<abstract xmlns="http://www.w3.org/1999/xhtml">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 <italic>ℛ</italic> admits an additive mapping <italic>D</italic> : <italic>ℛ→ℛ</italic>satisfying either <italic>D</italic>(<italic>xyx</italic>) = <italic>D</italic>(<italic>xy</italic>)<italic>x*</italic>+ <italic>xyD</italic>(<italic>x</italic>) for all <italic>x,y</italic> ∈ <italic>ℛ</italic>, or <italic>D</italic>(<italic>xyx</italic>) = <italic>D</italic>(<italic>x</italic>)<italic>y*x*</italic>+ <italic>xD</italic>(<italic>yx</italic>) for all pairs <italic>x, y</italic> ∈ <italic>ℛ</italic>, then <italic>D</italic> is a <italic>*</italic>-derivation. Moreover this result makes it possible to prove that if <italic>ℛ</italic> satis es 2<italic>D</italic>(<italic>xn</italic>) = <italic>D</italic>(<italic>xn−</italic>1)<italic>x*</italic> + <italic>xn−</italic>1<italic>D</italic>(<italic>x</italic>) + <italic>D</italic>(<italic>x</italic>)(<italic>x*</italic>)<italic>n−</italic>1 + <italic>xD</italic>(<italic>xn−</italic>1) for all <italic>x</italic> ∈ <italic>ℛ</italic> and some xed integer <italic>n ≥</italic> 2, then <italic>D</italic> is a Jordan <italic>*</italic>-derivation under some torsion restrictions. Finally, we apply these purely ring theoretic results to standard operator algebras <italic>𝒜</italic>(<italic>ℋ</italic>). In particular, we prove that if <italic>ℋ</italic> be a real or complex Hilbert space, with <italic>dim</italic>(<italic>ℋ</italic>) <italic>></italic> 1, admitting a linear mapping <italic>D</italic> : <italic>𝒜</italic>(<italic>ℋ</italic>) <italic>→ ℬ</italic>(<italic>ℋ</italic>) (where <italic>ℬ</italic>(<italic>ℋ</italic>) stands for the bounded linear operators) such that<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_puma-2015-0024_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mn>2</mn><mi>D</mi><mrow><mo>(</mo><mrow><msup><mi>A</mi><mi>n</mi></msup></mrow><mo>)</mo></mrow><mo>=</mo><mi>D</mi><mrow><mo>(</mo><mrow><msup><mi>A</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><mo>)</mo></mrow><msup><mi>A</mi><mo>*</mo></msup><mo>+</mo><msup><mi>A</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>D</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>+</mo><mi>D</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>A</mi><mo>*</mo></msup></mrow><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mi>A</mi><mi>D</mi><mrow><mo>(</mo><mrow><msup><mi>A</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><mo>)</mo></mrow></mrow></math><tex-math>$$2D\left( {A^n } \right) = D\left( {A^{n - 1} } \right)A^* + A^{n - 1} D\left( A \right) + D\left( A \right)\left( {A^* } \right)^{n - 1} + AD\left( {A^{n - 1} } \right)$$</tex-math></alternatives></disp-formula>for all <italic>A</italic>∈<italic>𝒜</italic>(<italic>ℋ</italic>). Then <italic>D</italic> is of the form <italic>D</italic>(<italic>A</italic>) = <italic>AB−BA*</italic> for all <italic>A</italic>∈<italic>𝒜</italic>(<italic>ℋ</italic>) and some fixed <italic>B</italic> ∈ <italic>ℬ</italic>(<italic>ℋ</italic>), which means that <italic>D</italic> is Jordan <italic>*</italic>-derivation.</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 that2D(An)=D(An−1)A*+An−1D(A)+D(A)(A*)n−1+AD(An−1)$$2D\left( {A^n } \right) = D\left( {A^{n - 1} } \right)A^* + A^{n - 1} D\left( A \right) + D\left( A \right)\left( {A^* } \right)^{n - 1} + AD\left( {A^{n - 1} } \right)$$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.

On certain functional equations related to Jordan <italic>*</italic>-derivations in semiprime <italic>*</italic>-rings and standard operator algebras

Pure Mathematics and Applications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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

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...

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

Published Online: Aug 06, 2018

Page range: 1 - 17

Received: Apr 17, 2017

DOI: https://doi.org/10.1515/puma-2015-0024

Keywords
prime ring, semiprime ring, standard operator algebra, Jordan derivation, Jordan triple -derivation

© 2018 Mohammad Ashraf et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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

Published Online: Aug 06, 2018

Page range: 1 - 17

Received: Apr 17, 2017

DOI: https://doi.org/10.1515/puma-2015-0024

Keywordsprime ring, semiprime ring, standard operator algebra, Jordan *derivation, Jordan triple *-derivation

© 2018 Mohammad Ashraf et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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

Keywords
prime ring, semiprime ring, standard operator algebra, Jordan derivation, Jordan triple -derivation