AUTOMORPHISMS ON A C -ALGEBRA AND ISOMORPHISMS BETWEEN LIE JC -ALGEBRAS ASSOCIATED WITH A GENERALIZED ADDITIVE MAPPING

Houston Journal of Mathematics c 2007 University of Houston Volume, No., 2007 AUTOMORPHISMS ON A C -ALGEBRA AND ISOMORPHISMS BETWEEN LIE JC -ALGEBRAS ASSOCIATED WITH A GENERALIZED ADDITIVE MAPPING CHOONKIL PARK Communicated by Vern I. Paulsen Abstract. Let X, Y be vector spaces, and let r be or. It is shown that if an odd mapping f : X Y satisfies the functional equation P d j= rf( x P j X d j= ) + rf( ( )ι(j) x j ) r ι(j)=0,, P r d j= ι(j)=l (0.) = ( d C l d C l + ) dx f(x j ), then the odd mapping f : X Y is additive, and we prove the Cauchy Rassias stability of the functional equation (0.) in Banach modules over a unital C -algebra. As an application, we show that every almost linear bijective mapping h : A A on a unital C -algebra A is an automorphism when h( n uy) = h( n u)h(y) for all unitaries u A, all y A and all n Z, and that every almost linear bijective mapping h : A B of a unital Lie JC -algebra A onto a unital Lie JC -algebra B is a Lie JC -algebra isomorphism when h( n u y) = h( n u) h(y) for all y A, all unitaries u A and all n Z. j= 2000 Mathematics Subject Classification. 9B52, 46L05, 47B48, 7B40. Key words and phrases. Banach module over C -algebra, Cauchy Rassias stability, linear functional equation, automorphism on C -algebra, Lie JC -algebra isomorphism. Supported by Korea Research Foundation Grant KRF-2005-070-C00009. The author would like to thank the referees for a number of valuable suggestions regarding a previous version of this paper. 85

86 C. PARK. 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, ) 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 for all x X. Găvruta [] generalized 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) = 2 j ϕ(2 j x, 2 j y) < for all x, y G. Suppose that f : G Y is a mapping satisfying j=0 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) ϕ(x, x) 2 for all x G. C. Park [6] applied Găvruta s result to linear functional equations in Banach modules over a C -algebra. Several functional equations have been investigated in [8] [5]. Throughout this paper, assume that r is or, and that d and l are integers with < l < d 2. Let dc l := d! l!(d l)!. In this paper, we solve the following functional equation d j= rf( x j d j= ) + ( )ι(j) x j ) r r (.) ι(j)=0,, P d j= ι(j)=l rf( = ( d C l d C l + ) f(x j ), which is called a generalized additive functional equation and every solution of the functional equation (.) is said to be a generalized additive mapping. Moreover, we prove the Cauchy Rassias stability of the functional equation (.) in Banach modules over a unital C -algebra. This result is applied to investigate j=

GENERALIZED ADDITIVE MAPPING 87 automorphisms on a unital C -algebra. We moreover prove that every almost linear bijective mapping h : A B of a unital Lie JC -algebra A onto a unital Lie JC -algebra B is a Lie JC -algebra isomorphism when h( n u y) = h( n u) h(y) for all y A, all unitaries u A and all n Z. 2. A generalized additive mapping in linear spaces Throughout this section, assume that X and Y are linear spaces. Lemma 2.. If an odd mapping f : X Y satisfies (.) for all x, x 2,, x d X, then f is Cauchy additive. Proof. Note that f(0) = 0 and f( x) = f(x) for all x X since f is an odd mapping. Putting x = x, x 2 = y and x = = x d = 0 in (.), we get (2.) d 2C l d 2 C l 2 + )rf( x + y ) r = ( d C l d C l + )(f(x) + f(y)) for all x, y X. Since d 2 C l d 2 C l 2 + = d C l d C l +, rf( x + y ) = f(x) + f(y) r for all x, y X. Letting y = 0 in (2.), we get rf( x r ) = f(x) for all x X. Hence f(x + y) = rf( x + y ) = f(x) + f(y) r for all x, y, X. Thus f is Cauchy additive.. Stability of generalized additive mappings in Banach modules over a unital 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 j= D u f(x,, x d ) : = rf( ux j ) + r ( d C l d C l + ) for all u U(A) and all x,, x d X. ι(j)=0,, P d j= ι(j)=l rf( uf(x j ) j= d j= ( )ι(j) ux j ) r

88 C. PARK Theorem.. Let r =, and f : X Y an odd mapping for which there is a function ϕ : X d [0, ) such that (.) ϕ(x,, x d ) : = j ϕ(j x,, j x d ) <, (.2) j=0 D u f(x,, x d ) ϕ(x,, x d ) for all u U(A) and all x,, x d X. Then there exists a unique A-linear generalized additive mapping L : X Y such that (.) for all x X. f(x) L(x) ϕ(x, x, x, 0,, 0 ) ( d C l d C l + ) }{{} Proof. Note that f(0) = 0 and f( x) = f(x) for all x X since f is an odd mapping. Let u = U(A). Putting x = x 2 = x = x and x 4 = = x d = 0 in (.2), we have f(x) f(x) ϕ(x, x, x, 0,, 0 ) d C l d C l + }{{} for all x X. So we get f(x) f(x) ϕ(x, x, x, 0,, 0 ) ( d C l d C l + ) }{{} for all x X. Hence n f(n x) (.4) n+ f(n+ x) n+ ( d C l d C l + ) ϕ(n x, n x, n x, 0,, 0 ) }{{} for all x X and all positive integers n. By (.4), we have m f(m x) n f(n x) n (.5) k+ ( d C l d C l + ) ϕ(k x, k x, k x, 0,, 0 ) }{{} k=m for all x X and all positive integers m and n with m < n. This shows that the sequence { n f( n x)} is a Cauchy sequence for all x X. Since Y is complete,

GENERALIZED ADDITIVE MAPPING 89 the sequence { f( n x)} converges for all x X. So we can define a mapping n L : X Y by L(x) := lim n f(n x) 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 D L(x,, x d ) = lim lim n D f( n x,, n x d ) n ϕ(n x,, n x d ) = 0 for all x,, x d X. So L is a generalized additive mapping. Putting m = 0 and letting n in (.5), we get (.). Now, let L : X Y be another generalized additive mapping satisfying (.). By Lemma 2., L and L are additive. So we have L(x) L (x) = n L(n x) L ( n x) n ( L(n x) f( n x) + L ( n x) f( n x) ) 2 n+ ( d C l d C l + ) ϕ(n x, n x, n x, 0,, 0 ), }{{} 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 for all x X. So D u L(x, 0,, 0 ) = lim }{{} n D uf( n x, 0,, 0 ) }{{} d times d times lim n ϕ(n x, 0,, 0 ) = 0 }{{} d times (.6) L(ux) = ul(x) 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 M < 4 < 2 =. By Theorem of [2], there exist three elements u, u 2, u U(A) such

820 C. PARK that a M = u + u 2 + u. So by (.6) L(ax) = L( M a M x) = M L( a M x) = M L( a M x) = M L(u x + u 2 x + u x) = M (L(u x) + L(u 2 x) + L(u x)) = M (u + u 2 + u )L(x) = M a M L(x) = al(x) 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. Corollary.2. Let θ and p < be positive real numbers. f : X Y an odd mapping such that D u f(x,, x d ) θ x j p j= Let r =, and for all u U(A) and all x,, x d X. Then there exists a unique A-linear generalized additive mapping L : X Y such that for all x X. f(x) L(x) θ ( p )( d C l d C l + ) x p Proof. Define ϕ(x,, x d ) = θ d j= x j p, and apply Theorem.. Theorem.. Let r =, and f : X Y an odd mapping for which there is a function ϕ : X d [0, ) satisfying (.2) such that (.7) ϕ(x,, x d ) := j ϕ( x j,, x d j ) < j= for all x,, x d X. Then there exists a unique A-linear generalized additive mapping L : X Y such that f(x) L(x) ϕ(x, x, x, 0,, 0 ) ( d C l d C l + ) }{{} for all x X.

GENERALIZED ADDITIVE MAPPING 82 Proof. Note that f(0) = 0 and f( x) = f(x) for all x X since f is an odd mapping. Let u = U(A). Putting x = x 2 = x = x and x 4 = = x d = 0 in (.2), we have f(x) f(x) ϕ(x, x, x, 0,, 0 ) d C l d C l + }{{} for all x X. So we get f(x) f( x ) d C l d C l + ϕ(x, x, x, 0, } {{, 0 ) } for all x X. The rest of the proof is similar to the proof of Theorem.. Corollary.4. Let θ and p > be positive real numbers. f : X Y an odd mapping such that D u f(x,, x d ) θ x j p j= Let r =, and for all u U(A) and all x,, x d X. Then there exists a unique A-linear generalized additive mapping L : X Y such that for all x X. f(x) L(x) θ ( p )( d C l d C l + ) x p Proof. Define ϕ(x,, x d ) = θ d j= x j p, and apply Theorem.. Theorem.5. Let r =, and f : X Y an odd mapping for which there is a function ϕ : X d [0, ) satisfying (.2) and (.7). Then there exists a unique A-linear generalized additive mapping L : X Y such that f(x) L(x) ϕ(x, x, x, 0,, 0 ) ( d C l d C l 2 ) }{{} 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 = U(A). Putting x = x 2 = x = x and x 4 = = x d = 0 in (.2), we have f( x ) f(x) ϕ(x, x, x, 0,, 0 ) 9( d C l d C l 2 ) }{{}

822 C. PARK for all x X. So we get f(x) f( x ) ϕ(x, x, x, 0,, 0 ) ( d C l d C l 2 ) }{{} for all x X. The rest of the proof is similar to the proof of Theorem.. Corollary.6. Let θ and p > be positive real numbers. f : X Y an odd mapping such that D u f(x,, x d ) θ x j p j= Let r =, and for all u U(A) and all x,, x d X. Then there exists a unique A-linear generalized additive mapping L : X Y such that θ f(x) L(x) ( p )( d C l d C l 2 ) x p for all x X. Proof. Define ϕ(x,, x d ) = θ d j= x j p, and apply Theorem.5. Theorem.7. Let r =, and f : X Y an odd mapping for which there is a function ϕ : X d [0, ) satisfying (.) and (.2). Then there exists a unique A-linear generalized additive mapping L : X Y such that f(x) L(x) ϕ(x, x, x, 0,, 0 ) ( d C l d C l 2 ) }{{} 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 = U(A). Putting x = x 2 = x = x and x 4 = = x d = 0 in (.2), we have f( x ) f(x) ϕ(x, x, x, 0,, 0 ) 9( d C l d C l 2 ) }{{} for all x X. So we get f(x) f(x) ϕ(x, x, x, 0,, 0 ) 9( d C l d C l 2 ) }{{} for all x X. The rest of the proof is similar to the proof of Theorem..

GENERALIZED ADDITIVE MAPPING 82 Corollary.8. Let θ and p < be positive real numbers. f : X Y an odd mapping such that D u f(x,, x d ) θ x j p j= Let r =, and for all u U(A) and all x,, x d X. Then there exists a unique A-linear generalized additive mapping L : X Y such that θ f(x) L(x) ( p )( d C l d C l 2 ) x p for all x X. Proof. Define ϕ(x,, x d ) = θ d j= x j p, and apply Theorem.7. 4. Automorphisms on a unital C -algebra Throughout this section, assume that A is a unital C -algebra with norm, unit e and unitary group U(A). We are going to investigate automorphisms on a unital C -algebra. Theorem 4.. Let r =, and h : A A an odd bijective mapping satisfying h( n uy) = h( n u)h(y) for all u U(A), all y A, and all n Z, for which there exists a function ϕ : A d [0, ) such that (4.) (4.2) (4.) ϕ(x,, x d ) : = j=0 D µ h(x,, x d ) ϕ(x,, x d ), h( n u ) h( n u) ϕ( n u,, n u) }{{} j ϕ(j x,, j x d ) <, for all µ T : = {λ C λ = }, all u U(A), n = 0,, 2,, and all x,, x d A. Assume that (4.4) lim n h(n e) is invertible. Then the odd bijective mapping h : A A is an automorphism. Proof. We can consider the C -algebra A as a left Banach module over the unital C -algebra C. By Theorem., there exists a unique C-linear generalized additive mapping H : A A such that (4.5) h(x) H(x) ϕ(x, x, x, 0,, 0 ) ( d C l d C l + ) }{{}

824 C. PARK for all x A. The generalized additive mapping H : A A is given by H(x) = lim n h(n x) for all x A. By (4.) and (4.), we get H(u ) = lim = H(u) n h(n u ) = lim n h(n u) = ( lim n h(n u)) for all u U(A). Since H is C-linear and each x A is a finite linear combination of unitary elements (cf. []), i.e., x = m j= λ ju j (λ j C, u j U(A)), H(x ) = H( λ j u j ) = λ j H(u j ) = λ j H(u j ) = ( λ j H(u j )) j= j= = H( λ j u j ) = H(x) j= for all x A. Since h( n uy) = h( n u)h(y) for all u U(A), all y A, and all n Z, (4.6) H(uy) = lim n h(n uy) = lim n h(n u)h(y) = H(u)h(y) for all u U(A) and all y A. By the additivity of H and (4.6), j= n H(uy) = H( n uy) = H(u( n y)) = H(u)h( n y) for all u U(A) and all y A. Hence (4.7) H(uy) = n H(u)h(n y) = H(u) n h(n y) for all u U(A) and all y A. Taking the limit in (4.7) as n, we obtain (4.8) H(uy) = H(u)H(y) 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= λ ju j (λ j C, u j U(A)), it follows from (4.8) that H(xy) = H( λ j u j y) = λ j H(u j y) = λ j H(u j )H(y) j= j= = H( λ j u j )H(y) = H(x)H(y) j= j= j=

GENERALIZED ADDITIVE MAPPING 825 for all x, y A. By (4.6) and (4.8), H(e)H(y) = H(ey) = H(e)h(y) for all y A. Since lim n h( n e) = H(e) is invertible, H(y) = h(y) for all y A. Therefore, the odd bijective mapping h : A A is an automorphism. Corollary 4.2. Let θ and p < be positive real numbers. Let r =, and h : A A an odd bijective mapping satisfying h( n uy) = h( n u)h(y) for all u U(A), all y A, and all n Z, such that D µ h(x,, x d ) θ x j p, j= h( n u ) h( n u) d pn θ for all µ T, all u U(A), n = 0,, 2,, and all x,, x d A. Assume that lim h( n e) is invertible. Then the odd bijective mapping h : A A is an n automorphism. Proof. Define ϕ(x,, x d ) = θ d j= x j p, and apply Theorem 4.. Theorem 4.. Let r =, and h : A A an odd bijective mapping satisfying h( n uy) = h( n u)h(y) for all u U(A), all y A, and all n Z, for which there exists a function ϕ : A d [0, ) such that (4.9) j ϕ( x j,, x d j ) <, (4.0) (4.) j= D µ h(x,, x d ) ϕ(x,, x d ), h( u n ) h( u n ) ϕ( u n,, u }{{ n ) } for all µ T, all u U(A), n = 0,, 2,, and all x,, x d A. Assume that (4.2) lim n h( e ) is invertible. n Then the odd bijective mapping h : A A is an automorphism. Proof. The proof is similar to the proof of Theorem 4..

826 C. PARK Corollary 4.4. Let θ and p > be positive real numbers. Let r =, and h : A A an odd bijective mapping satisfying h( n uy) = h( n u)h(y) for all u U(A), all y A, and all n Z, such that D µ h(x,, x d ) θ h( u n ) h( u n ) x j p, j= d pn θ for all µ T, all u U(A), n = 0,, 2,, and all x,, x d A. Assume that lim n h( e ) is invertible. Then the odd bijective mapping h : A A is n an automorphism. Proof. Define ϕ(x,, x d ) = θ d j= x j p, and apply Theorem 4.. Theorem 4.5. Let r =, and h : A A an odd bijective mapping satisfying h( n uy) = h( n u)h(y) for all u U(A), all y A, and all n Z, for which there exists a function ϕ : A d [0, ) satisfying (4.9), (4.0), (4.) and (4.2). Then the odd bijective mapping h : A A is an automorphism. Proof. We can consider the C -algebra A as a left Banach module over the unital C -algebra C. By Theorem.5, there exists a unique C-linear generalized additive mapping H : A A such that h(x) H(x) ϕ(x, x, x, 0,, 0 ) ( d C l d C l 2 ) }{{} for all x A. The generalized additive mapping H : A A is given by H(x) = lim n h( x n ) for all x A. The rest of the proof is similar to the proof of Theorem 4.. Corollary 4.6. Let θ and p > be positive real numbers. Let r =, and h : A A an odd bijective mapping satisfying h( n uy) = h( n u)h(y) for all u U(A), all y A, and all n Z, such that D µ h(x,, x d ) θ h( u n ) h( u n ) x j p, j= d pn θ

GENERALIZED ADDITIVE MAPPING 827 for all µ T, all u U(A), n = 0,, 2,, and all x,, x d A. Assume that lim n h( e ) is invertible. Then the odd bijective mapping h : A A is n an automorphism. Proof. Define ϕ(x,, x d ) = θ d j= x j p, and apply Theorem 4.5. Theorem 4.7. Let r =, and h : A A an odd bijective mapping satisfying h( n uy) = h( n u)h(y) for all u U(A), all y A, and all n Z, for which there exists a function ϕ : A d [0, ) satisfying (4.), (4.2), (4.) and (4.4). Then the odd bijective mapping h : A A is an automorphism. Proof. The proof is similar to the proofs of Theorems 4. and 4.5. Corollary 4.8. Let θ and p < be positive real numbers. Let r =, and h : A A an odd bijective mapping satisfying h( n uy) = h( n u)h(y) for all u U(A), all y A, and all n Z, such that D µ h(x,, x d ) θ x j p, j= h( n u ) h( n u) d pn θ for all µ T, all u U(A), n = 0,, 2,, and all x,, x d A. Assume that lim h( n e) is invertible. Then the odd bijective mapping h : A A is an n automorphism. Proof. Define ϕ(x,, x d ) = θ d j= x j p, and apply Theorem 4.7. 5. Isomorphisms between Lie JC -algebras The original motivation for introducing the class of nonassociative algebras known as Jordan algebras came from quantum mechanics (see [6]). 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 92, Jordan observed that L(H) is a (nonassociative) algebra via the anticommutator product x y := xy+yx 2. A commutative algebra X with product x y is called a Jordan algebra. A unital Jordan C -subalgebra of a C -algebra, endowed with the anticommutator product, is called a JC -algebra. A unital C -algebra C, endowed with the Lie product [x, y] = xy yx 2 on C, is called a Lie C -algebra. A unital C -algebra C, endowed with the Lie product [, ] and the anticommutator product, is called a Lie JC -algebra if (C, ) is a JC -algebra and (C, [, ]) is a Lie C -algebra (see [4], [5]).

828 C. PARK Throughout this paper, let A be a unital Lie JC -algebra with norm, unit e and unitary group U(A) = {u A uu = u u = e}, and B a unital Lie JC -algebra with norm and unit e. Definition. A C-linear bijective mapping H : A B is called a Lie JC - algebra isomorphism if H : A B satisfies for all x, y A. H(x y) = H(x) H(y), H([x, y]) = [H(x), H(y)], H(x ) = H(x) Remark. A C-linear mapping H : A B is a C -algebra isomorphism if and only if the mapping H : A B is a Lie JC -algebra isomorphism. Assume that H is a Lie JC -algebra isomorphism. Then H(xy) = H([x, y] + x y) = H([x, y]) + H(x y) = [H(x), H(y)] + H(x) H(y) = H(x)H(y) for all x, y A. So H is a C -algebra isomorphism. Assume that H is a C -algebra isomorphism. Then xy yx H(x)H(y) H(y)H(x) H([x, y] = H( ) = 2 2 xy + yx H(x)H(y) + H(y)H(x) H(x y) = H( ) = 2 2 for all x, y A. So H is a Lie JC -algebra isomorphism. = [H(x), H(y)], = H(x) H(y) Given a mapping h : X Y, we set F µ h(x,, x d, z, w) : = rh( ( d j= µx j) + [z, w] ) r ( d + ι(j)=0,, P d j= ι(j)=l rh( ( d C l d C l + )( j= ( )ι(j) µx j ) + [z, w] r µh(x j ) [h(z), h(w)]) for all µ T and all x,, x d, z, w A. We investigate Lie JC -algebra isomorphisms between Lie JC -algebras. j= )

GENERALIZED ADDITIVE MAPPING 829 Theorem 5.. Let r =, and h : A B an odd bijective mapping satisfying h( n u y) = h( n u) h(y) for all y A, all u U(A) and all n Z, for which there exists a function ϕ : A d+2 [0, ) such that (5.) (5.2) (5.) ϕ(x,, x d, z, w) : = j=0 F µ h(x,, x d, z, w) ϕ(x,, x d, z, w), h( n u ) h( n u) ϕ( n u,, n u, 0, 0) }{{} j ϕ(j x,, j x d, j z, j w) <, for all µ T, all u U(A), all x,, x d, z, w A and n = 0,, 2,. Assume that h( n e) (5.4) lim n = e. Then the bijective mapping h : A B is a Lie JC -algebra isomorphism. Proof. Put z = w = 0 and µ = T in (5.2). By the same reasoning as in the proofs of Theorems. and 4., there exists a unique C-linear generalized additive involutive mapping H : A B such that h(x) H(x) ϕ(x, x, x, 0,, 0 ) ( d C l d C l + ) }{{} d times for all x A. The generalized additive mapping H : A B is given by (5.5) H(x) = lim n h(n x) for all x A. Since h( n u y) = h( n u) h(y) for all y A, all u U(A) and all n Z, (5.6) H(u y) = lim = H(u) h(y) n h(n u y) = lim n h(n u) h(y) for all y A and all u U(A). By the additivity of H and (5.6), n H(u y) = H( n u y) = H(u ( n y)) = H(u) h( n y) for all y A and all u U(A). Hence (5.7) H(u y) = n H(u) h(n y) = H(u) n h(n y) for all y A and all u U(A). Taking the limit in (5.7) as n, we obtain (5.8) H(u y) = H(u) H(y)

80 C. PARK for all y A and all u U(A). Since H is C-linear and each x A is a finite linear combination of unitary elements, i.e., x = m j= λ ju j (λ j C, u j U(A)), H(x y) = H( λ j u j y) = λ j H(u j y) = λ j H(u j ) H(y) j= j= = H( λ j u j ) H(y) = H(x) H(y) j= for all x, y A. By (5.4), (5.6) and (5.8), for all y A. So for all y A. It follows from (5.5) that (5.9) j= H(y) = H(e y) = H(e) h(y) = e h(y) = h(y) H(y) = h(y) h( 2n x) H(x) = lim for all x A. Let µ = and x = = x d = 0 in (5.2). Then we get ( d C l d C l + )(h([z, w]) [h(z), h(w)]) ϕ(0,, 0, z, w) }{{} for all z, w A. So (5.0) 2n h([n z, n w]) [h( n z), h( n w)] 2n ( d C l d C l + ) ϕ(0, } {{, 0, n z, n w) } 2n n ( d C l d C l + ) ϕ(0, } {{, 0, n z, n w) } for all z, w A. By (5.), (5.9), and (5.0), H([z, w]) = h( 2n [z, w]) lim for all z, w A. = lim = [H(z), H(w)] 2n = lim h([ n z, n w]) 2n 2n [h(n z), h( n z) w)] = lim [h(n n, h(n w) n ]

GENERALIZED ADDITIVE MAPPING 8 Therefore, the bijective mapping h : A B is a Lie JC -algebra isomorphism, as desired. Corollary 5.2. Let θ and p < be positive real numbers. Let r =, and h : A B an odd bijective mapping satisfying h( n u y) = h( n u) h(y) for all u U(A), all y A, and all n Z, such that F µ h(x,, x d, z, w) θ( x j p + z p + w p ), j= h( n u ) h( n u) d pn θ for all µ T, all u U(A), n = 0,, 2,, and all x,, x d, z, w A. Assume that lim h( n e) = e. Then the odd bijective mapping h : A B is a Lie n JC -algebra isomorphism. Proof. Define ϕ(x,, x d, z, w) = θ( d j= x j p + z p + w p ), and apply Theorem 5.. Theorem 5.. Let r =, and h : A B an odd bijective mapping satisfying h( n u y) = h( n u) h(y) for all y A, all u U(A) and all n Z, for which there exists a function ϕ : A d+2 [0, ) satisfying (5.), (5.2), (5.) and (5.4). Then the bijective mapping h : A B is a Lie JC -algebra isomorphism. Proof. Put z = w = 0 and µ = T in (5.2). By the same reasoning as in the proofs of Theorems.,.7 and 4., there exists a unique C-linear generalized additive involutive mapping H : A B such that h(x) H(x) ϕ(x, x, x, 0,, 0 ) ( d C l d C l 2 ) }{{} d times for all x A. The generalized additive mapping H : A B is given by (5.) H(x) = lim n h(n x) for all x A. It follows from (5.) that (5.2) h( 2n x) H(x) = lim for all x A. Let µ = and x = = x d = 0 in (5.2). Then we get ( d C l d C l + )(h( 2n [z, w] ) [h(z), h(w)]) ϕ(0,, 0, z, w) }{{}

82 C. PARK for all z, w A. So (5.) z, n w] 2n h([n ) [h( n z), h( n w)] 2n ( d C l d C l + ) ϕ(0, } {{, 0, n z, n w) } n ( d C l d C l + ) ϕ(0, } {{, 0, n z, n w) } for all z, w A. By (5.), (5.2), and (5.), [z, w] H( ) = lim h( 2n [z,w] ) = lim = [H(z), H(w)] 2n = lim h( [n z, n w] ) 2n 2n [h(n z), h( n z) w)] = lim [h(n n, h(n w) n ] for all z, w A. Since H is additive, H([z, w]) = H( [z, w] ) = [H(z), H(w)] for all z, w A. The rest of the proof is similar to the proof of Theorem 5.. Corollary 5.4. Let θ and p < be positive real numbers. Let r =, and h : A B an odd bijective mapping satisfying h( n u y) = h( n u) h(y) for all u U(A), all y A, and all n Z, such that F µ h(x,, x d, z, w) θ( x j p + z p + w p ), j= h( n u ) h( n u) d pn θ for all µ T, all u U(A), n = 0,, 2,, and all x,, x d, z, w A. Assume that lim h( n e) = e. Then the odd bijective mapping h : A B is a Lie n JC -algebra isomorphism. Proof. Define ϕ(x,, x d, z, w) = θ( d j= x j p + z p + w p ), and apply Theorem 5.. Theorem 5.5. Let r =, and h : A B an odd bijective mapping satisfying h( n u y) = h( n u) h(y) for all y A, all u U(A) and all n Z, for which

GENERALIZED ADDITIVE MAPPING 8 there exists a function ϕ : A d+2 [0, ) such that (5.4) (5.5) (5.6) j= 9 j ϕ( x j,, x d j, z j, w j ) <, F µ h(x,, x d, z, w) ϕ(x,, x d, z, w), h( u n ) h( u n ) ϕ( u n,, u }{{ n, 0, 0) } for all µ T, all u U(A), all x,, x d, z, w A and n = 0,, 2,. Assume that (5.7) lim n h( e n ) = e. Then the bijective mapping h : A B is a Lie JC -algebra isomorphism. Proof. Put z = w = 0 and µ = T in (5.5). By the same reasoning as in the proofs of Theorems.,. and 4., there exists a unique C-linear generalized additive involutive mapping H : A B such that h(x) H(x) for all x A, where ϕ(x,, x d, z, w) = ϕ(x, x, x, 0,, 0 ) ( d C l d C l + ) }{{} d times j= j ϕ( x j,, x d j, z j, w j ) for all x,, x d, z, w A. The generalized additive mapping H : A B is given by (5.8) for all x A. It follows from (5.8) that (5.9) H(x) = lim n h( x n ) H(x) = lim 2n h( x 2n ) for all x A. Let µ = and x = = x d = 0 in (5.5). Then we get ( d C l d C l + )(h([z, w]) [h(z), h(w)]) ϕ(0,, 0, z, w) }{{}

84 C. PARK for all z, w A. So (5.20) 2n h([ z n, w z ]) [h( n n ), h( w n )] 2n ( d C l d C l + ) ϕ(0, } {{, 0, z } n, w n ) for all z, w A. By (5.4), (5.9), and (5.20), H([z, w]) = [z, w] lim 2n h( ) = lim 2n 2n h([ z n, w n ]) = lim 2n [h( z n ), h( w )] = lim n [n h( z n ), n h( w n )] = [H(z), H(w)] for all z, w A. The rest of the proof is similar to the proof of Theorem 5.. Corollary 5.6. Let θ and p > 2 be positive real numbers. Let r =, and h : A B an odd bijective mapping satisfying h( n u y) = h( n u) h(y) for all u U(A), all y A, and all n Z, such that F µ h(x,, x d, z, w) θ( x j p + z p + w p ), h( u n ) h( u d n ) pn θ for all µ T, all u U(A), n = 0,, 2,, and all x,, x d, z, w A. Assume that lim n h( e ) = e. Then the odd bijective mapping h : A B is a Lie n JC -algebra isomorphism. Proof. Define ϕ(x,, x d, z, w) = θ( d j= x j p + z p + w p ), and apply Theorem 5.5. Theorem 5.7. Let r =, and h : A B an odd bijective mapping satisfying h( n u y) = h( n u) h(y) for all y A, all u U(A) and all n Z, for which there exists a function ϕ : A d+2 [0, ) satisfying (5.4), (5.5), (5.6) and (5.7). Then the bijective mapping h : A B is a Lie JC -algebra isomorphism. Proof. Put z = w = 0 and µ = T in (5.5). By the same reasoning as in the proofs of Theorems.,.7 and 4., there exists a unique C-linear generalized additive involutive mapping H : A B such that h(x) H(x) ϕ(x, x, x, 0,, 0 ) ( d C l d C l 2 ) }{{} d times j=

GENERALIZED ADDITIVE MAPPING 85 for all x A, where ϕ(x,, x d, z, w) = j= j ϕ( x j,, x d j, z j, w j ) for all x,, x d, z, w A. The generalized additive mapping H : A B is given by (5.2) for all x A. It follows from (5.2) that (5.22) H(x) = lim n h( x n ) H(x) = lim 2n h( x 2n ) for all x A. Let µ = and x = = x d = 0 in (5.5). Then we get ( d C l d C l + )(h( for all z, w A. So (5.2) [z, w] ) [h(z), h(w)]) ϕ(0,, 0, z, w) }{{} 2n h( [ z, w n ] n ) [h( z n ), h( w n )] 2n ( d C l d C l + ) ϕ(0, } {{, 0, z } n, w n ) for all z, w A. By (5.4), (5.22), and (5.2), [z, w] H( ) = lim [z, w] 2n+ h( ) = lim 2n+ 2n+ h( [ z, w n ] n ) = lim 2n [h( z n ), h( w )] = lim n [n h( z n ), n h( w n )] = [H(z), H(w)] for all z, w A. Since H is additive, H([z, w]) = H( [z, w] ) = [H(z), H(w)] for all z, w A. The rest of the proof is similar to the proof of Theorem 5..

86 C. PARK Corollary 5.8. Let θ and p > 2 be positive real numbers. Let r =, and h : A B an odd bijective mapping satisfying h( n u y) = h( n u) h(y) for all u U(A), all y A, and all n Z, such that F µ h(x,, x d, z, w) θ( h( u n ) h( u n ) x j p + z p + w p ), j= d pn θ for all µ T, all u U(A), n = 0,, 2,, and all x,, x d, z, w A. Assume that lim n h( e ) = e. Then the odd bijective mapping h : A B is a Lie n JC -algebra isomorphism. Proof. Define ϕ(x,, x d, z, w) = θ( d j= x j p + z p + w p ), and apply Theorem 5.7. References [] P. Găvruta, A generalization of the Hyers Ulam Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 84 (994), 4 46. [2] R.V. Kadison and G. Pedersen, Means and convex combinations of unitary operators, Math. Scand. 57 (985), 249 266. [] R.V. Kadison and J.R. Ringrose, Fundamentals of the Theory of Operator Algebras: Elementary Theory, Academic Press, New York, 98. [4] S. Oh, C. Park and Y. Shin, Quantum n-space and Poisson n-space, Commun. Algebra 0 (2002), 497 4209. [5] S. Oh, C. Park and Y. Shin, A Poincaré Birkhoff Witt theorem for Poisson enveloping algebras, Commun. Algebra 0 (2002), 4867 4887. [6] C. Park, On the stability of the linear mapping in Banach modules, J. Math. Anal. Appl. 275 (2002), 7 720. [7] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (978), 297 00. [8] Th.M. Rassias, On the stability of the quadratic functional equation and its applications, Studia Univ. Babes-Bolyai XLIII () (998), 89 24. [9] Th.M. Rassias, The problem of S.M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246 (2000), 52 78. [0] Th.M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), 2 0. [] Th.M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 25 (2000), 264 284. [2] Th.M. Rassias and P. Semrl, On the behavior of mappings which do not satisfy Hyers Ulam stability, Proc. Amer. Math. Soc. 4 (992), 989 99. [] Th.M. Rassias and P. Semrl, On the Hyers Ulam stability of linear mappings, J. Math. Anal. Appl. 7 (99), 25 8.

GENERALIZED ADDITIVE MAPPING 87 [4] Th.M. Rassias and K. Shibata, Variational problem of some quadratic functionals in complex analysis, J. Math. Anal. Appl. 228 (998), 24 25. [5] S.M. Ulam, Problems in Modern Mathematics, Wiley, New York, 960. [6] H. Upmeier, Jordan Algebras in Analysis, Operator Theory, and Quantum Mechanics: Regional Conference Series in Mathematics No. 67, Amer. Math. Soc., Providence, 987. Received August 2, 2004 Revised version received December 9, 2005 Department of Mathematics, Hanyang University, Seoul 79, Republic of Korea E-mail address: baak@hanyang.ac.kr