ACTA issues

Characterization of two-sided generalized derivations

Amin Hosseini

Acta Sci. Math. (Szeged) 86:3-4(2020), 577-600

Abstract. Let $\mathcal {A}$ be a unital semiprime, complex normed $\ast $-algebra and let $f, g, h :\mathcal {A} \rightarrow \mathcal {A}$ be linear mappings such that $f$ and $g + h$ are continuous. Under certain conditions, we prove that if $f(p \circ p) = g(p) \circ p + p \circ h(p)$ holds for any projection $p$ of $\mathcal {A}$, then $f$ and $g + h$ are two-sided generalized derivations, where $a \circ b = a b + ba$. We present some consequences of this result. Moreover, we show that if $\mathcal {A}$ is a semiprime algebra with the unit element $\textbf {e}$ and $n > 1$ is an integer such that the linear mappings $f, g\colon \mathcal {A} \rightarrow \mathcal {A}$ satisfy $f(x^n) = \sum _{j = 1}^{n}x^{n - j}g(x) x^{j - 1}$ for all $x \in \mathcal {A}$ and further $g(\textbf {e}) \in Z(\mathcal {A})$, then $f$ and $g$ are two-sided generalized derivations associated with the same derivation. Also, we show that if $\mathcal {A}$ is a unital, semiprime Banach algebra and $F, G\colon \mathcal {A} \rightarrow \mathcal {A}$ are linear mappings satisfying $F(b) = - b G(b^{-1}) b$ for all invertible elements $b \in \mathcal {A}$, then $F$ and $G$ are two-sided generalized derivations. Some other related results are also discussed.

DOI: 10.14232/actasm-020-295-8

AMS Subject Classification (1991): 47B47; 47B48, 39B05

Keyword(s): two-sided generalized derivation, generalized derivation, derivation, functional equation, normed algebra

received 5.4.2020, revised 21.7.2020, accepted 18.8.2020. (Registered under 45/2020.)