Characterizations of *-antiderivable mappings on operator algebras

Abstract

Abstract Let A {\mathcal{A}} be a ∗ \ast -algebra, ℳ {\mathcal{ {\mathcal M} }} be a ∗ \ast - A {\mathcal{A}} -bimodule, and δ \delta be a linear mapping from A {\mathcal{A}} into ℳ {\mathcal{ {\mathcal M} }} . δ \delta is called a ∗ \ast -derivation if δ ( A B ) = A δ ( B ) + δ ( A ) B \delta \left(AB)=A\delta \left(B)+\delta \left(A)B and δ ( A ∗ ) = δ ( A ) ∗ \delta \left({A}^{\ast })=\delta {\left(A)}^{\ast } for each A , B A,B in A {\mathcal{A}} . Let G G be an element in A {\mathcal{A}} , δ \delta is called a ∗ \ast -antiderivable mapping at G G if A B ∗ = G ⇒ δ ( G ) = B ∗ δ ( A ) + δ ( B ) ∗ A A{B}^{\ast }=G\Rightarrow \delta \left(G)={B}^{\ast }\delta \left(A)+\delta {\left(B)}^{\ast }A for each A , B A,B in A {\mathcal{A}} . We prove that if A {\mathcal{A}} is a C ∗ {C}^{\ast } -algebra, ℳ {\mathcal{ {\mathcal M} }} is a Banach ∗ \ast - A {\mathcal{A}} -bimodule and G G in A {\mathcal{A}} is a separating point of ℳ {\mathcal{ {\mathcal M} }} with A G = G A AG=GA for every A A in A {\mathcal{A}} , then every ∗ \ast -antiderivable mapping at G G from A {\mathcal{A}} into ℳ {\mathcal{ {\mathcal M} }} is a ∗ \ast -derivation. We also prove that if A {\mathcal{A}} is a zero product determined Banach ∗ \ast -algebra with a bounded approximate identity, ℳ {\mathcal{ {\mathcal M} }} is an essential Banach ∗ \ast - A {\mathcal{A}} -bimodule and δ \delta is a continuous ∗ \ast -antiderivable mapping at the point zero from A {\mathcal{A}} into ℳ {\mathcal{ {\mathcal M} }} , then there exists a ∗ \ast -Jordan derivation Δ \Delta from A {\mathcal{A}} into ℳ ♯ ♯ {{\mathcal{ {\mathcal M} }}}^{\sharp \sharp } and an element ξ \xi in ℳ ♯ ♯ {{\mathcal{ {\mathcal M} }}}^{\sharp \sharp } such that δ ( A ) = Δ ( A ) + A ξ \delta \left(A)=\Delta \left(A)+A\xi for every A A in A {\mathcal{A}} . Finally, we show that if A {\mathcal{A}} is a von Neumann algebra and δ \delta is a ∗ \ast -antiderivable mapping (not necessary continuous) at the point zero from A {\mathcal{A}} into itself, then there exists a ∗ \ast -derivation Δ \Delta from A {\mathcal{A}} into itself such that δ ( A ) = Δ ( A ) + A δ ( I ) \delta \left(A)=\Delta \left(A)+A\delta \left(I) for every A A in A {\mathcal{A}} .