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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2403.18773 |
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| _version_ | 1866916432923590656 |
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| author | Hosseini, Amin Peralta, Antonio M. Su, Shanshan |
| author_facet | Hosseini, Amin Peralta, Antonio M. Su, Shanshan |
| contents | Let $\mathcal{M}$ be a Banach bimodule over an associative Banach algebra $\mathcal{A}$, and let $F: \mathcal{A}\to \mathcal{M}$ be a linear mapping. Three main uses of the term \emph{generalized derivation} are identified in the available literature, namely,
($\checkmark$) $F$ is a generalized derivation of the first type if there exists a derivation $ d : \mathcal{A}\to \mathcal{M}^{**}$ satisfying $F(a b ) = F(a) b + a d(b),$ for all $a,b\in \mathcal{A}$.
($\checkmark$) $F$ is a generalized derivation of the second type if there exists an element $ξ\in \mathcal{M}^{**}$ satisfying $F(a b ) = F(a) b + a F(b) - a ξb,$ for all $a,b\in \mathcal{A}$.
($\checkmark$) $F$ is a generalized derivation of the third type if there exist two (non-necessarily linear) mappings $G,H : \mathcal{A}\to \mathcal{M}$ satisfying $F(a b ) = G(a) b + a H(b),$ for all $a,b\in \mathcal{A}$.
These three types of maps are not, in general, equivalent. Although the first two notions are well studied when $\mathcal{A}$ is a C$^*$-algebra, their connections with the third one have not yet been explored. In this note we prove that every generalized derivation of the third type from a C$^*$-algebra $\mathcal{A}$ to a Banach $\mathcal{A}$-bimodule $\mathcal{M}$ is automatically continuous. We also show that every (continuous) generalized derivation of the third type from $\mathcal{A}$ to $\mathcal{M}$ is a generalized derivation of the first and second type. Consequently, the three notions coincide in this case. We also explore some concepts of generalized Jordan derivations on a C$^*$-algebra and establish some continuity properties for them. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18773 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the equivalence of all notions of generalized derivations whose domain is a C$^{\ast}$-algebra Hosseini, Amin Peralta, Antonio M. Su, Shanshan Operator Algebras Let $\mathcal{M}$ be a Banach bimodule over an associative Banach algebra $\mathcal{A}$, and let $F: \mathcal{A}\to \mathcal{M}$ be a linear mapping. Three main uses of the term \emph{generalized derivation} are identified in the available literature, namely, ($\checkmark$) $F$ is a generalized derivation of the first type if there exists a derivation $ d : \mathcal{A}\to \mathcal{M}^{**}$ satisfying $F(a b ) = F(a) b + a d(b),$ for all $a,b\in \mathcal{A}$. ($\checkmark$) $F$ is a generalized derivation of the second type if there exists an element $ξ\in \mathcal{M}^{**}$ satisfying $F(a b ) = F(a) b + a F(b) - a ξb,$ for all $a,b\in \mathcal{A}$. ($\checkmark$) $F$ is a generalized derivation of the third type if there exist two (non-necessarily linear) mappings $G,H : \mathcal{A}\to \mathcal{M}$ satisfying $F(a b ) = G(a) b + a H(b),$ for all $a,b\in \mathcal{A}$. These three types of maps are not, in general, equivalent. Although the first two notions are well studied when $\mathcal{A}$ is a C$^*$-algebra, their connections with the third one have not yet been explored. In this note we prove that every generalized derivation of the third type from a C$^*$-algebra $\mathcal{A}$ to a Banach $\mathcal{A}$-bimodule $\mathcal{M}$ is automatically continuous. We also show that every (continuous) generalized derivation of the third type from $\mathcal{A}$ to $\mathcal{M}$ is a generalized derivation of the first and second type. Consequently, the three notions coincide in this case. We also explore some concepts of generalized Jordan derivations on a C$^*$-algebra and establish some continuity properties for them. |
| title | On the equivalence of all notions of generalized derivations whose domain is a C$^{\ast}$-algebra |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2403.18773 |