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mmr 2022, 8(1): 224-234 |
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Ghahranmani H, Fadaee B, Fallahi K. Linear maps on von-Neumann algebras behaving like anti-derivations at orthogonal elements. mmr 2022; 8 (1) :224-234
URL: http://mmr.khu.ac.ir/article-1-3147-en.html
URL: http://mmr.khu.ac.ir/article-1-3147-en.html
1- University of Kurdistan , h.ghahramani@uok.ac.ir
2- University of Kurdistan
3- Department of Mathematics, Payam Noor University of Technology
2- University of Kurdistan
3- Department of Mathematics, Payam Noor University of Technology
Abstract: (1101 Views)
Introduction
Through this paper all algebras and linear spaces are on the complex fieldd : d xy = yd x + d y x ( x , y ∈ d d xy = xd y + d x y ( x , y ∈ δ : d x 2 = xd x + d x x ( x ∈
Bersar studied in [3] additive maps on prime ring contain a non-trivial idempotent satisfying
x , y ∈ 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 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 mapδ :
x y * =0 ⟹ xδ y * + δ x y * =0, ( x , y ∈
x y * =0⟹ x * δ y + xδ y * =0, x , y ∈
We consider following conditions on continuous linear map on von Neumann algebras:
xy yδ x δ y x , ( x y
x y y δ x δ y x x y
x y yδ x δ y x x y
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. Letδ : δ y δ x + δ y x =0 x y xy =0 μ , ν ∈ δ x = x μ - νx μ - ν ∈ Z ( x , y , μ ]+2 x , y μ - ν =0 x y
Theorem. Letδ : δ y δ x δ y x x y x y μ , ν ∈ δ x = νx - μx μ ∈ Z (
x , y , μ + ν - μ * x , y + x , y ν - μ =0
for allx y
Theorem. Letδ : δ δ x * + yδ x * x y x μ , ν ∈ δ x = xμ - νx μ ∈ Z (
x , y , μ + x , y μ - ν * + μ - ν x , y =0
for allx y
Conclusion
Letδ : δ δ
We guess that the results obtained can also be proved on standard operator algebras.
Through this paper all algebras and linear spaces are on the complex field
Bersar studied in [3] additive maps on prime ring contain a non-trivial idempotent satisfying
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
We consider following conditions on continuous linear map on von Neumann algebras:
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
Theorem. Let
for all
Theorem. Let
for all
Conclusion
Let
We guess that the results obtained can also be proved on standard operator algebras.
Type of Study: S |
Subject:
Operator theorey
Received: 2020/11/9 | Revised: 2023/06/18 | Accepted: 2021/01/31 | Published: 2022/05/14 | ePublished: 2022/05/14
Received: 2020/11/9 | Revised: 2023/06/18 | Accepted: 2021/01/31 | Published: 2022/05/14 | ePublished: 2022/05/14
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