2022/08/02 by Antonio M. Peralta, Peralta, Antonio M.
Physics and Astronomy · Computer Science · Mathematics · #Quantum Mechanics and Applications #Advanced Algebra and Logic #Advanced Topology and Set Theory
paper · pdf · doi:10.48550/arxiv.2208.01464
We prove that every bijection preserving triple transition pseudo-probabilities between the sets of minimal tripotents of two atomic JBW^*-triples automatically preserves orthogonality in both directions. Consequently, each bijection preserving triple transition pseudo-probabilities between the sets of minimal tripotents of two atomic JBW^*-triples is precisely the restriction of a (complex-)linear triple isomorphism between the corresponding JBW^*-triples. This result can be regarded as triple version of the celebrated Wigner theorem for Wigner symmetries on the posets of minimal projections in B(H). We also present a Tingley type theorem by proving that every surjective isometry between the sets of minimal tripotents in two atomic JBW^*-triples admits an extension to a real linear surjective isometry between these two JBW^*-triples. We also show that the class of surjective isometries between the sets of minimal tripotents in two atomic JBW^*-triples is, in general, strictly wider than the set of bijections preserving triple transition pseudo-probabilities.