A Wigner-type theorem on symmetry transformations in Banach spaces (Q2714386)

The paper continues the author's series of papers on extensions of Wigner's theorem on symmetry transformations. The basic idea behind these papers is the author's discovery of a new, algebraic and elegant approach to the proof of Wigner's theorem. NEWLINENEWLINENEWLINEThree theorems are proved, and all the three proofs use the results from the theory of linear preservers. The first result can be viewed as a Banach space analogue of Wigner's therorem. It characterizes bijective functions \(\varphi:I_1(X)\to I_1(X)\), where \(I_1(X)\) denotes the set of all bounded rank-one idempotents on a (real or complex) Banach space \(X\), such that \(\text{tr} \varphi (P)\varphi(Q)=\text{tr} PQ\) for all \(P,Q\in I_1(X)\) (here, tr stands for the usual trace-functional). The other two results consider similar functions in a finite dimensional setting.