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

Authors
Hoger Ghahranmani 1
Behrooz Fadaee 1
Kamal Fallahi 2
1 University of Kurdistan
2 Department of Mathematics, Payam Noor University of Technology
Abstract
Introduction

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 dxy=ydx+dyx (x,y∈A). Also, d is called a derivation if dxy=xdy+dxy (x,y∈A). The linear mapping δ:A→M is a Jordan derivation if dx2=xdx+dxx (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 ⟹δxy+xδy=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 anti-derivations 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.

Material and methods

In this article, the subsequent conditions on a continuous linear map δ:A→A where A is a *-algebra has been considered:

xy*=0⟹xδy*+δxy*=0, (x , y∈A);

xy*=0⟹x*δy+xδy*=0, x , y∈A.

We consider following conditions on continuous linear map on von Neumann algebras:

xy=0⟹yδx+δyx=0, (x , y∈A);

xy*=0⟹y*δx+δy*x=0, (x , y∈A);

x*y=0⟹yδx*+δyx*=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 δx+δyx=0 for all x , y∈A with xy=0 if only if there are elements μ,ν∈A such that δx=x μ-νx, where μ-ν∈Z (A) and [x,y,μ]+2x,yμ-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*δx+δy*x=0 for all x , y∈A with xy*=0 if only if there are elements μ,ν∈A such that δx=νx-μx, where Reμ∈Z (A) and

x,y,μ+ν-μ*x,y+x,yν-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 δyx*+yδx*=0 for all x , y∈A with x*y=0 if only if there are elements μ,ν∈A such that δx=xμ-νx, where Reμ∈Z (A) and

x,y,μ+x,yμ-ν*+μ-νx,y=0,

for all x , y∈A.

Conclusion

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.
Keywords
anti-derivations
orthogonal elements
von-Neumann algebras
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Mathematical Research
Volume 8, Issue 1 - Serial Number 20
Winter 2022
Pages 224-234