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Maps on \({\mathcal B}({\mathcal H})\) preserving involution. (English) Zbl 1185.47037

This paper belongs to a recent spate of papers (see, for example, [Linear Algebra Appl.431, No.5–7, 833–842 (2009; Zbl 1183.47031); ibid., 974–984 (2009; Zbl 1183.15017)] in the same issue as the paper being reviewed, and references therein). The common thread of these papers is to characterize maps on Hilbert space satisfying a certain property (preservers), without assuming linearity. For instance, in the paper under review, the following is proven:
Given an infinite-dimensional Hilbert space \({\mathcal H}\), let \(\Gamma=\{A\in{\mathcal B}({\mathcal H}): A^2= \text{id}_{\mathcal H}\}\), and let \(\varphi:{\mathcal B}({\mathcal H})\to{\mathcal B}({\mathcal H})\), such that
\[ A-\lambda B\in\Gamma\iff \varphi(A)-\lambda\varphi(B)\in\Gamma\text{ for all }A,B\in{\mathcal B}({\mathcal H}), \quad \lambda\in\mathbb C. \]
Then either:
(i)
\(\varphi(A)=\pm TAT^{-1}\), with \(T\in\text{GL}({\mathcal H})\), or
(ii)
\(\varphi(A)=\pm TA^*T^{-1}\), with \(T\) invertible and conjugate linear.

MSC:

47B49 Transformers, preservers (linear operators on spaces of linear operators)
15A04 Linear transformations, semilinear transformations
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