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Characterization of two-sided generalized derivations

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Abstract

Let A be a unital semiprime, complex normed *-algebra and let f,g,h: A → A be linear mappings such that f and g + h are continuous. Under certain conditions, we prove that if f (p ○ p) = g(p) ○ p + p ○ h(p) holds for any projection p of A, then f and g + h are two-sided generalized derivations, where a ○ b = ab + ba. We present some consequences of this result. Moreover, we show that if A is a semiprime algebra with the unit element e and n > 1 is an integer such that the linear mappings f,g: A → A satisfy f (xn) = √j = 1nxn-jg(x)xj-1 for all x £ A and further g(e) ∈ Z(A), then f and g are two-sided generalized derivations associated with the same derivation. Also, we show that if A is a unital, semiprime Banach algebra and F, G: A → A are linear mappings satisfying F(b) = —bG(b-1)b for all invertible elements b ∈ A, then F and G are two-sided generalized derivations. Some other related results are also discussed.

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Authors and Affiliations

  1. Department of Mathematics, Kashmar Higher Education Institute, Kashmar, Iran

    Amin Hosseini

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  1. Amin Hosseini
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Correspondence to Amin Hosseini.

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Communicated by L. Molnár

Acknowledgments.

The author thanks the referee for his/her careful reading of the article and suggesting valuable comments that improved the quality of this work. Moreover, this research was supported by a grant from Kashmar Higher Education Institute [grant number 28/1348/97/21213].

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Hosseini, A. Characterization of two-sided generalized derivations. ActaSci.Math. 86, 577–600 (2020). https://doi.org/10.14232/actasm-020-295-8

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  • DOI: https://doi.org/10.14232/actasm-020-295-8

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