scholarly journals Maps preserving 2-idempotency of certain products of operators

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3909-3916
Author(s):  
Hossein Khodaiemehr ◽  
Fereshteh Sady
Keyword(s):  
Banach Spaces ◽  
Operator Algebras ◽  
Triple Product ◽  
Jordan Product ◽  
Standard Operator ◽  
Scalar Multiple ◽  
Standard Operator Algebras ◽  
Jordan Triple Product ◽  
Usual Product

Let A,B be standard operator algebras on complex Banach spaces X and Y of dimensions at least 3, respectively. In this paper we give the general form of a surjective (not assumed to be linear or unital) map ? : A ? B such that ?2 : M2(C)?A ? M2(C)?B defined by ?2((sij)2x2) = (?(sij))2x2 preserves nonzero idempotency of Jordan product of two operators in both directions. We also consider another specific kinds of products of operators, including usual product, Jordan semi-triple product and Jordan triple product. In either of these cases it turns out that ? is a scalar multiple of either an isomorphism or a conjugate isomorphism.

scholarly journals Maps Preserving k-Jordan Products on Operator Algebras

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 814
Author(s):  
Xiaofei Qi ◽  
Miaomiao Wang
Keyword(s):  
Positive Integer ◽  
Operator Algebras ◽  
Jordan Product ◽  
Standard Operator ◽  
Standard Operator Algebras ◽  
Nonlinear Maps ◽  
Jordan Products ◽  
Additive Maps

For any positive integer k, the k-Jordan product of a , b in a ring R is defined by { a , b } k = { { a , b } k − 1 , b } 1 , where { a , b } 0 = a and { a , b } 1 = a b + b a . A map f on R is k-Jordan zero-product preserving if { f ( a ) , f ( b ) } k = 0 whenever { a , b } k = 0 for a , b ∈ R ; it is strong k-Jordan product preserving if { f ( a ) , f ( b ) } k = { a , b } k for all a , b ∈ R . In this paper, strong k-Jordan product preserving nonlinear maps on general rings and k-Jordan zero-product preserving additive maps on standard operator algebras are characterized, generalizing some known results.

scholarly journals Maps Completely Preserving Involutions and Maps Completely Preserving Drazin Inverse

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Hongmei Yao ◽  
Baodong Zheng ◽  
Gang Hong
Keyword(s):  
Banach Spaces ◽  
Operator Algebras ◽  
Drazin Inverse ◽  
Complex Field ◽  
The Real ◽  
Infinite Dimensional ◽  
Standard Operator ◽  
Standard Operator Algebras ◽  
Preserver Problems

Let X and Y be infinite dimensional Banach spaces over the real or complex field 𝔽, and let 𝒜 and ℬ be standard operator algebras on X and Y, respectively. In this paper, the structures of surjective maps from 𝒜 onto ℬ that completely preserve involutions in both directions and that completely preserve Drazin inverse in both direction are determined, respectively. From the structures of these maps, it is shown that involutions and Drazin inverse are invariants of isomorphism in complete preserver problems.

A characterization of an isomorphism

2018 ◽  
Vol 11 (02) ◽  
pp. 1850022
Author(s):  
Ali Taghavi ◽  
Roja Hosseinzadeh ◽  
Efat Nasrollahi
Keyword(s):  
Banach Spaces ◽  
Operator Algebras ◽  
Standard Operator ◽  
Linear Maps ◽  
Standard Operator Algebras

Let [Formula: see text] and [Formula: see text] be some standard operator algebras on complex Banach spaces [Formula: see text] and [Formula: see text], respectively, and [Formula: see text] be a polynomial with no repeated roots and [Formula: see text], such that [Formula: see text]. We characterize the forms of surjective linear maps [Formula: see text] which preserve the nonzero products of operators that annihilated by [Formula: see text].

Maps preserving strong 2-Jordan product on some algebras

2017 ◽  
Vol 10 (03) ◽  
pp. 1750044 ◽  
Author(s):  
Ali Taghavi ◽  
Farzaneh Kolivand
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Operator Algebras ◽  
Jordan Product ◽  
Von Neumann ◽  
Standard Operator ◽  
Nest Algebras ◽  
Concrete Form ◽  
Standard Operator Algebras ◽  
Nonzero Scalar

Let [Formula: see text] be a surjective map between some operator algebras such that [Formula: see text] for all [Formula: see text], where [Formula: see text] defined by [Formula: see text] and [Formula: see text] is Jordan product, i.e. [Formula: see text]. In this paper, we determine the concrete form of map [Formula: see text] on some operator algebras. Such operator algebras include standard operator algebras, properly infinite von Neumann algebras and nest algebras. Particularly, if [Formula: see text] is a factor von Neumann algebra that satisfies [Formula: see text] for all [Formula: see text] and idempotents [Formula: see text] then there exists nonzero scalar [Formula: see text] with [Formula: see text] such that [Formula: see text] for all [Formula: see text]

Mappings which Preserve Idempotents, Local Automorphisms, and Local Derivations

1993 ◽  
Vol 45 (3) ◽  
pp. 483-496 ◽  
Author(s):  
Matej Brešar ◽  
Peter Šemrl
Keyword(s):  
Local Ring ◽  
Operator Algebras ◽  
Banach Algebras ◽  
Continuous Linear ◽  
Matrix Algebras ◽  
Standard Operator ◽  
Weakly Continuous ◽  
Standard Operator Algebras ◽  
Jordan Homomorphisms

AbstractIt is proved that linear mappings of matrix algebras which preserve idempotents are Jordan homomorphisms. Applying this theorem we get some results concerning local derivations and local automorphisms. As an another application, the complete description of all weakly continuous linear surjective mappings on standard operator algebras which preserve projections is obtained. We also study local ring derivations on commutative semisimple Banach algebras.

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