A generalization of a theorem of Chernoff on standard operator algebras

2021 ◽  
Author(s):  
Irena Kosi-Ulbl ◽  
Ángel Rodríguez Palacios ◽  
Joso Vukman
Keyword(s):  
Operator Algebras ◽  
Standard Operator ◽  
Standard Operator Algebras

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.

Elementary Operators on Standard Operator Algebras

2002 ◽  
Vol 50 (4) ◽  
pp. 315-319 ◽  
Author(s):  
Lajos Molnár ◽  
Peter Šemrl
Keyword(s):  
Operator Algebras ◽  
Standard Operator ◽  
Elementary Operators ◽  
Standard Operator Algebras

scholarly journals On certain functional equations related to Jordan *-derivations in semiprime *-rings and standard operator algebras

2018 ◽  
Vol 27 (1) ◽  
pp. 1-17
Author(s):  
Mohammad Ashraf ◽  
Bilal Ahmad Wani
Keyword(s):  
Operator Algebras ◽  
Additive Mapping ◽  
Complex Hilbert Space ◽  
Linear Mapping ◽  
Linear Operators ◽  
Bounded Linear ◽  
Standard Operator ◽  
Semiprime Rings ◽  
Standard Operator Algebras ◽  
Torsion Free

Abstract The purpose of this paper is to investigate identities with Jordan *-derivations in semiprime *-rings. Let ℛ be a 2-torsion free semiprime *-ring. In this paper it has been shown that, if ℛ admits an additive mapping D : ℛ→ℛsatisfying either D(xyx) = D(xy)x*+ xyD(x) for all x,y ∈ ℛ, or D(xyx) = D(x)y*x*+ xD(yx) for all pairs x, y ∈ ℛ, then D is a *-derivation. Moreover this result makes it possible to prove that if ℛ satis es 2D(xn) = D(xn−1)x* + xn−1D(x) + D(x)(x*)n−1 + xD(xn−1) for all x ∈ ℛ and some xed integer n ≥ 2, then D is a Jordan *-derivation under some torsion restrictions. Finally, we apply these purely ring theoretic results to standard operator algebras 𝒜(ℋ). In particular, we prove that if ℋ be a real or complex Hilbert space, with dim(ℋ) > 1, admitting a linear mapping D : 𝒜(ℋ) → ℬ(ℋ) (where ℬ(ℋ) stands for the bounded linear operators) such that $$2D\left( {A^n } \right) = D\left( {A^{n - 1} } \right)A^* + A^{n - 1} D\left( A \right) + D\left( A \right)\left( {A^* } \right)^{n - 1} + AD\left( {A^{n - 1} } \right)$$ for all A∈𝒜(ℋ). Then D is of the form D(A) = AB−BA* for all A∈𝒜(ℋ) and some fixed B ∈ ℬ(ℋ), which means that D is Jordan *-derivation.

Linear Maps on Standard Operator Algebras Characterized by Action on Zero Products

2019 ◽  
Vol 45 (5) ◽  
pp. 1573-1583
Author(s):  
Amin Barari ◽  
Behrooz Fadaee ◽  
Hoger Ghahramani
Keyword(s):  
Operator Algebras ◽  
Standard Operator ◽  
Linear Maps ◽  
Standard Operator Algebras

Characterization of linear mappings on (Banach) ⋆-algebras by similar properties to derivations

2020 ◽  
Vol 70 (4) ◽  
pp. 1003-1011
Author(s):  
Behrooz Fadaee ◽  
Kamal Fallahi ◽  
Hoger Ghahramani
Keyword(s):  
Banach Algebra ◽  
Complex Variable ◽  
Operator Algebras ◽  
Banach Algebras ◽  
Jordan Derivation ◽  
Continuous Linear ◽  
Linear Map ◽  
Standard Operator ◽  
Standard Operator Algebras

AbstractLet 𝓐 be a ⋆-algebra, δ : 𝓐 → 𝓐 be a linear map, and z ∈ 𝓐 be fixed. We consider the condition that δ satisfies xδ(y)⋆ + δ(x)y⋆ = δ(z) (x⋆δ(y) + δ(x)⋆y = δ(z)) whenever xy⋆ = z (x⋆y = z), and under several conditions on 𝓐, δ and z we characterize the structure of δ. In particular, we prove that if 𝓐 is a Banach ⋆-algebra, δ is a continuous linear map, and z is a left (right) separating point of 𝓐, then δ is a Jordan derivation. Our proof is based on complex variable techniques. Also, we describe a linear map δ satisfying the above conditions with z = 0 on two classes of ⋆-algebras: zero product determined algebras and standard operator algebras.

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