Preservers of triple transition pseudo-probabilities in connection with orthogonality preservers and surjective isometries

Abstract

We prove that every bijection preserving triple transition pseudo-probabilities between the sets of minimal tripotents of two atomic JBW${}^{\ast }$-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${}^{\ast }$-triples is precisely the restriction of a (complex-)linear triple isomorphism between the corresponding JBW${}^{\ast }$-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${}^{\ast }$-triples admits an extension to a real linear surjective isometry between these two JBW${}^{\ast }$-triples. We also show that the class of surjective isometries between the sets of minimal tripotents in two atomic JBW${}^{\ast }$-triples is, in general, strictly wider than the set of bijections preserving triple transition pseudo-probabilities.