Linear Preserves of BP-quasi invertible elements in JB*-algebras

Abstract

In this note, we study one of the main outcomes of the Russo-Dye Theorem of JB*-algebra: a linear operator that preserves Brown-Pedersen-quasi invertible elements between two JB*-algebras is characterized by a Jordan ∗-homomorphism. Earlier, in C*-setting of algebras, Russo and Dye gave a characterization of any linear operator that maps unitary elements into unitary elements; namely a Jordan ∗-homomorphism. Special sorts of linear preservers between C*-algebras and between JB*-triples were introduced by Burgos et al. As a result, if G is a linear operator between two JB*-algebras having non-empty sets of extreme points of the closed unit sphere that preserves extreme points, then there exists a Jordan ∗-homomorphism Φ which also preserves extreme points and characterizes the linear operator G. We also explore the connection between linear operators that strongly preserve Brown-Pedersen-quasi nvertible elements between two JB*-triples and the λ-property of both JB*-triples. Other geometric properties, such as extremally richness and the Bade property of two JB*-algebras or triples under linear preservers, are to be elaborated on in forthcoming research.