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A ternary characterization of automorphisms of \({\mathbb B}({\mathcal H})\). (English) Zbl 1226.47037

Let \({\mathcal B}(H)\) be the algebra of all bounded linear operators on a complex Hilbert space \(H\) and let \(\sigma(T)\) stand for the spectrum of \(T\in{\mathcal B}(H)\). The authors show that a \(*\)-surjective mapping \(\varphi\) on \({\mathcal B}(H)\) is either an algebra automorphism or an algebra anti-automorphism provided that one of the following conditions holds:
(1) \(\sigma(\varphi(A)\varphi(B)\varphi(A)^*)=\sigma(ABA^*)\) for all \(A,B\in{\mathcal B}(H)\).
(2) \(\sigma(|\varphi(A)|^2\varphi(B))=\sigma(|A|^2B)\) for all \(A,B\in{\mathcal B}(H)\).

MSC:

47B49 Transformers, preservers (linear operators on spaces of linear operators)
46L05 General theory of \(C^*\)-algebras
47A10 Spectrum, resolvent
47L30 Abstract operator algebras on Hilbert spaces
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References:

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