The structure of Banach algebras \(A\), satisfying \(xAx=x^2 Ax^2\) for every \(x\in A\) (Q2740970)
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scientific article; zbMATH DE number 1642183
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | The structure of Banach algebras \(A\), satisfying \(xAx=x^2 Ax^2\) for every \(x\in A\) |
scientific article; zbMATH DE number 1642183 |
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9 September 2001
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The structure of Banach algebras \(A\), satisfying \(xAx=x^2 Ax^2\) for every \(x\in A\) (English)
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Let \(A\) be a Banach algebra. The authors show that the following are equivalent:NEWLINENEWLINENEWLINE(1) \(xAx= x^2Ax^2\) for every \(x\in A\),NEWLINENEWLINENEWLINE(2) \(A= B\oplus R\), where \(B\) is a subalgebra of \(A\) isomorphic to \(\mathbb{C}^n\) for some \(n\geq 0\), \(xAx= (0)\) for every \(x\in R\) and \(ayx= yxa\) for every \(a\in B\) and \(x,y\in R\).NEWLINENEWLINENEWLINEAs a result, it follows that if \(Ax= Ax^2\) for every \(x\in A\) then (1) is valid.
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