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Maps on ${\cal B}({\cal H})$ 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 ${\cal H}$, let $\Gamma=\{A\in{\cal B}({\cal H}): A^2= \text{id}_{\cal H}\}$, and let $\varphi:{\cal B}({\cal H})\to{\cal B}({\cal H})$, such that $$ A-\lambda B\in\Gamma\iff \varphi(A)-\lambda\varphi(B)\in\Gamma\text{ for all }A,B\in{\cal B}({\cal H}), \quad \lambda\in\Bbb C. $$ Then either: {\parindent=7mm \item{(i)} $\varphi(A)=\pm TAT^{-1}$, with $T\in\text{GL}({\cal H})$, or \item{(ii)} $\varphi(A)=\pm TA^*T^{-1}$, with $T$ invertible and conjugate linear. \par}

47B49Transformers, preservers (operators on spaces of operators)
15A04Linear transformations, semilinear transformations (linear algebra)
Full Text: DOI
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