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

Abstract

We study new classes of linear preservers between ${\mathrm{C}}^{\ast }$-algebras and between ${\mathrm{JB}}^{\ast }$-triples. Let $E$ and $F$ be ${\mathrm{JB}}^{\ast }$-triples with ${\partial }_e\left(E_1\right)\ne \varnothing$. 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 ${\mathrm{C}}^{\ast }$-algebras $A$ and $B$, for each linear map $T$ strongly preserving Brown–Pedersen quasi-invertible elements, there exists a Jordan ${}^{\ast }$-homomorphism $S:A\to B$ satisfying $T\left(x\right)=T\left(1\right)S\left(x\right)$ 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 ${\mathrm{C}}^{\ast }$-algebras like Bergmann-zero pairs preservers, Brown–Pedersen quasi-invertibility preservers, and extreme points preservers.