## Banach Journal of Mathematical Analysis

### Linear maps between $\mathrm{C}^{*}$-algebras preserving extreme points and strongly linear preservers

#### Abstract

We study new classes of linear preservers between $\mathrm{C}^{*}$-algebras and between $\mathrm{JB}^{*}$-triples. Let $E$ and $F$ be $\mathrm{JB}^{*}$-triples with $\partial_{e}(E_{1})\neq\emptyset$. 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 $\mathrm{C}^{*}$-algebras $A$ and $B$, for each linear map $T$ strongly preserving Brown–Pedersen quasi-invertible elements, 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 $\mathrm{C}^{*}$-algebras like Bergmann-zero pairs preservers, Brown–Pedersen quasi-invertibility preservers, and extreme points preservers.

#### Article information

Source
Banach J. Math. Anal., Volume 10, Number 3 (2016), 547-565.

Dates
Accepted: 11 November 2015
First available in Project Euclid: 22 July 2016

https://projecteuclid.org/euclid.bjma/1469199409

Digital Object Identifier
doi:10.1215/17358787-3607288

Mathematical Reviews number (MathSciNet)
MR3528347

Zentralblatt MATH identifier
06621469

#### Citation

Burgos, María J.; Márquez-García, Antonio C.; Morales-Campoy, Antonio; Peralta, Antonio M. Linear maps between $\mathrm{C}^{*}$ -algebras preserving extreme points and strongly linear preservers. Banach J. Math. Anal. 10 (2016), no. 3, 547--565. doi:10.1215/17358787-3607288. https://projecteuclid.org/euclid.bjma/1469199409

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