Characterizations of *-antiderivable mappings on operator algebras

1 Introduction

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

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

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

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 ℳ is called an antiderivable mapping at G if

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

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 ℳ is called a ∗ - A -bimodule if ℳ equipped with a ∗ -mapping from ℳ into itself, such that

whenever A in A , M , N in ℳ 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 ℳ be a ∗ - A -bimodule. A derivation δ from A into ℳ 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 ℳ 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, ℳ be a ∗ - A -bimodule and G be an element in A . A linear mapping δ from A into ℳ is called a ∗ -derivable mapping at G if

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

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 ℳ is an essentially Banach A -bimodule, and proved that if δ is a continuous ∗ -antiderivable mapping at the point zero from A into ℳ , then there exists a ∗ -derivation Δ from A into ℳ ♯ ♯ and ξ in ℳ ♯ ♯ such that δ ( A ) = Δ ( A ) + A ξ for every A in A , where ℳ ♯ ♯ is the second dual of ℳ . 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 ℳ , we call an element G in A a left (right) separating point of ℳ if G M = 0 ( M G = 0 ) implies M = 0 for every M in ℳ . It is easy to see that every left(right) invertible element in A is a left(right) separating point of ℳ . If G ∈ A is both the left and right separating point, then G is called a separating point of ℳ .

In Section 2, we prove that if A is a C ∗ -algebra, ℳ is a Banach ∗ - A -bimodule and G in A is a separating point of ℳ with A G = G A for every A ∈ A , then every ∗ -antiderivable mapping at G from A into ℳ 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, ℳ is an essential Banach ∗ - A -bimodule and δ is a continuous ∗ -antiderivable mapping at the point zero from A into ℳ , then there exists a ∗ -Jordan derivation Δ from A into ℳ ♯ ♯ and ξ in ℳ ♯ ♯ , such that δ ( A ) = Δ ( A ) + A ξ for every A in A , where ℳ ♯ ♯ 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.

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

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

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

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

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

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 ℳ is a Banach- A -bimodule. ℳ is called an essential Banach A -bimodule if

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 .

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

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