Linear maps between C*-algebras preserving extreme points and strongly linear preservers

Abstract

We study new classes of linear preservers between C${}^{\ast }$-algebras and JB${}^{\ast }$-triples. Let $E$ and $F$ be JB${}^{\ast }$-triples with ${\partial }_e(E_1)$. We prove that every linear map $T:E\to F$ strongly preserving Brown-Pedersen quasi-invertible elements is a triple homomorphism. Among the consequences, we establish that, given two unital C${}^{\ast }$-algebras $A$ and $B,$ for each linear map $T$ strongly preserving Brown-Pedersen quasi-invertible elements, then there exists a Jordan ${}^{\ast }$-homomorphism $S:A\to B$ satisfying $T(x)=T(1)S(x)$, for every $x\in A$. We also study the connections between linear maps strongly preserving Brown-Pedersen quasi-invertibility and other clases of linear preservers between C${}^{\ast }$-algebras like Bergmann-zero pairs preservers, Brown-Pedersen quasi-invertibility preservers and extreme points preservers.