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A generalization of peripherally-multiplicative surjections between standard operator algebras. (English) Zbl 1197.47051

Let \({\mathcal A}\subseteq B(X)\) and \({\mathcal B}\subseteq B(Y)\) be (possibly non-closed and non-unital) subalgebras of bounded operators on complex Banach spaces \(X\) and \(Y\), both containing the ideal of finite-rank operators. The authors classify pairs of surjective maps \(\phi,\psi:{\mathcal A}\to{\mathcal B}\) with the property that
\[ \sigma_{\pi}(\phi(S)\psi(T))=\sigma_{\pi}(ST),\quad S,T\in{\mathcal A}, \]
where \(\sigma_{\pi}(A):=\{\lambda\in\mathbb C\mid \exists (\lambda-A)^{-1}\in B(X)\text{ and } |\lambda|=\lim\|A^n\|^{1/n}\}\) is a peripheral spectrum with respect to the whole algebra \(B(X)\) (which can be replaced with any closed unital subalgebra in \(B(X)\)). In particular, it is shown that such maps are automatically linear, bounded, and bijective, and if, in addition, \(1\in{\mathcal A}\) and \(\phi(1)=1\), then it follows that \(\phi=\psi\) is an isomorphism or an anti-isomorphism.
The main idea is the following reduction to rank-one operators. Given a nonzero operator \(T\in{\mathcal A}\), we have rank\(\,T=1\) if and only if \(\sigma_{\pi}(ST)\) is a singleton for each \(S\in{\mathcal A}\).

MSC:

47B49 Transformers, preservers (linear operators on spaces of linear operators)
47L10 Algebras of operators on Banach spaces and other topological linear spaces
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