2022/04/07 by Peralta, Antonio M.
#17C65 #46L60 #47N50 Secondary 81R15 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Primary 47B49
paper · doi:10.48550/arxiv.2204.03463
Let e and v be minimal tripotents in a JBW^*-triple M. We introduce the notion of triple transition pseudo-probability from e to v as the complex number TTP(e,v)= φv(e), where φv is the unique extreme point of the closed unit ball of M_* at which v attains its norm. In the case of two minimal projections in a von Neumann algebra, this correspond to the usual transition probability. We prove that every bijective transformation Φ preserving triple transition pseudo-probabilities between the lattices of tripotents of two atomic JBW^*-triples M and N admits an extension to a bijective \rm(complex\rm) linear mapping between the socles of these JBW^*-triples. If we additionally assume that Φ preserves orthogonality, then Φ can be extended to a surjective (complex-)linear \rm(isometric\rm) triple isomorphism from M onto N. In case that M and N are two spin factors or two type 1 Cartan factors we show, via techniques and results on preservers, that every bijection preserving triple transition pseudo-probabilities between the lattices of tripotents of M and N automatically preserves orthogonality, and hence admits an extension to a triple isomorphism from M onto N.