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On rings and Banach algebras with skew derivations. (English) Zbl 1464.16041

In this article, the author assumes that \(A\) represents a Banach algebra over the complex field, \(Z(A)\) denotes the center of \(A\) and \(M\) is a closed linear subspace of \(A\). The aim of this paper is to investigate the commutativity of a prime Banach algebra with skew derivations and prove that if \(A\) is prime Banach algebra and \(A\) has a nonzero continuous linear skew derivation \(F\) from \(A\) to \(A\) such that \([F(xm), F(yn)]- [xm, yn]\) in \(Z(A)\) for an integers \(m = m(x, y) > 1\) and \(n = n(x,y) > 1\) and sufficiently many \(x, y\), then \(A\) is commutative. Hence, the main result of this paper is:
Theorem 3. Let \(A\) be a prime Banach algebra and \(F\) be a continuous linear skew derivation. Suppose that there are non-empty open subsets \(G1\) and \(G2\) of \(A\) such that \([F(xm), F(yn)] - [xm, yn]\) in \(Z(A)\) for each \(x\) in \(G1\) and \(y\) in \(G2\). Then A is commutative.
To deduce the above result, the author obtains the following key theorem.
Theorem 2. Let \(R\) be a prime ring with characteristic different from two. If \(R\) admits a skew derivation \(F\) that satisfies \([F(xm), F(yn)] - [xm, yn]\) in \(Z(R)\) for all \(x, y\) in \(R\), where \(m\) and \(n\) are fixed positive integers, then \(R\) is commutative.

MSC:

16W25 Derivations, actions of Lie algebras
46J10 Banach algebras of continuous functions, function algebras
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References:

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