Period Math Hung (2014) 69:139–148 DOI 10.1007/s10998-014-0052-1

A characterization of generalized Jordan derivations on Banach algebras B. Arslan · H. Inceboz

Published online: 24 September 2014 © Akadémiai Kiadó, Budapest, Hungary 2014

Abstract Generalized Jordan left derivations and generalized (σ, τ )-Jordan derivations on Banach algebras are characterized and some results of [6] and [7] are extended. Keywords Generalized Jordan left derivation · Generalized (σ, τ )-Jordan derivation · von Neumann algebra Mathematics Subject Classification

46L10 · 46H25 · 46L57

1 Introduction Throughout this paper A will represent an associative algebra over the field F. Recall that a Banach space M which is also an A-module is called a Banach A-module if the Amodule maps (a, x)  → a.x; A × M → M and (x, a)  → x.a; M × A → M, satisfy max{a.x, x.a} ≤ ax for all a ∈ A and x ∈ M. Suppose that A is a unital Banach algebra and M is a Banach A-module. We denote the identity of A by 1. A Banach A-module M is called unital provided that 1.x = x = x.1 for each x ∈ M. A linear mapping d : A → M is called a derivation (resp. left derivation) if d(ab) = d(a)b + ad(b) for all a, b ∈ A (resp. d(ab) = ad(b) + bd(a) for all a, b ∈ A); and is called a Jordan derivation (resp. Jordan left derivation) if d(a ◦ b) = d(a) ◦ b + a ◦ d(b) for all a, b ∈ A (resp. d(a 2 ) = 2ad(a) for any a ∈ A), where a ◦ b denotes the Jordan product ab + ba.

B. Arslan (B) · H. Inceboz Department of Mathematics, Science and Art Faculty, Adnan Menderes University, 09010 Aydin, Turkey e-mail: [email protected] H. Inceboz e-mail: [email protected]

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Obviously, every derivation is a Jordan derivation. The converse, in general, is not true except the case when the algebra is semiprime [2]. In [4], Bre˜sar and Vukman have proved that the existence of a nonzero Jordan left derivation on a prime ring R of char R  = 2, 3 forces R to be commutative. More related results concerning the commutativity of prime rings and semiprime rings on left derivations and Jordan left derivations had been obtained by several authors (see for example; [1] and [15]). Also Jung and Park [9,13] obtained some results on Banach algebras concerning Jordan derivations and Jordan left derivations. Their research is based on the Singer-Wermer theorem [14] which states that every bounded derivation of a commutative Banach algebra has its range in the Jacobson radical. Linear maps which satisfy the derivation equation (that is d(ab) = d(a)b + ad(b)) for special pairs of a and b have been studied by a number of mathematicians in the last fifteen years. For example, linear maps which satisfy the derivation equation when ab = 0 were investigated in [3,5,8,16,17]. In [18–20], the authors analysed linear maps which satisfy the derivation equation when ab is a fixed invertible element. Recently, Lu in [12] studied continuous linear maps from a Banach algebra into its Banach bimodule which satisfy the derivation equation when ab is a left (or right) invertible element and showed that this map is a Jordan derivation. Later, Li and Zhou generalized this result in [11]. In [6], Ebadian and Gordji pursue the same line of investigation in case of Banach algebras and von Neumann algebras to characterize the Jordan left derivations. Let σ, τ ∈ B L(A), the algebra of all bounded linear operators acting on A. A linear mapping d : A → M is called a (σ, τ )-derivation if d(ab) = d(a)τ (b) + σ (a)d(b) (a, b ∈ A). By this definition, every (1, 1)-derivation is a derivation, where 1 means the identity map of A. A (σ, τ )-Jordan derivation is a linear map which satisfies the above equation when a = b. Gordji [7] showed that for bounded linear maps σ, τ on A, every bounded linear map f from a von Neumann algebra A into a Banach A-module M, which satisfies f (ab) = σ (a) f (b) + f (a)τ (b) for all a, b ∈ A with ab = S, is a (σ, τ )-derivation if τ (S) is left invertible for fixed S ∈ A. A linear mapping g : A → M satisfying g(ab) = g(a)b + ad(b)

(r esp. g(a 2 ) = ag(a) + ad(a))

for all a, b ∈ A, where d : A → M is a derivation (resp. Jordan left derivation), is said to be a generalized derivation (resp. generalized Jordan left derivation and is denoted by (g, d). The concept of generalized derivation has been introduced by Bre˜sar [2]. In the same manner the concept of generalized derivation is also extended to generalized (σ, τ )-derivations as follows: Let σ, τ ∈ B L(A). A linear mapping g : A → M satisfying g(ab) = g(a)τ (b) + σ (a)d(b) for any a, b ∈ A, where d : A → M is a (σ, τ )-derivation, is said to be a generalized (σ, τ )derivation; and is called a generalized (σ, τ )-Jordan derivation if g(a 2 ) = g(a)τ (a) + σ (a)d(a) for any a ∈ A where d is a (σ, τ )-Jordan derivation. The purpose of the present paper is to characterize generalized Jordan left derivations and generalized (σ, τ )-Jordan derivations on von Neumann algebras. Throughout this paper we assume that A is a unital Banach algebra with identity element 1 and M is a Banach A-module, unless stated otherwise. We say that S in A is a left (or right)

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Generalized Jordan derivations on Banach algebras

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separating point of M if Sm = 0 (or m S = 0) for m ∈ M implies m = 0. It is easy to see that left (right) invertible elements in A are left (right) separating points of M.

2 Main results Theorem 2.1 Suppose A is an unital Banach algebra and M is a Banach A-module. Let S be in Z (A), the center of A, such that S is a left separating point of M. If g : A → M and d : A → M are bounded linear maps, then the following assertions are equivalent: (a) g(ab) = ag(b) + bd(a) for all a, b ∈ A with ab = ba = S. (b) (g, d) is a generalized Jordan left derivation which satisfies g(aS) = ag(S) + Sd(a) for all a ∈ A. Proof Suppose that (a) holds. Then we get g(S) = g(1S) = 1g(S) + Sd(1) = g(S) + Sd(1) so d(1) = 0, since S is a left separating point of M. Let a ∈ A. For scalars λ with |λ| < 1 a , 1 − λa is invertible in A, since 1 − λa − 1 =  − λa = |λ|a < 1. So (1 − λa)−1 =

∞ 

∞ 

n=0

n=0

(1 − (1 − λa))n =

λn a n . It follows that

g(S) = g((1 − λa)(1 − λa)−1 S) = (1 − λa)g((1 − λa)−1 S) + (1 − λa)−1 Sd(1 − λa) ∞  ∞   n n = (1 − λa)g λ a S + λn a n S(d(1) − λd(a)) n=0

n=0

= g(S + λaS + λ2 a 2 S + · · · ) − λag(S + λaS + λ2 a 2 S + · · · ) + (S + λaS + λ2 a 2 S + · · · )(−λd(a)) = g(S) + λg(aS) + λ2 g(a 2 S) + λ3 g(a 3 S) + · · · − λag(S) − λ2 ag(aS) − λ3 ag(a 2 S) − · · · − λSd(a) − λ2 aSd(a) − λ3 a 2 Sd(a) − · · · ∞  = g(S) + λn [g(a n S) − ag(a n−1 S) − a n−1 Sd(a)]. n=1

Then

∞ 

λn [g(a n S) − ag(a n−1 S) − a n−1 Sd(a)] = 0 for all λ with |λ| <

1 a .

Conse-

n=1

quently, g(a n S) − ag(a n−1 S) − a n−1 Sd(a) = 0

(2.1)

for all n ∈ N. Putting n = 1 in (2.1) we get g(aS) = ag(S) + Sd(a)

(2.2)

for all a ∈ A. On the other hand, if we put n = 2 in (2.1) then we have g(a 2 S) = ag(aS) + aSd(a) = a(ag(S) + Sd(a)) + aSd(a) = a 2 g(S) + 2aSd(a).

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Replacing a by a 2 in (2.2) results in g(a 2 S) = a 2 g(S) + Sd(a 2 ). So it follows from the last two equations that S(d(a 2 ) − 2ad(a)) = 0

(2.3)

for all a ∈ A. Since S is a left separating point of M, then d is a Jordan left derivation. Also we know that condition (a) holds. This means that g(a 2 ) = ag(a) + ad(a) for all a ∈ A, where d is a Jordan left derivation. So (g, d) is a generalized Jordan left derivation which satisfies (2.2). Now suppose that (b) holds. Denote a ◦ b := ab + ba for all a, b ∈ A. Since (g, d) is a generalized Jordan left derivation then we get g(a ◦ b) = g(ab + ba) = g((a + b)2 − a 2 − b2 ) = g((a + b)2 ) − g(a 2 ) − g(b2 ) = (a + b)g(a + b) + (a + b)d(a + b) − ag(a) − ad(a) − bg(b) − bd(b) = ag(b) + ad(b) + bg(a) + bd(a)

(2.4)

for all a, b ∈ A. On the other hand, we see that a ◦ (a ◦ b) = a ◦ (ab + ba) = a(ab + ba) + (ab + ba)a = a 2 ◦ b + 2aba for all a, b ∈ A. So it follows from (2.4) that 2g(aba) = g(a ◦ (a ◦ b)) − g(a 2 ◦ b) = ag(a ◦ b) + ad(a ◦ b) + (a ◦ b)g(a) + (a ◦ b)d(a) − a 2 g(b) − a 2 d(b) − bg(a 2 ) − bd(a 2 ) = a(ag(b) + ad(b) + bg(a) + bd(a)) + 2a(ad(b) + bd(a)) + (ab + ba)(g(a) + d(a)) − a 2 (g(b) + d(b)) − ba(g(a) + d(a)) − 2bad(a) = 2abg(a) + 4abd(a) − 2bad(a) + 2a 2 d(b). Therefore g(aba) = abg(a) + 2abd(a) − bad(a) + a 2 d(b)

(2.5)

for all a, b ∈ A. Now suppose that ab = ba = S. Then g(aS) = abg(a) + abd(a) + a 2 d(b).

(2.6)

Multiplying both sides of the equation (2.6) by b gives S(g(S) − bg(a) − ad(b)) = 0 since S ∈ Z (A). Hence, by assumption, we see that g(S) − bg(a) − ad(b) = 0 for all a, b ∈ A. Since ba = S, we have g(ba) = bg(a) + ad(b)

123

(2.7)

Generalized Jordan derivations on Banach algebras

143

for all a, b ∈ A. Now, we characterize the generalized Jordan left derivations on von Neumann algebras. Theorem 2.2 Let A be a von Neumann algebra and let M be a Banach A-module and g : A → M, d : A → M be bounded linear maps with the property that g( p 2 ) = pg( p)+ pd( p) for every projection p ∈ A. Then (g, d) is a generalized Jordan left derivation. Proof Suppose that p, q ∈ A are orthogonal projections. So p + q is a projection. Therefore by assumption, pg( p) + pd( p) + qg(q) + qd(q) = g( p) + g(q) = g( p + q) = ( p + q)g( p + q) + ( p + q)d( p + q) = pg( p) + pg(q) + qg( p) + qg(q) + pd( p) + pd(q) + qd( p) + qd(q). It follows that pg(q) + qg( p) + pd(q) + qd( p) = 0 Let a =

n 

(2.8)

λ j p j be a combination of mutually orthogonal projections p1 , p2 , . . . , pn ∈ A.

j=1

Then we get pi g( p j ) + p j g( pi ) + pi d( p j ) + p j d( pi ) = 0

(2.9)

for all i, j ∈ 1, 2, . . . , n with i  = j. So ⎛ ⎞ n n   g(a 2 ) = g ⎝ λ2j p j ⎠ = λ2j g( p j ) j=1

=

n 

j=1

λ2j ( p j g( p j ) + p j d( p j )).

j=1

On the other hand by (2.9), we see that  n  n  n  n     λi pi λ j g( p j ) + λi pi λ j d( p j ) ag(a) + ad(a) = i=1

=

n  j=1

j=1

λ2j p j g( p j ) +

i=1 n 

j=1

λ2j p j d( p j ).

j=1

g(a 2 )

= ag(a) + ad(a). It follows from the last two equations, that By the spectral theorem (see Theorem 5.2.2 of [10]), every self adjoint element a ∈ Asa is the norm-limit of a sequence of finite combinations of mutually orthogonal projections. Since g and d are bounded, we have that g(a 2 ) = ag(a) + ad(a)

(2.10)

for all a ∈ Asa . If a is replaced by a + b in (2.10), then we obtain that g((a + b)2 ) = (a + b)g(a + b) + (a + b)d(a + b) = ag(a) + ag(b) + bg(a) + bg(b) + ad(a) + ad(b) + bd(a) + bd(b).

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On the other hand we get g((a + b)2 ) = g(a 2 + b2 + ab + ba) = g(a 2 ) + g(b2 ) + g(ab + ba) = ag(a) + ad(a) + bg(b) + bd(b) + g(ab + ba). Comparing the last two equations, we have g(ab + ba) = ag(b) + bg(a) + ad(b) + bd(a)

(2.11)

for all a, b ∈ Asa . Let a ∈ A. Then there are a1 , a2 ∈ Asa such that a = a1 + ia2 . So it follows from (2.11) that g(a 2 ) = g(a12 − a22 + i(a1 a2 + a2 a1 )) = a1 g(a1 ) + a1 d(a1 ) − a2 g(a2 ) − a2 d(a2 ) +i[a1 g(a2 ) + a2 g(a1 ) + a1 d(a2 ) + a2 d(a1 )] = (a1 + ia2 )g(a1 + ia2 ) + (a1 + ia2 )d(a1 + ia2 ) = ag(a) + ad(a). Corollary 2.3 Let A be a von Neumann algebra and M be a Banach A-module. Let g : A → M and d : A → M be bounded linear maps. Then the following assertions are equivalent: (a) ag(a −1 ) + a −1 d(a) = g(1) for all invertible a ∈ A. (b) (g, d) is a generalized Jordan left derivation. (c) g( p 2 ) = pg( p) + pd( p) for every projection p ∈ A. Proof (a) ⇔ (b) follows from Theorem 2.1, and (b) ⇔ (c) follows from Theorem 2.2. For the generalized (σ, τ )-Jordan derivations, we have the following results. Theorem 2.4 Let A be a unital Banach algebra and M be a Banach A-module. Let S be in A and σ, τ ∈ H om(A) be bounded homomorphisms with the properties that τ (S) is a right separating point of M, σ (S) is a left separating point of M and σ (1) = τ (1) = 1. Let g : A → M and d : A → M be bounded linear maps. Then the following assertions are equivalent: (a) g(ab) = g(a)τ (b) + σ (a)d(b) for all a, b ∈ A with ab = S. (b) (g, d) is a generalized (σ, τ )-Jordan derivation which satisfies g(Sa) = g(S)τ (a) + σ (S)d(a) for all a ∈ A. Proof Suppose that (a) holds. So we have g(S) = g(S1) = g(S)τ (1) + σ (S)d(1) = g(S) + σ (S)d(1). Since σ (S) is a left separating point of M, then we obtain that d(1) = 0. Let a ∈ A. For 1 scalars λ with |λ| < a , 1 − λa is invertible in A. Indeed, (1 − λa)−1 =

∞  n=0

123

λn a n .

Generalized Jordan derivations on Banach algebras

145

Hence g(S) = g[S(1 − λa)−1 (1 − λa)] = g(S(1 − λa)−1 )τ (1 − λa) + σ (S(1 − λa)−1 )d(1 − λa) ∞ ∞   = g(S λn a n )τ (1 − λa) + σ (S λn a n )d(1 − λa) n=0

= g(S) +

n=0 ∞ 

λn [g(Sa n ) − g(Sa n−1 )τ (a) − σ (Sa n−1 )d(a)].

n=1

So we see that ∞ 

λn [g(Sa n ) − g(Sa n−1 )τ (a) − σ (Sa n−1 )d(a)] = 0

n=1

for all λ with |λ| <

1 a .

Then we get

g(Sa n ) − g(Sa n−1 )τ (a) − σ (Sa n−1 )d(a) = 0

(2.12)

for all n ∈ N. If we put n = 1 in (2.12), then we have g(Sa) = g(S)τ (a) + σ (S)d(a)

(2.13)

for all a ∈ A. If we put n = 2 in (2.12), then we get g(Sa 2 ) = g(Sa)τ (a) + σ (Sa)d(a) = g(S)τ (a)2 + σ (S)d(a)τ (a) + σ (Sa)d(a) by using the equation (2.13). Replacing a by a 2 in (2.13), we get g(Sa 2 ) = g(S)τ (a 2 ) + σ (S)d(a 2 ). Combining the last two equations, we have σ (S)(d(a)τ (a) + σ (a)d(a) − d(a 2 )) = 0. Hence, by hypothesis, we get that d(a 2 ) = d(a)τ (a) + σ (a)d(a). So d is a (σ, τ )-Jordan derivation. Therefore, g is a generalized (σ, τ )-Jordan derivation which satisfies the property given by (2.13). Now suppose that (b) holds. Let a, b ∈ A be such that ab = S. Using the fact that ab + ba = (a + b)2 − a 2 − b2 , it is easy to show that the identity of the generalized (σ, τ )-Jordan derivation g is equivalent to g(ab + ba) = g(a)τ (b) + g(b)τ (a) + σ (a)d(b) + σ (b)d(a) for all a, b ∈ A. Then, g(Sa) = g(aba) = =

1 [g(a(ab + ba) + (ab + ba)a) − g(a 2 b + ba 2 )] 2

1 [g(a)τ (ab + ba) + g(ab + ba)τ (a) + σ (a)d(ab + ba) + σ (ab + ba)d(a) 2 − g(a 2 )τ (b) − g(b)τ (a 2 ) − σ (a 2 )d(b) − σ (b)d(a 2 )]

= g(a)τ (b)τ (a) + σ (a)d(b)τ (a) + σ (a)σ (b)d(a).

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On the other hand, we have g(Sa) = g(S)τ (a) + σ (S)d(a) for all a ∈ A. Combining the last two equations we have (g(ab) − g(a)τ (b) − σ (a)d(b))τ (a) = 0. So [g(ab) − g(a)τ (b) − σ (a)d(b)]τ (a)τ (b) = [g(ab) − g(a)τ (b) − σ (a)d(b)]τ (S) = 0. Since τ (S) is a right separating point of M, then g(ab) = g(a)τ (b) + σ (a)d(b) for all a, b ∈ A. The following result characterizes generalized (σ, τ )-Jordan derivations on von Neumann algebras. Theorem 2.5 Let A be a von Neumann algebra, σ, τ be bounded homomorphisms on A and let M be a Banach A-module. Let g : A → M and d : A → M be bounded linear maps which satisfy the identity g( p 2 ) = g( p)τ ( p) + σ ( p)d( p) for every projection p ∈ A. Then (g, d) is a generalized (σ, τ )-Jordan derivation. Proof Suppose that p, q ∈ A are orthogonal projections in A. Then p + q is a projection and so, by assumption, g( p)τ ( p) + σ ( p)d( p) + g(q)τ (q) + σ (q)d(q) = g( p) + g(q) = g( p + q) = g( p + q)τ ( p + q) + σ ( p + q)d( p + q) = g( p)τ ( p) + g( p)τ (q) + g(q)τ ( p) + g(q)τ (q) + σ ( p)d( p) + σ ( p)d(q) + σ (q)d( p) + σ (q)d(q). Thus g( p)τ (q) + g(q)τ ( p) + σ ( p)d(q) + σ (q)d( p) = 0. Let a =

n 

(2.14)

λ j p j be a combination of mutually orthogonal projections p1 , p2 , . . . , pn ∈ A.

j=1

Then we have g( pi )τ ( p j ) + g( p j )τ ( pi ) + σ ( pi )d( p j ) + σ ( p j )d( pi ) = 0 for all i, j ∈ 1, 2, . . . , n with i  = j. So ⎛ ⎞ n n   2 2 λj pj⎠ = λ2j g( p j ) g(a ) = g ⎝ j=1

=

n  j=1

123

j=1

λ2j (g( p j )τ ( p j ) + σ ( p j )d( p j )).

(2.15)

Generalized Jordan derivations on Banach algebras

On the other hand by (2.15), we get g(a)τ (a) + σ (a)d(a) =

n 

147

⎛ ⎞ n  λ j g( p j )τ ⎝ λj pj⎠

j=1

j=1

n n   +σ( λj pj) λ j d( p j ) j=1

=

n 

j=1

λ2j (g( p j )τ ( p j ) + σ ( p j )d( p j )).

j=1

Combining the last two equations gives g(a 2 ) = g(a)τ (a) + σ (a)d(a). By the spectral theorem (see Theorem 5.2.2 of [10]), every self adjoint element a ∈ Asa is the norm-limit of finite combinations of mutually orthogonal projections. Then for all a ∈ Asa , we have g(a 2 ) = g(a)τ (a) + σ (a)d(a)

(2.16)

since g, d, σ and τ are bounded. Replacing a by a + b in (2.16), we have g((a + b)2 ) = g(a 2 + b2 + ab + ba) = g(a 2 ) + g(b2 ) + g(ab + ba) = g(a)τ (a) + σ (a)d(a) + g(b)τ (b) + σ (b)d(b) + g(ab + ba). Also we have g((a + b)2 ) = g(a + b)τ (a + b) + σ (a + b)d(a + b) = g(a)τ (a) + g(a)τ (b) + g(b)τ (a) + g(b)τ (b) + σ (a)d(a) + σ (a)d(b) + σ (b)d(a) + σ (b)d(b). Combining last two equations to get g(ab + ba) = g(a)τ (b) + g(b)τ (a) + σ (a)d(b) + σ (b)d(a)

(2.17)

for all a, b ∈ Asa . Let a ∈ A. Then there are a1 , a2 ∈ Asa such that a = a1 + ia2 . So it follows from (2.17) that g(a 2 ) = g(a12 − a22 + i(a1 a2 + a2 a1 )) = g(a12 ) − g(a22 ) + ig(a1 a2 + a2 a1 ) = g(a1 )τ (a1 ) + σ (a1 )d(a1 ) − g(a2 )τ (a2 ) − σ (a2 )d(a2 ) +i(g(a1 )τ (a2 ) + g(a2 )τ (a1 ) + σ (a1 )d(a2 ) + σ (a2 )d(a1 )) = g(a1 + ia2 )τ (a1 + ia2 ) + σ (a1 + ia2 )d(a1 + ia2 ) = g(a)τ (a) + σ (a)d(a) for all a ∈ A. This means that (g, d) is a generalized (σ, τ )-Jordan derivation from A into M. Corollary 2.6 Let A be a von Neumann algebra and let σ, τ be bounded homomorphisms on A satisfying σ (1) = τ (1) = 1. Let M be a Banach A-module and g : A → M, d : A → M be bounded linear maps. Then the following are equivalent: (a) g( p 2 ) = g( p)τ ( p) + σ ( p)d( p) for every projection p ∈ A. (b) g(a)τ (a −1 ) + σ (a)d(a −1 ) = g(1) for all invertible a ∈ A. (c) (g, d) is a generalized (σ, τ )-Jordan derivation.

123

148 Acknowledgments ments.

B. Arslan, H. Inceboz The authors would like to thank the referee for his/her valuable suggestions and com-

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