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Fixed product preserving mappings on Banach algebras. (English) Zbl 1518.47063

Let \(\mathcal{A}\) and \(\mathcal{B}\) be two complex unital Banach algebras and let \(\Phi: \mathcal{A} \to \mathcal{B}\) be a bijective linear map. Given \(c \in \mathcal{A}\) and \(d \in \mathcal{B}\), the author investigates the following general property \[ ab=c \Rightarrow \phi (a)\phi (b)=d. \] Typical examples are scalar multiples of isomorphisms. The case where \(\mathcal{A}= \mathcal{B}=M_n(\mathbb{C})\) was treated in [L. Catalano and H. Julius, J. Algebra 575, 220–232 (2021; Zbl 1475.15036); C. Costara, J. Algebra 587, 336–343 (2021; Zbl 1478.15004)]. In the present paper, the author provides a complete description of the following cases:
(1)
\(\mathcal{A}= \mathcal{B}(X)\), the algebra formed by all bounded operators on an infinite-dimensional complex Banach space \(X\), \(\mathcal{B}\) is a prime algebra, and the operator \(c\) has finite rank.
(2)
\(d\) is invertible.
Moreover, the author investigates the relation between the elements \(c\) and \(d\) in the case where \(\Phi\) exists.

MSC:

47B48 Linear operators on Banach algebras
47B49 Transformers, preservers (linear operators on spaces of linear operators)
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