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

TL;DR

This paper characterizes linear maps between C*-algebras and JB*-triples that strongly preserve certain invertibility properties, showing they are essentially Jordan *-homomorphisms or triple homomorphisms, with implications for structure-preserving maps.

Contribution

It introduces new classes of linear preservers and proves they are triple or Jordan *-homomorphisms, extending understanding of structure-preserving maps in operator algebras.

Findings

Linear maps strongly preserving Brown-Pedersen quasi-invertible elements are triple homomorphisms.

Such maps correspond to Jordan *-homomorphisms when between unital C*-algebras.

Connections established between various classes of linear preservers.

Abstract

We study new classes of linear preservers between C$^*$-algebras and JB$^*$-triples. Let $E$ and $F$ be JB$^*$-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$^*$-algebras $A$ and $B,$ for each linear map $T$ strongly preserving Brown-Pedersen quasi-invertible elements, then there exists a Jordan $^*$-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$^*$-algebras like Bergmann-zero pairs preservers, Brown-Pedersen quasi-invertibility preservers and extreme points preservers.