for which there exists a function [phi] : [A.sup.2] [right arrow] [0, [infinity]) satisfying (4.i), (4.ii), and (4.iii) such that

||h([[2.sup.n]u, y]) - [h([2.sup.n]u), y] - [[2.sup.n]u, h(y)]|| [less than or equal to] [phi]([2.sup.n]u, y)

for all u [member of] U(A), n = 0, 1,..., and all y [member of] A. Then there exists a unique C-linear *-derivation D : A [right arrow] A satisfying the inequality (4.v).

Proof. The proof is similar to the proofs of Theorems 2.1 and 4.1. [square]

Corollary 5.4. Let A be a unital Lie C*-algebra. Let h : A [right arrow] A be a mapping with h(0) = 0 for which there exist constants [theta] [greater than or equal to] 0 and p [member of] [0, 1) such that

||h([mu]x + [mu]y) - [mu]h(x) - [mu]h(y)|| [less than or equal to] [theta](||x||[.sup.p] + ||y||[.sup.p]),

||h([2.sup.n]u*) - h([2.sup.n]u)*|| [less than or equal to] 2 x [2.sup.np][theta],

||h([[2.sup.n]u, y]) - [h([2.sup.n]u), y] - [[2.sup.n]u, h(y)]|| [less than or equal to] [theta]([2.sup.np] + ||y||[.sup.p])

for all [mu] [member of] [T.sup.1], all u [member of] U(A), n = 0, 1,..., and all x, y [member of] A. Then there exists a unique C-linear *-derivation D : A [right arrow] A such that

||h(x) - D(x)|| [less than or equal to] [2[theta]/[2-[2.sup.p]]]||x||[.sup.p]

for all x [member of] A.

Proof. Define [psi](x, y) = [theta](||x||[.sup.p] + ||y||[.sup.p]), and apply Theorem 5.3. [square]

On Lie C*-algebras, one can obtain similar results to Theorems 4.3 and 4.5, and Corollaries 4.4 and 4.6.

References

[1] P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431-436.

[2] K. Jun and Y. Lee, A generalization of the Hyers-Ulam-Rassias stability of Jensen's equation, J. Math. Anal. Appl. 238 (1999), 305-315.

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

[4] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Elementary Theory, New York, 1983.

[5] C. Park, On the stability of the linear mapping in Banach modules, J. Math. Anal. Appl. 275 (2002), 711-720.

[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 and W. Park, On the Jensen's equation in Banach modules, Taiwanese J. Math. 6 (2002), 523-531.

[8] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.

[9] 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.

[10] H. Upmeier, Jordan Algebras in Analysis, Operator Theory and Quantum Mechanics, Regional Conference Series in Mathematics, Amer. Math. Soc. 67 (1987), Providence.

Chun-Gil Park ([dagger])

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

Received July 29, 2005, Accepted October 5, 2005.

*2000 mathematics Subject classification. 47C10, 39B52, 17Cxx, 46L05

([dagger]) E-mail: cgparkmath.cnu.ac.kr
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