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BY 4.0 license Open Access Published by De Gruyter Open Access June 29, 2022

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.

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Received: 2021-06-30
Revised: 2022-02-16
Accepted: 2022-03-05
Published Online: 2022-06-29

© 2022 Guangyu An et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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