Maps preserving triple transition pseudo-probabilities

TL;DR

This paper characterizes transformations preserving a generalized transition probability between tripotents in JBW*-triples, showing they extend to (isometric) triple isomorphisms under certain conditions.

Contribution

It introduces the notion of triple transition pseudo-probability in JBW*-triples and characterizes bijections preserving this quantity as extendable to (isometric) triple isomorphisms.

Findings

Preservers of triple transition pseudo-probabilities extend to linear or isometric triple isomorphisms.

In spin factors and Cartan factors, such preservers automatically preserve orthogonality.

The results unify and generalize transition probability preservation in operator algebra contexts.

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 $TTP(e,v)= \varphi_v(e),$ where $\varphi_v$ 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 $\Phi$ 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 $\Phi$ preserves orthogonality, then $\Phi$ 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 $N$.