Maps preserving triple transition pseudo-probabilities

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 T T P (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 .

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

“…Here, instead of combining classical tools on preservers for concrete Cartan factors, we shall turn our point of view to a completely newfangled strategy with arguments and tools taken from abstract theory of JB * -triples. As we shall see in section 2, the achievements in [39,Theorem 2.3] prove that each bijective transformation Φ preserving triple transition pseudo-probabilities between the sets of minimal tripotents of two atomic JBW * -triples M and N , admits an extension to a bijective (complex) linear mapping T 0 from the socle of M onto the socle of N whose restriction to U min (M ) is Φ, where the socle of a JB * -triple is the subspace linearly generated by its minimal tripotents. If we additionally assume that Φ preserves orthogonality, then Φ admits an extension to a surjective (complex-)linear (isometric) triple isomorphism from M onto N (cf.…”

“…This problem has been positively solved when M and N are both Cartan factors of type 1 (i.e. Banach spaces B(H, K) of bounded linear operators between complex Hilbert spaces) or when M and N are both type 4 or spin Cartan factors (see [39,Theorems 4.4 and 3.2]). It is worth to note that the proof of the results is built upon classic theorems on preservers and concrete tools for operator spaces and Hilbert spaces.…”

Preservers of Triple Transition Pseudo-Probabilities in Connection with Orthogonality Preservers and Surjective Isometries

“…If we fix a minimal partial isometry e in B(H), the functional ϕ e (x) = tr(e * x) is the unique extreme point of the closed unit ball of B(H) * , the predual of B(H), at which e attains its norm, so tr(e * v) = ϕ e (v). This is the crucial point to consider the notion of triple transition pseudo-probability from a minimal tripotent to another minimal tripotent in an arbitrary JBW * -triple as introduced in the recent reference [39]. More concretely, for each minimal tripotent e in a JBW * -triple, M, there exists a unique pure atom (i.e.…”

“…As we shall see in Sect. 2, the achievements in [39,Theorem 2.3] prove that each bijective transformation Φ preserving triple transition pseudo-probabilities between the sets of minimal tripotents of two atomic JBW * -triples M and N , admits an extension to a bijective (complex) linear mapping T 0 from the socle of M onto the socle of N whose restriction to U min (M ) is Φ, where the socle of a JB * -triple is the subspace linearly generated by its minimal tripotents. If we additionally assume that Φ preserves orthogonality, then Φ admits an extension to a surjective (complex-)linear (isometric) triple isomorphism from M onto N (cf.…”

“…If we additionally assume that Φ preserves orthogonality, then Φ admits an extension to a surjective (complex-)linear (isometric) triple isomorphism from M onto N (cf. [39,Corollary 2.5]). In Theorem 3.2 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.…”