Maps preserving triple transition pseudo-probabilities (Research on preserver problems on Banach algebras and related topics)

Abstract

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 (complex) 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 (isometric) 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.