Linear maps on von-Neumann algebras behaving like anti-derivations at orthogonal elements

Through this paper all algebras and linear spaces are on the complex field ℂ. Let 𝒜 be an algebra and ℳ be an 𝒜-bimodule. The linear mapping 𝑑: 𝒜 → ℳ is called an anti-derivation if 𝑑(𝑥𝑦) = 𝑦𝑑(𝑥) + 𝑑(𝑦)𝑥 (𝑥, 𝑦 ∈ 𝒜). Also, 𝑑 is called a derivation if 𝑑(𝑥𝑦) = 𝑥𝑑(𝑦) + 𝑑(𝑥)𝑦 (𝑥, 𝑦 ∈ 𝒜). The linear mapping 𝛿: 𝒜 → ℳ is a Jordan derivation if 𝑑(𝑥 2 ) = 𝑥𝑑(𝑥) + 𝑑(𝑥)𝑥 (𝑥 ∈ 𝒜). Any anti-derivation and derivation is a Jordan derivation, but the converse is not necessarily true. Jordan in [1] has shown that every continuous Jordan derivation on C*-algebra 𝒜 into any Banach 𝒜-bimodule is a derivation. Derivations and anti-derivations are important classes of mappings on algebras which have been used to study of structure of algebras. We refer to [2] and the references there in. Bersar studied in [3] additive maps on prime ring contain a non-trivial idempotent satisfying 𝑥, 𝑦 ∈ 𝒜, 𝑥𝑦 = 0 ⟹ 𝛿(𝑥)𝑦 + 𝑥𝛿(𝑦) = 0 . Later, many studies have been done in this case and different results were obtained, for instance, see [4, 5, 6, 7, 8, 9] and the references therein. Recently [10, 11, 12, 13], the problem of characterizing continuous linear maps behaving like derivations or antiderivations at orthogonal elements for several types of orthogonality conditions on *- algebras have been studied. In this paper we study the above problems on von Neumann algebra.

In this article, the subsequent conditions on a continuous linear map 𝛿: 𝒜 → 𝒜 where 𝒜 is a *-algebra has been considered: 𝑥𝑦 ∗ = 0 ⟹ 𝑥𝛿(𝑦) ∗ + 𝛿(𝑥)𝑦 ∗ = 0, (𝑥 , 𝑦 ∈ 𝒜); 𝑥𝑦 ∗ = 0 ⟹ 𝑥 ∗𝛿(𝑦) + 𝑥𝛿(𝑦) ∗ = 0, (𝑥 , 𝑦 ∈ 𝒜). We consider following conditions on continuous linear map on von Neumann algebras: 𝑥𝑦 = 0 ⟹ 𝑦𝛿(𝑥) + 𝛿(𝑦)𝑥 = 0, (𝑥 , 𝑦 ∈ 𝒜); 𝑥𝑦 ∗ = 0 ⟹ 𝑦 ∗𝛿(𝑥) + 𝛿(𝑦) ∗𝑥 = 0, (𝑥 , 𝑦 ∈ 𝒜); 𝑥 ∗𝑦 = 0 ⟹ 𝑦𝛿(𝑥) ∗ + 𝛿(𝑦)𝑥 ∗ = 0, (𝑥 , 𝑦 ∈ 𝒜). Over methods are based on structure of von Neumann algebras and the fact that every derivation on von Neumann algebras is inner. Main Results The followings are the main results of our paper. Theorem. Let 𝒜 be a von Neumann algebra and 𝛿: 𝒜 → 𝒜 is a continuous linear map. Then 𝛿 satisfies 𝑦 𝛿(𝑥) + 𝛿(𝑦)𝑥 = 0 for all 𝑥 , 𝑦 ∈ 𝒜 with 𝑥𝑦 = 0 if only if there are elements 𝜇, 𝜈 ∈ 𝒜 such that 𝛿(𝑥) = 𝑥 𝜇 − 𝜈𝑥, where 𝜇 − 𝜈 ∈ 𝑍 (𝒜) and [[𝑥, 𝑦], 𝜇] + 2[𝑥, 𝑦](𝜇 − 𝜈) = 0 for all 𝑥 , 𝑦 ∈ 𝒜. Theorem. Let 𝒜 be a von Neumann algebra and 𝛿: 𝒜 → 𝒜 is a continuous linear map. Then 𝛿 satisfies 𝑦 ∗𝛿(𝑥) + 𝛿(𝑦) ∗𝑥 = 0 for all 𝑥 , 𝑦 ∈ 𝒜 with 𝑥𝑦 ∗ = 0 if only if there are elements 𝜇, 𝜈 ∈ 𝒜 such that 𝛿(𝑥) = 𝜈𝑥 − 𝜇𝑥, where Re𝜇 ∈ 𝑍 (𝒜) and [[𝑥, 𝑦], 𝜇] + (𝜈 − 𝜇) ∗ [𝑥, 𝑦] + [𝑥, 𝑦](𝜈 − 𝜇) = 0, for all 𝑥 , 𝑦 ∈ 𝒜. Theorem. Let 𝒜 be a von Neumann algebra and 𝛿: 𝒜 → 𝒜 is a continuous linear map. Then 𝛿 satisfies 𝛿(y)𝑥 ∗ + 𝑦𝛿(𝑥) ∗ = 0 for all 𝑥 , 𝑦 ∈ 𝒜 with 𝑥 ∗y = 0 if only if there are elements 𝜇, 𝜈 ∈ 𝒜 such that 𝛿(𝑥) = 𝑥𝜇 − 𝜈𝑥, where Re𝜇 ∈ 𝑍 (𝒜) and [[𝑥, 𝑦], 𝜇] + [𝑥, 𝑦](𝜇 − 𝜈) ∗ + (𝜇 − 𝜈)[𝑥, 𝑦] = 0, for all 𝑥 , 𝑦 ∈ 𝒜.

Let 𝒜 be a von Neumann algebra and 𝛿: 𝒜 → 𝒜 be a continuous linear map. Let 𝛿 be anti-derivation at orthogonal elements. We characterized the structure of 𝛿 according to the )generalized) inner derivation. We guess that the results obtained can also be proved on standard operator algebras.