Characterizations of *-antiderivable mappings on operator algebras

Let A be a*-algebra, A be a*-A-bimodule, and delta be a linear mapping from A into M. delta is called a*-derivation if delta(AB) = A delta(B) + delta(A)B and delta(A*) = delta(A)* for each A, B in A. Let G be an element in A, delta is called a *-antiderivable mapping at G if AB* = G double right arrow delta(G) = B*delta(A) + delta(B)*A for each A, B in A. We prove that if A is a *C-algebra, A is a Banach*- A-bimodule andG in A is a separating point of A with AG = GA for every A in A, then every *-antiderivable mapping at G from A into A is a *-derivation. We also prove that if A is a zero product determined Banach *-algebra with a bounded approximate identity, A is an essential Banach *- A-bimodule and d is a continuous *-antiderivable mapping at the point zero from A into A, then there exists a *-Jordan derivation. from A into A.. and an element xi in M-# such that delta(A) = Delta(A) + A xi for every A in A. Finally, we show that if A is a von Neumann algebra and d is a *-antiderivable mapping (not necessary continuous) at the point zero from A into itself, then there exists a *-derivation. from A into itself such that delta(A) = Delta(A) + A delta(I) for every A in A.

Automated summary

This article discusses the relationship between *-derivations and *-antiderivable mappings, as well as their properties and existence under different conditions. The research results show that under certain conditions, *-antiderivable mappings can be proven as *-derivations, and there is a connection between derivations and Jordan derivations. In addition, we also prove the existence of *-derivations in von Neumann algebras.