This paper deals with some identities on Banach $ ^* $-algebras that are equipped with linear generalized derivations. As an application of one of our results, we describe the structure of the underlying algebras. Precisely, we prove that for a linear generalized derivation $ F $ on a Banach $ ^* $-algebra $ A $, either we obtain the existence of a central idempotent element $ e\in Q $, for which $ F = 0 $ on $ eQ $ and $ (1-e)Q $ satisfies $ s_{4} $, or the set of elements $ u\in A $ such that the identity $ [F(u)^n, F(u^*)^nF(u)^n]\in Z(A) $ holds for no positive integer $ n $ turns out to be dense. In addition to this we consider an identity satisfied by a semisimple Banach $ ^* $-algebra and look for its commutativity. Moreover, some related results are also established.
Citation: Shakir Ali, Ali Yahya Hummdi, Mohammed Ayedh, Naira Noor Rafiquee. Linear generalized derivations on Banach $ ^* $-algebras[J]. AIMS Mathematics, 2024, 9(10): 27497-27511. doi: 10.3934/math.20241335
This paper deals with some identities on Banach $ ^* $-algebras that are equipped with linear generalized derivations. As an application of one of our results, we describe the structure of the underlying algebras. Precisely, we prove that for a linear generalized derivation $ F $ on a Banach $ ^* $-algebra $ A $, either we obtain the existence of a central idempotent element $ e\in Q $, for which $ F = 0 $ on $ eQ $ and $ (1-e)Q $ satisfies $ s_{4} $, or the set of elements $ u\in A $ such that the identity $ [F(u)^n, F(u^*)^nF(u)^n]\in Z(A) $ holds for no positive integer $ n $ turns out to be dense. In addition to this we consider an identity satisfied by a semisimple Banach $ ^* $-algebra and look for its commutativity. Moreover, some related results are also established.
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Shakir Ali, Ali Yahya Hummdi, Mohammed Ayedh, Naira Noor Rafiquee. Linear generalized derivations on Banach $ ^* $-algebras[J]. AIMS Mathematics, 2024, 9(10): 27497-27511. doi: 10.3934/math.20241335
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