About the image of strongly generalized derivations of order $n$

Amin Hosseini

doi:10.5269/bspm.64567

Amin Hosseini Kashmar Institute

Résumé

Let A and B be two algebras. A linear mapping 𝜟:A → B is called a strongly generalized
derivation of order n, if there exist the families {𝐸_𝑘: 𝐴 → 𝐵}_{𝑘 = 1}^{𝑛}, {𝐹_𝑘: 𝐴 → 𝐵}_{𝑘 = 1}^{𝑛}, {𝐺_𝑘: 𝐴 → 𝐵}_{𝑘 = 1}^{𝑛} 𝑎𝑛𝑑 {𝐻_𝑘: 𝐴 → 𝐵}_{𝑘 = 1}^{𝑛} of linear mappings which satisfy 𝛥(𝑎𝑏) = Σ [𝐸_𝑘(𝑎) 𝐹_𝑘(𝑏) + 𝐺_𝑘(𝑎)𝐻_𝑘(𝑏)] 𝑛𝑘 =1 for all 𝑎, 𝑏 ⋲ 𝐴.
The main purpose of this paper is to study the image of such derivations. Our main result on
the image of strongly generalized derivations of order one reads as follows: Let A be a unital,
commutative Banach algebra and let 𝛥: 𝐴 → 𝐴 be a continuous strongly generalized
derivation of order one; that is, there exist the linear mappings 𝐸, 𝐹, 𝐺, 𝐻: 𝐴 → 𝐴 satisfying
𝐷(𝑎𝑏) = 𝐸(𝑎) 𝐹(𝑏) + 𝐺(𝑎) 𝐻(𝑏) for all 𝑎, 𝑏 ⋲ 𝐴. Let 𝐸, 𝐹, 𝐺 and 𝐻 be continuous linear
mappings. We prove that, under certain conditions, 𝐻 (𝐴), 𝐸(𝐴) 𝑎𝑛𝑑 𝛥(𝐴) are contained in
the Jacobson radical of A. This result generalizes Singer-Wermer theorem about the image of
continuous derivations on commutative Banach algebras.

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