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Maps on () 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 , let Γ={A():A 2 =id }, and let ϕ:()(), such that

A-λBΓϕ(A)-λϕ(B)ΓforallA,B(),λ·

Then either:

ϕ(A)=±TAT -1 , with TGL(), or

ϕ(A)=±TA * T -1 , with T invertible and conjugate linear.

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