# Characterizations of *-antiderivable mappings on operator algebras

Guangyu An, Xueli Zhang and Jun He
From the journal Open Mathematics

# Abstract

Let A be a -algebra, be a - A -bimodule, and δ be a linear mapping from A into . δ is called a -derivation if δ ( A B ) = A δ ( B ) + δ ( A ) B and δ ( A ) = δ ( A ) for each A , B in A . Let G be an element in A , δ is called a -antiderivable mapping at G if A B = G δ ( G ) = B δ ( A ) + δ ( B ) A for each A , B in A . We prove that if A is a C -algebra, is a Banach - A -bimodule and G in A is a separating point of with A G = G A for every A in A , then every -antiderivable mapping at G from A into is a -derivation. We also prove that if A is a zero product determined Banach -algebra with a bounded approximate identity, is an essential Banach - A -bimodule and δ is a continuous -antiderivable mapping at the point zero from A into , then there exists a -Jordan derivation Δ from A into and an element ξ in such that δ ( A ) = Δ ( A ) + A ξ for every A in A . Finally, we show that if A is a von Neumann algebra and δ is a -antiderivable mapping (not necessary continuous) at the point zero from A into itself, then there exists a -derivation Δ from A into itself such that δ ( A ) = Δ ( A ) + A δ ( I ) for every A in A .

MSC 2010: 46L57; 47B47; 47C15; 16E50

## 1 Introduction

Throughout this paper, let A be an associative algebra over the complex field C and be an A -bimodule. A linear mapping δ from A into is called a derivation if

δ ( A B ) = A δ ( B ) + δ ( A ) B

for each A , B in A ; and δ is called a Jordan derivation if

δ ( A 2 ) = A δ ( A ) + δ ( A ) A

for every A in A . It follows from [1, Corollary 17] that every Jordan derivation from a C -algebra A into a Banach A -bimodule is a derivation.

Let G be an element in A , δ is called a derivable mapping at G if

A B = G δ ( G ) = A δ ( B ) + δ ( A ) B

for each A , B in A . In [2,3,4, 5,6,7, 8,9], the authors investigated derivable mappings at the point zero. In [10,11, 12,13,14, 15,16], the authors investigated derivable mappings at nonzero points.

A linear mapping δ from A into is called an antiderivable mapping at G if

A B = G δ ( G ) = B δ ( A ) + δ ( B ) A

for each A , B in A . In [6,7] and [17,18], the authors characterized antiderivable mappings at the point zero on properly infinite von Neumann algebras, C -algebras and group algebras.

By an involution on an algebra A , we mean a mapping from A into itself, such that

( λ A + μ B ) = λ ¯ A + μ ¯ B , ( A B ) = B A and ( A ) = A ,

whenever A , B in A , λ , μ in C and λ ¯ , μ ¯ denote the conjugate complex numbers. An algebra A equipped with an involution is called a -algebra. Moreover, let A be a -algebra, an A -bimodule is called a - A -bimodule if equipped with a -mapping from into itself, such that

( λ M + μ N ) = λ ¯ M + μ ¯ N , ( A M ) = M A , ( M A ) = A M and ( M ) = M ,

whenever A in A , M , N in and λ , μ in C .

An element A in a -algebra A is called Hermitian if A = A ; an element P in A is called an idempotent if P 2 = P ; and P is called a projection if P is both a self-adjoint element and an idempotent.

In [19], Kishimoto studied the -derivations on a C -algebra and proved that the closure of a normal -derivation on a UHF algebra satisfying a special condition is a generator of a one-parameter group of -automorphisms. Let A be a -algebra and be a - A -bimodule. A derivation δ from A into is called a -derivation if δ ( A ) = δ ( A ) for every A in A . Obviously, every derivation δ is a linear combination of two -derivations. In fact, we can define a linear mapping δ from A into by δ ( A ) = δ ( A ) for every A in A ; therefore, δ = δ 1 + i δ 2 , where δ 1 = 1 2 ( δ + δ ) and δ 2 = 1 2 i ( δ δ ) . It is easy to show that δ 1 and δ 2 are both -derivations.

Similar to derivable and antiderivable mappings, we can consider -derivable and -antiderivable mappings. Let A be a -algebra, be a - A -bimodule and G be an element in A . A linear mapping δ from A into is called a -derivable mapping at G if

A B = G δ ( G ) = A δ ( B ) + δ ( A ) B

for each A , B in A and δ is called a -antiderivable mapping at G if

A B = G δ ( G ) = B δ ( A ) + δ ( B ) A

for each A , B in A .

In [6], Ghahramani supposed that G is a locally compact group, L 1 ( G ) and M ( G ) denote the the group algebra and the measure convolution algebra of G , respectively, and showed that if δ is a -derivable mapping or a -antiderivable mapping at the point zero from L 1 ( G ) into M ( G ) , then there exist two elements B , C in M ( G ) such that δ ( A ) = A B C A for every A in L 1 ( G ) . In [7], Ghahramani and Pan supposed that A is a properly infinite W -algebra or a simple C -algebra with a nontrivial idempotent, and proved that if δ is a -derivable mapping at the point zero from A into itself, then there exist two elements B , C in A such that δ ( A ) = A B C A for every A in A ; if δ is a -antiderivable mapping at the point zero from A into itself, then δ ( A ) = δ ( I ) A for every A in A . In [17], Abulhamil et al. supposed that A is a C -algebra and is an essentially Banach A -bimodule, and proved that if δ is a continuous -antiderivable mapping at the point zero from A into , then there exists a -derivation Δ from A into and ξ in such that δ ( A ) = Δ ( A ) + A ξ for every A in A , where is the second dual of . In [18], Fadaee and Ghahramani supposed that A is a von Neumann algebra or a simple unital C -algebra, and proved that if δ is a -derivable mapping or a -antiderivable mapping at the point zero from A into itself, then there exist two elements B , C in A such that δ ( A ) = A B C A for every A in A .

For an algebra A and an A -bimodule , we call an element G in A a left (right) separating point of if G M = 0 ( M G = 0 ) implies M = 0 for every M in . It is easy to see that every left(right) invertible element in A is a left(right) separating point of . If G A is both the left and right separating point, then G is called a separating point of .

In Section 2, we prove that if A is a C -algebra, is a Banach - A -bimodule and G in A is a separating point of with A G = G A for every A A , then every -antiderivable mapping at G from A into is a -derivation.

In Section 3, we investigate -antiderivable mappings at the point zero and prove that if A is a zero product determined Banach -algebra with a bounded approximate identity, is an essential Banach - A -bimodule and δ is a continuous -antiderivable mapping at the point zero from A into , then there exists a -Jordan derivation Δ from A into and ξ in , such that δ ( A ) = Δ ( A ) + A ξ for every A in A , where stands for the second dual of M . Thus, we generalize [6, Theorem 3.2(2)] and [17, Theorem 9]. Finally, we prove that every -antiderivable mapping at the point zero from a von Neumann algebra A into itself satisfies that δ ( A ) = Δ ( A ) + A δ ( I ) for every A in A , where Δ is a -derivation from A into itself.

## 2 ∗ -Antiderivable mappings at a separating points

Before we give the main result in this section, we need to prove the following proposition.

## Proposition 2.1

Suppose that A is a unital Banach algebra, is a unital Banach A -bimodule and G in A is a separating point of with A G = G A for every A in A . If δ and τ are two linear mappings from A into such that

A B = G δ ( G ) = B δ ( A ) + τ ( B ) A

for each A , B in A , then τ is a Jordan derivation, δ is a generalized Jordan derivation, that is, for every A in A , δ ( A 2 ) = A δ ( A ) + δ ( A ) A A δ ( I ) A . Moreover, the following identities hold:

τ ( A G ) = τ ( G ) A + G δ ( A ) A G δ ( I )

and

δ ( G A ) = A δ ( G ) + τ ( A ) G

for every A in A .

## Proof

By I G = G I = G , we have that

(2.1) δ ( G ) = G δ ( I ) + τ ( G )

and

(2.2) δ ( G ) = δ ( G ) + τ ( I ) G .

Since G is a separating point for , by (2.2), we have τ ( I ) = 0 . Let T be a invertible element in A . By G T 1 T = T 1 G T = G , we obtain

(2.3) δ ( G ) = T δ ( G T 1 ) + τ ( T ) G T 1

and

(2.4) δ ( G ) = T 1 G δ ( T ) + τ ( T 1 G ) T .

Multiplying by T 1 from the left-hand side of (2.3), we can obtain that

(2.5) δ ( G T 1 ) = T 1 δ ( G ) T 1 τ ( T ) G T 1 .

Multiplying by T 1 from the right-hand side of (2.4), we have that

(2.6) τ ( T 1 G ) = δ ( G ) T 1 T 1 G δ ( T ) T 1 .

Let A be in A , n be a positive integer with n > ( A + 1 ) and B = n I + A . Then, both B and I B are invertible in A . By replacing T with B in (2.3), by (2.5) and τ ( I ) = 0 , we obtain

τ ( B ) G B 1 = δ ( G ) B δ ( G B 1 ) = δ ( G ) B δ ( G B 1 ( I B ) + G ) = ( I B ) δ ( G ) B δ ( G B 1 ( I B ) ) = ( I B ) δ ( G ) B [ B 1 ( I B ) δ ( G ) B 1 ( I B ) τ ( ( I B ) 1 B ) G B 1 ( I B ) ]

= ( I B ) τ ( ( I B ) 1 B ) G B 1 ( I B ) = ( I B ) τ ( ( I B ) 1 I ) G B 1 ( I B ) = ( I B ) τ ( ( I B ) 1 ) G B 1 ( I B ) .

Since A G = G A for every A in A , it follows that

(2.7) τ ( B ) B 1 G = ( I B ) τ ( ( I B ) 1 ) B 1 ( I B ) G .

By replacing T with B in (2.4), we obtain

B 1 G δ ( B ) = δ ( G ) τ ( B 1 G ) B = δ ( G ) τ ( B 1 ( I B ) G + G ) B = δ ( G ) τ ( G ) B τ ( B 1 ( I B ) G ) B .

By (2.6), it implies that

B 1 G δ ( B ) = δ ( G ) τ ( G ) B τ ( B 1 ( I B ) G ) B = δ ( G ) τ ( G ) B [ δ ( G ) B 1 ( I B ) B 1 ( I B ) G δ ( ( I B ) 1 B ) B 1 ( I B ) ] B = δ ( G ) τ ( G ) B δ ( G ) ( I B ) + B 1 ( I B ) G δ ( ( I B ) 1 B ) ( I B ) = ( δ ( G ) τ ( G ) ) B + B 1 ( I B ) G δ ( ( I B ) 1 I ) ( I B ) ,

and by (2.1), it follows that

B 1 G δ ( B ) = ( δ ( G ) τ ( G ) ) B + B 1 ( I B ) G δ ( ( I B ) 1 I ) ( I B ) = G δ ( I ) B B 1 ( I B ) G δ ( I ) ( I B ) + B 1 ( I B ) G δ ( ( I B ) 1 ) ( I B ) .

By A G = G A for every A in A , we can obtain that

(2.8) G B 1 δ ( B ) = G [ δ ( I ) B B 1 ( I B ) δ ( I ) ( I B ) + B 1 ( I B ) δ ( ( I B ) 1 ) ( I B ) ] .

Since G is a separating point of , by (2.7) and (2.8), we obtain

(2.9) τ ( B ) B 1 = ( I B ) τ ( ( I B ) 1 ) B 1 ( I B )

and

(2.10) B 1 δ ( B ) = δ ( I ) B B 1 ( I B ) δ ( I ) ( I B ) + B 1 ( I B ) δ ( ( I B ) 1 ) ( I B ) .

Multiplying by B from the right-hand side of (2.9) and from the left-hand side of (2.10), we can obtain that

(2.11) τ ( B ) = ( I B ) τ ( ( I B ) 1 ) ( I B )

and

(2.12) δ ( B ) = B δ ( I ) B ( I B ) δ ( I ) ( I B ) + ( I B ) δ ( ( I B ) 1 ) ( I B ) .

Multiplying by G from the right-hand side of (2.11) and by A G = G A , it follows that

τ ( B ) G = ( I B ) τ ( ( I B ) 1 ) ( I B ) G = ( I B ) τ ( ( I B ) 1 ) G ( I B ) .

By (2.3),

(2.13) τ ( B ) G = ( I B ) [ δ ( G ) ( I B ) 1 δ ( G ( I B ) ) ] = ( I B ) δ ( G ) δ ( G G B ) = δ ( G B ) B δ ( G ) .

Multiplying by G from the left of (2.12) and by A G = G A , it follows that

G δ ( B ) = G B δ ( I ) B G ( I B ) δ ( I ) ( I B ) + G ( I B ) δ ( ( I B ) 1 ) ( I B ) = G B δ ( I ) B G ( I B ) δ ( I ) ( I B ) + ( I B ) G δ ( ( I B ) 1 ) ( I B ) ,

and by (2.4),

G δ ( B ) = G B δ ( I ) B G ( I B ) δ ( I ) ( I B ) + [ δ ( G ) τ ( ( I B ) G ) ( I B ) 1 ] ( I B ) = G B δ ( I ) B G ( I B ) δ ( I ) + G ( I B ) δ ( I ) B + δ ( G ) δ ( G ) B τ ( G ) + τ ( B G ) = G B δ ( I ) B G δ ( I ) + G B δ ( I ) + G δ ( I ) B G B δ ( I ) B + δ ( G ) δ ( G ) B τ ( G ) + τ ( B G ) = G B δ ( I ) + [ δ ( G ) τ ( G ) G δ ( I ) ] + [ G δ ( I ) δ ( G ) ] B + τ ( B G ) ,

and by (2.1), it implies that

(2.14) G δ ( B ) = G B δ ( I ) τ ( G ) B + τ ( B G ) .

By (2.13), (2.14) and A G = G A , we have that

δ ( G B ) = B δ ( G ) + τ ( B ) G

and

τ ( B G ) = G δ ( B ) + τ ( G ) B B G δ ( I ) .

Since B = n I + A , we have the following two equations:

(2.15) δ ( G A ) = A δ ( G ) + τ ( A ) G

and

(2.16) τ ( A G ) = τ ( G ) A + G δ ( A ) A G δ ( I ) .

By (2.15), we know that for every invertible element T in A , it follows that

δ ( G ) = δ ( G T T 1 ) = T 1 δ ( G T ) + τ ( T 1 ) G T = T 1 [ T δ ( G ) + τ ( T ) G ] + τ ( T 1 ) T G = δ ( G ) + T 1 τ ( T ) G + τ ( T 1 ) T G .

Since G is a separating point,

(2.17) T 1 τ ( T ) + τ ( T 1 ) T = 0 .

By (2.16), we know that for every invertible element T in A , it follows that

δ ( G ) = δ ( T 1 T G ) = T G δ ( T 1 ) + τ ( T G ) T 1 = T G δ ( T 1 ) + [ τ ( G ) T + G δ ( T ) T G δ ( I ) ] T 1 = T G δ ( T 1 ) + τ ( G ) + G δ ( T ) T 1 T G δ ( I ) T 1 = G T δ ( T 1 ) + τ ( G ) + G δ ( T ) T 1 G T δ ( I ) T 1 .

Thus,

δ ( G ) τ ( G ) = G T δ ( T 1 ) + G δ ( T ) T 1 G T δ ( I ) T 1 .

By (2.1), we have that

G δ ( I ) = G T δ ( T 1 ) + G δ ( T ) T 1 G T δ ( I ) T 1 .

Since G is a separating point, we know that

(2.18) δ ( I ) = T δ ( T 1 ) + δ ( T ) T 1 T δ ( I ) T 1 .

It follows from (2.17), (2.18) and [13, Lemma 2.1] that τ and Δ ( A ) δ ( A ) A δ ( I ) both are Jordan derivations, and hence, δ is a generalized Jordan derivation.□

Let G = I in Proposition 2.1, we have the following result.

## Corollary 2.2

Suppose A is a unital Banach algebra and is a unital Banach A -bimodule. If δ and τ are two linear mappings from A into , such that

A B = I δ ( I ) = B δ ( A ) + τ ( B ) A

for each A , B in A , then τ is a Jordan derivation and δ is a generalized Jordan derivation. Moreover, for every A in A , we have that

δ ( A ) = A δ ( I ) + τ ( A ) .

For every -antiderivable mapping at unit element from a unital Banach -algebra into its unital Banach - A -bimodule, we have the following result.

## Corollary 2.3

Suppose that A is a unital Banach -algebra and is a unital Banach - A -bimodule. If δ is a linear mapping from A into such that

A B = I δ ( I ) = B δ ( A ) + δ ( B ) A

for each A , B in A , then δ is a -Jordan derivation.

## Proof

Let τ be the linear mapping from A into such that for every A in A ,

δ ( A ) = δ ( A ) .

It follows that for each A , B in A , we have that

A B = I = A ( B ) = I δ ( I ) = B δ ( A ) + δ ( B ) A δ ( I ) = B δ ( A ) + δ ( B ) A .

It follows from Proposition 2.1 that δ is a Jordan derivation, and hence, δ is also a Jordan derivation.

Finally, we prove that δ is a -Jordan derivation, that is, δ ( A ) = δ ( A ) for every A in A . In fact, by δ ( I ) = 0 and Corollary 2.2, we have that δ ( A ) = δ ( A ) = δ ( A ) . It implies that δ ( A ) = δ ( A ) for every A in A .□

For every -antiderivable mapping from a unital C -algebra into its Banach - A -bimodule, we have the following theorem.

## Theorem 2.4

Suppose that A is a unital C -algebra, is a unital Banach - A -bimodule and G in A is a separating point of with A G = G A for every A in A . If δ is a linear mapping from A into such that

A B = G δ ( G ) = B δ ( A ) + δ ( B ) A

for each A , B in A , then δ is a -derivation.

## Proof

Let τ be a linear mapping from A into such that for every A in A

τ ( A ) = δ ( A ) .

It follows that for each A , B in A , we have that

A B = G = A ( B ) = G δ ( G ) = B δ ( A ) + δ ( B ) A δ ( G ) = B δ ( A ) + τ ( B ) A .

By Proposition 2.1, τ is a Jordan derivation, and hence, δ is also a Jordan derivation. Since A is a C -algebra, δ is a derivation.

Finally, we show that δ is a -derivation, that is, δ ( A ) = δ ( A ) for every A in A . Let A be an invertible element in A , by G A ( ( A 1 ) ) = G , we have that

δ ( G ) = A 1 δ ( G A ) + δ ( ( A 1 ) ) G A .

Since δ is a derivation and A G = G A , it follows that

δ ( G ) = δ ( ( A 1 ) ) A G + A 1 ( A δ ( G ) + δ ( A ) G ) ,

that is,

δ ( ( A 1 ) ) A G + A 1 δ ( A ) G = 0 .

Since G is a separating point, we have that

(2.19) δ ( ( A 1 ) ) A + A 1 δ ( A ) = 0 .

On the other hand, we can obtain that

(2.20) δ ( A 1 ) A + A 1 δ ( A ) = δ ( I ) = 0 .

By (2.19) and (2.20), we know that δ ( ( A 1 ) ) A = δ ( A 1 ) A , that is, δ ( ( A 1 ) ) = δ ( A 1 ) . Thus, for every invertible element A A , we have showed that δ ( A ) = δ ( A ) .

Since every element in a unital C -algebra is a linear combination of four unitaries [20], it follows that δ ( A ) = δ ( A ) for every A A .□

In particular, let G = I in Theorem 2.4, the following corollary holds.

## Corollary 2.5

Suppose A is a unital C -algebra and is a unital Banach - A -bimodule. If δ is a linear mapping from A into such that

A B = I δ ( I ) = B δ ( A ) + δ ( B ) A

for each A , B in A , then δ is a -derivation.

## Remark 2.6

Suppose that A is a unital -algebra, is a unital Banach - A -bimodule and δ is a linear mapping from A into . We should notice that the following two conditions are not equivalent:

1. A , B A , A B = G B δ ( A ) + δ ( B ) A = δ ( G ) ;

2. A , B A , A B = G B δ ( A ) + δ ( B ) A = δ ( G ) .

Hence, we also can define a -derivable mapping at G in A from A into by

A , B A , A B = G B δ ( A ) + δ ( B ) A = δ ( G ) .

Through the minor modifications, we can obtain the corresponding results.

## 3 ∗ -Antiderivable mappings at the point zero

A (Banach) algebra A is said to be zero product determined if every (continuous) bilinear mapping ϕ from A × A into any (Banach) linear space X satisfying

ϕ ( A , B ) = 0 , whenever A B = 0

can be written as ϕ ( A , B ) = T ( A B ) , for some (continuous) linear mapping T from A into X . In [21], Brešar showed that if A = J ( A ) , then A is a zero product determined, where J ( A ) is the subalgebra of A generated by all idempotents in A , and in [2], the authors proved that every C -algebra A is zero product determined.

Suppose that A is a Banach algebra and is a Banach- A -bimodule. is called an essential Banach A -bimodule if

= span ¯ { A N B : A , B A , N } ,

where span ¯ { } denotes the norm closure of the linear span of the set { } .

Let A be a Banach -algebra, a bounded approximate identity for A is a net ( e i ) i Γ of self-adjoint elements in A such that lim i A e i A = lim i e i A A = 0 for every A in A and sup i Γ e i K for some K > 0 .

## Theorem 3.1

Suppose A is a zero product determined Banach -algebra with a bounded approximate identity and is an essential Banach - A -bimodule. If δ is a continuous linear mapping from A into such that

A B = 0 B δ ( A ) + δ ( B ) A = 0

for each A , B in A , then there are a -Jordan derivation Δ from A into and an element ξ in , such that

δ ( A ) = Δ ( A ) + A ξ

for every A in A . Furthermore, ξ can be chosen in in each of the following cases:

1. A has an identity.

2. is a dual - A -bimodule.

In [17, Section 4] and in [22, p. 720], the authors showed that is also a Banach - A -bimodule, where is the second dual space of . But, for the sake of completeness, we recall the argument here.

In fact, since is a Banach - A -bimodule, turns into a dual Banach A -bimodule with the operation defined by

A M = lim μ A M μ and M A = lim μ M μ A

for every A in A and every M in , where ( M μ ) is a net in with M μ M and ( M μ ) M in the weak -topology σ ( , ) .

We define an involution in by

( M ) ( ρ ) = M ( ρ ) ¯ , ρ ( M ) = ρ ( M ) ¯ ,

where M in , ρ in and M in . Moreover, if ( M μ ) is a net in and M is an element in such that M μ M in σ ( , ) , then for every ρ in , we have that

ρ ( M μ ) = M μ ( ρ ) M ( ρ ) .

It follows that

( M μ ) ( ρ ) = ρ ( M μ ) = ρ ( M μ ) ¯ M ( ρ ) ¯ = ( M ) ( ρ )

for every ρ in . It means that the involution in is continuous in σ ( , ) . Thus, we can obtain that

( A M ) = ( lim μ A M μ ) = lim μ M μ A = ( M ) A .

Similarly, we can show that ( M A ) = A ( M ) . It implies that is a Banach - A -bimodule.

In the following, we prove that Theorem 3.1.

## Proof

Let ( e i ) i Γ be a bounded approximate identity of A . Since δ is a continuous mapping, ( δ ( e i ) ) i Γ is bounded in . Moreover, ( δ ( e i ) ) i Γ is also bounded in . By the Alaoglu-Bourbaki theorem, we may assume that ( δ ( e i ) ) i Γ converges to the element ξ in with the weak -topology σ ( , ) .

Since is an essential Banach- A -bimodule, M e i converges to M with respect to the weak -topology σ ( , ) for every M in . In fact, since = span ¯ { A N B : A , B A , N } , there exists a sequence M n = Σ k = 1 m n A k n N k n B k n converging to M in the norm topology, where A k n , B k n A and N k n , k = 1 , 2 , , m n , n = 1 , 2 , . Since ( A N B e i ) converges to A N B in the norm topology for each A , B A and N , it follows that M e i converges to in the norm topology for every M in .

Define a continuous bilinear mapping from A × A into by

ϕ ( A , B ) = δ ( B ) A + B δ ( A )

for every A , B in A . It follows that

A B = 0 ϕ ( A , B ) = 0 .

Since A is a zero product determined Banach algebra, there exists a continuous linear mapping T : A such that ϕ ( A , B ) = T ( A B ) for every A , B A . Moreover, for every A , B , C A , we have that

ϕ ( A B , C ) = ϕ ( A , B C ) .

That is,

(3.1) δ ( C ) A B + C δ ( A B ) = δ ( C B ) A + B C δ ( A ) .

Let A = e i in (3.1) and take the limit on both sides with the weak -topology σ ( , ) , we can obtain that

(3.2) δ ( C ) B + C δ ( B ) = δ ( C B ) + B C ξ .

Take the involution on both sides in (3.2), it implies that

(3.3) B δ ( C ) + δ ( B ) C = δ ( C B ) + ξ C B .

Let C = e i in (3.3) and take the limit on both sides of (3.3) with the weak -topology σ ( , ) , we can obtain that

B ξ + δ ( B ) = δ ( B ) + ξ B ,

that is,

(3.4) δ ( B ) B ξ = δ ( B ) ξ B .

Define a linear mapping Δ from A into by

Δ ( A ) = δ ( A ) A ξ

for every A in A . Next, we prove that Δ is a -Jordan derivation. By (3.4), we have that Δ ( A ) = Δ ( A ) for every A A .

By replacing C , B with A , B in (3.3), respectively, we can obtain that

B δ ( A ) + δ ( B ) A = δ ( A B ) + ξ A B ,

that is,

(3.5) δ ( A B ) = B δ ( A ) + δ ( B ) A ξ A B .

In the following, we prove that

Δ ( A 2 ) = A Δ ( A ) + Δ ( A ) A

for every A in A . By the definition of Δ and (3.5), we have the following two equations:

(3.6) Δ ( A 2 ) = δ ( A 2 ) A 2 ξ = A δ ( A ) + δ ( A ) A ξ A 2 A 2 ξ

and

(3.7) A Δ ( A ) + Δ ( A ) A = A ( δ ( A ) A ξ ) + ( δ ( A ) A ξ ) A = A δ ( A ) A 2 ξ + δ ( A ) A A ξ A .

By Δ ( A ) = Δ ( A ) , it implies that δ ( A ) A ξ = ( δ ( A ) A ξ ) , and

(3.8) δ ( A ) ξ A = δ ( A ) A ξ .

Multiplying by A from the right side of (3.8), we have that

(3.9) ( δ ( A ) ξ A ) A = ( δ ( A ) A ξ ) A .

Finally, by (3.6), (3.7), and (3.9), it follows that Δ ( A 2 ) = A Δ ( A ) + Δ ( A ) A . Thus, Δ is a -Jordan derivation.

Suppose that A is a unital Banach algebra, we can assume that ξ = δ ( I ) .

Suppose that is a dual essential Banach - A -bimodule and is the pre-dual space of , since δ is continuous, we can assume that the net ( δ ( e i ) ) i Γ converges to element ξ with the weak -topology σ ( , ) .□

Let G be a locally compact group. The group algebra and the measure convolution algebra of G , are denoted by L 1 ( G ) and M ( G ) , respectively. The convolution product is denoted by , and the involution is denoted by . It is well known that M ( G ) is a unital Banach -algebra, and L 1 ( G ) is a closed ideal in M ( G ) with a bounded approximate identity. By [23, Lemma 1.1], we know that L 1 ( G ) is zero product determined. By [24, Theorem 3.3.15(ii)], it follows that M ( G ) with respect to convolution product is the dual of C 0 ( G ) as a Banach M ( G ) -bimodule.

Since L 1 ( G ) is a semisimple algebra, we know from [25] that every continuous Jordan derivation from L 1 ( G ) into itself is a derivation. By [26, Corollary 1.2], we know that every continuous derivation Δ from L 1 ( G ) into M ( G ) is an inner derivation, that is, there exists μ in M ( G ) such that Δ ( f ) = f μ μ f for every f in L 1 ( G ) . Thus, by Theorem 3.1, we can rediscover [6, Theorem 3.2(ii)] as follows:

## Corollary 3.2

[6, Theorem 3.2(ii)] Let G be a locally compact group. If δ is a continuous linear mapping from L 1 ( G ) into M ( G ) such that

f g = 0 δ ( g ) f + g δ ( f ) = 0 ,

for each f , g in L 1 ( G ) , then there exist two-element μ , ν M ( G ) such that

δ ( f ) = f ν μ f

for every f in L 1 ( G ) and Re μ Z ( M ( G ) ) .

## Proof

By Theorem 3.1, we know that there exist a -derivation Δ from L 1 ( G ) into M ( G ) and an element ξ in M ( G ) such that

δ ( f ) = Δ ( f ) + ξ f

for every f in L 1 ( G ) . By [26, Corollary 1.2], it follows that there exists μ in M ( G ) such that Δ ( f ) = f μ μ f . Since Δ ( f ) = Δ ( f ) , we have that

f μ μ f = μ f f μ

for every f in L 1 ( G ) . By [23, Lemma 1.3(ii)], we know Re μ = 1 2 ( μ + μ ) Z ( M ( G ) ) . Let ν = μ ξ , from the definition of Δ , we have that δ ( f ) = f μ ν f for every f in L 1 ( G ) .□

In [2], the authors proved that every C -algebra A is zero product determined, and by [27, Corollary 7.5], we know that A has a bounded approximate identity. Thus, by Theorem 3.1, we can obtain a new proof of [17, Theorem 9] as follows:

## Corollary 3.3

[17, Theorem 9] Let A be a C -algebra and an essential Banach - A -bimodule. If δ is a continuous linear mapping from A into such that

A B = 0 B δ ( A ) + δ ( B ) A = 0

for every A , B in A , then there exists a -derivation Δ from A into and ξ in such that

δ ( A ) = Δ ( A ) + A ξ

for every A in A . Furthermore, ξ can be chosen in in each of the following cases:

1. A has an identity.

2. is a dual - A -bimodule.

Suppose that A is a zero product determined unital -algebra and δ is a -antiderivable mapping from A into a - A -bimodule. Let ( e i ) i Γ = I and ξ = δ ( I ) in Theorem 3.1, we can obtain the following conclusion.

## Corollary 3.4

Let A be a zero product determined unital -algebra and be a - A -bimodule. If δ is a linear mapping (continuity is not necessary) from A into such that

A B = 0 B δ ( A ) + δ ( B ) A = 0

for each A , B in A , then there exists a -Jordan derivation Δ from A into such that

δ ( A ) = Δ ( A ) + A δ ( I )

for every A in A .

Finally, we investigate -antiderivable mappings at the zero point on a von Neumann algebra. The following result is the second main theorem in this section.

## Theorem 3.5

Let A be a von Neumann algebra. If δ is a linear mapping from A into itself, such that

A B = 0 B δ ( A ) + δ ( B ) A = 0

for each A , B in A , then there exists a -derivation Δ from A into such that

δ ( A ) = Δ ( A ) + A δ ( I )

for every A in A . In particular, δ is a -derivation when δ ( I ) = 0 .

## Proof

Suppose that is a commutative von Neumann subalgebra of A . For each A , B in , we have that

A B = 0 A B = 0 A B = 0 A B = 0 .

Let A , B , C be in satisfying A B = B C = 0 . Since A B = 0 , we obtain B δ ( A ) + δ ( B ) A = 0 . By multiplying the previous identity by C form the left-hand side, we have C δ ( B ) A = 0 , equivalently, A δ ( B ) C = 0 . Therefore, [28, Theorem 2.12] implies that δ is automatically continuous, and by [17, Theorem 9], we can prove this theorem.□

## Remark 3.6

Let A be a unital -algebra and be a unital Banach - A -bimodule. δ is a linear mapping from A into such that

A B = 0 B δ ( A ) + δ ( B ) A = 0 .

Through the minor modifications of Theorems 3.1 and 3.5, we can obtain the corresponding results.

# Acknowledgements

The authors thank the referee for his or her suggestions. This research was partly supported by the National Natural Science Foundation of China (Grant Nos. 11801342 and 11801005); Natural Science Foundation of Shaanxi Province (Grant No. 2020JQ-693); Scientific research plan projects of Shannxi Education Department (Grant No. 19JK0130).

1. Conflict of interest: The author states no conflict of interest.

### References

[1] A. Peralta and B. Russo, Automatic continuity of derivations on C∗-algebras and JB∗-triples, J. Algebra 399 (2014), no. 2, 960–977, https://doi.org/10.1016/j.jalgebra.2013.10.017. Search in Google Scholar

[2] J. Alaminos, M. Bressssar, J. Extremera, and A. Villena, Maps preserving zero products, Studia Math. 193 (2009), no. 2, 131–159, https://doi.org/10.4064/sm193-2-3. Search in Google Scholar

[3] J. Alaminos, M. Bresssar, J. Extremera, and A. Villena, Characterizing Jordan maps on C∗-algebras through zero products, Proc. Edinburgh Math. Soc. 53 (2010), no. 3, 543–555, https://doi.org/10.1017/S0013091509000534. Search in Google Scholar

[4] M. Bresssar, Characterizing homomorphisms, derivations and multipliers in rings with idempotents, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), no. 1, 9–21, https://doi.org/10.1017/S0308210504001088. Search in Google Scholar

[5] H. Ghahramani, On derivations and Jordan derivations through zero products, Oper. Matrices 8 (2014), no. 3, 759–771, https://doi.org/10.7153/oam-08-42. Search in Google Scholar

[6] H. Ghahramani, Linear maps on group algebras determined by the action of the derivations or anti-derivations on a set of orthogonal elements, Results Math. 73 (2018), no. 4, 133, https://doi.org/10.1007/s00025-018-0898-2. Search in Google Scholar

[7] H. Ghahramani and Z. Pan, Linear maps on ∗-algebras acting on orthogonal elements like derivations or anti-derivations, Filomat 13 (2018), no. 13, 4543–4554, https://doi.org/10.2298/FIL1813543G. Search in Google Scholar

[8] M. Jiao and J. Hou, Additive maps derivable or Jordan derivable at zero point on nest algebras, Linear Algebra Appl. 432 (2010), no. 11, 2984–2994, https://doi.org/10.1016/j.laa.2010.01.009. Search in Google Scholar

[9] M. Kosssan, T. Lee, and Y. Zhou, Bilinear forms on matrix algebras vanishing on zero products of xy and yx, Linear Algebra Appl. 453 (2014), 110–124, https://doi.org/10.1016/j.laa.2014.04.004. Search in Google Scholar

[10] R. An and J. Hou, Characterizations of Jordan derivations on rings with idempotent, Linear Multilinear Algebra 58 (2010), no. 6, 753–763, https://doi.org/10.1080/03081080902992047. Search in Google Scholar

[11] J. He, J. Li, and W. Qian, Characterizations of centralizers and derivations on some algebras, J. Korean Math. Soc. 54 (2017), no. 2, 685–696, https://doi.org/10.4134/JKMS.j160265. Search in Google Scholar

[12] J. Hou and R. An, Additive maps on rings behaving like derivations at idempotent-product elements, J. Pure Appl. Algebra 215 (2011), no. 8, 1852–1862, https://doi.org/10.1016/j.jpaa.2010.10.017. Search in Google Scholar

[13] J. Li and J. Zhou, Characterizations of Jordan derivations and Jordan homomorphisms, Linear Multilinear Algebra 59 (2011), no. 2, 193–204, https://doi.org/10.1080/03081080903304093. Search in Google Scholar

[14] F. Lu, Characterizations of derivations and Jordan derivations on Banach algebras, Linear Algebra Appl. 430 (2009), no. 8, 2233–2239, https://doi.org/10.1016/j.laa.2008.11.025. Search in Google Scholar

[15] S. Zhao and J. Zhu, Jordan all-derivable points in the algebra of all upper triangular matrices, Linear Algebra Appl. 433 (2010), no. 11–12, 1922–1938, https://doi.org/10.1016/j.laa.2010.07.006. Search in Google Scholar

[16] J. Zhu and C. Xiong, Derivable mappings at unit operator on nest algebras, Linear Algebra Appl. 422 (2007), no. 2–3, 721–735, https://doi.org/10.1016/j.laa.2006.12.002. Search in Google Scholar

[17] D. Abulhamil, F. Jamjoom, and A. Peralta, Linear maps which are anti-derivable at zero, Bull. Malays. Math. Sci. Soc. 43 (2020), no. 2, 4315–4334, https://doi.org/10.1007/s40840-020-00918-7. Search in Google Scholar

[18] B. Fadaee and H. Ghahramani, Linear maps behaving like derivations or anti-derivations at orthogonal elements on C∗-algebras, Bull. Malays. Math. Sci. Soc. 43 (2020), no. 7, 2851–2859, https://doi.org/10.1007/s40840-019-00841-6. Search in Google Scholar

[19] A. Kishimoto, Dissipations and derivations, Commun. Math. Phys. 47 (1976), no. 1, 25–32, DOI: https://doi.org/10.1007/BF01609350. 10.1007/BF01609350Search in Google Scholar

[20] R. Kadison and J. Ringrose, Fundamentals of the theory of operator algebras, Vol. I, Elementary theory, Pure and Applied Mathematics, 100, Academic Press, New York, 1983. Search in Google Scholar

[21] M. Bresssar, Multiplication algebra and maps determined by zero products, Linear Multilinear Algebra 60 (2012), no. 7, 763–768, https://doi.org/10.1080/03081087.2011.564580. Search in Google Scholar

[22] M. Burgos, F. Fernández-Polo, and A. Peralta, Local triple derivations on C∗-algebras and JB∗-triples, Bull. Lond. Math. Soc. 46 (2014), no. 4, 709–724, https://doi.org/10.1112/blms/bdu024. Search in Google Scholar

[23] J. Alaminos, M. Bresssar, J. Extremera, and A. Villena, Orthogonality preserving linear maps on group algebras, Math. Proc. Camb. Phil. Soc. 158 (2015), no. 3, 493–504, https://doi.org/10.1017/S0305004115000110. Search in Google Scholar

[24] H. Dales, Banach Algebras and Automatic Continuity, London Mathematical Society Monographs Series 24, Oxford Univ. Press, New York, 2000. Search in Google Scholar

[25] A. Sinclair, Jordan automorphisms on a semisimple Banach algebra, Proc. Amer. Math. Soc. 25 (1970), no. 3, 526–528, https://doi.org/10.1090/S0002-9939-1970-0259604-2. Search in Google Scholar

[26] V. Losert, The derivation problem for group algebras, Ann. Math. 168 (2008), no. 1, 221–246, DOI: https://doi.org/10.4007/annals.2008.168.221. 10.4007/annals.2008.168.221Search in Google Scholar

[27] M. Takesaki, Theory of Operator Algebras, Springer-Verlag, New York, 2001. Search in Google Scholar

[28] A. Ben and M. Peralta, Linear maps on C∗-algebras which are derivations or triple derivations at a point, Linear Algebra Appl. 538 (2018), no. 1, 1–21, https://doi.org/10.1016/j.laa.2017.10.009. Search in Google Scholar