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

TL;DR

This paper characterizes bijections preserving triple transition pseudo-probabilities in atomic JBW*-triples, showing they are restrictions of linear triple isomorphisms, and extends isometries between minimal tripotents to the entire triples.

Contribution

It establishes that such bijections are exactly restrictions of linear triple isomorphisms and extends Tingley-type theorems to atomic JBW*-triples.

Findings

Bijections preserving triple transition pseudo-probabilities are orthogonality-preserving.

Such bijections are restrictions of complex-linear triple isomorphisms.

Surjective isometries extend to real linear isometries between the triples.

Abstract

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