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

Abstract

We study new classes of linear preservers between C * -algebras and JB * -triples. Let E and F be JB * -triples with ∂ e (E 1 ). We prove that every linear map T : E → 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 → B satisfying T (x) = T (1)S(x), for every x ∈ 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 quasiinvertibility preservers and extreme points preservers.2010 Mathematics Subject Classification. 47B49, 15A09, 46L05, 47B48.between Banach algebras is a Jordan homomorphism if T (a 2 ) = T (a) 2 , for all a ∈ A (equivalently, T (ab + ba) = T (a)T (b) + T (b)T (a), for all a, b ∈ A). If A and B are unital, T is called unital if T (1) = 1. If A and B are C * -algebras and T (a * ) = T (a) * , for every a ∈ A, then T is called symmetric. Symmetric Jordan homomorphisms are named Jordan * -homomorphisms.Consequently, the problem of studying the linear maps T : A → B such that T (∂ e (A 1 )) ⊆ ∂ e (B 1 ) is a more general challenge, which is directly motivated by the just mentioned consequence of the Russo-Dye theorem. We only know partial answers to this problem. Concretely, V. Mascioni and L. Molnár studied the linear maps on a von Neumann factor M which preserve the extreme points of the unit ball of M. They prove that if M is infinite, then every linear mapping T on M preserving extreme points admits a factorization of the form T (a) = uS(a) (a ∈ M), where u is a (fixed) unitary in M and S either is a unital * -homomorphism or a unital * -anti-homomorphism (see [30, Theorem 1]). Theorem 2 in [30] states that, for a finite von Neumann algebra M, a linear map T : M → M preserves extreme points of the unit ball of M if and only if there exist a unitary operator u ∈ M and a unital Jordan * -homomorphism S : M → M such that T (a) = uS(a) (a ∈ A). In [29], L.E. Labuschagne and V.Mascioni study linear maps between C * -algebras whose adjoints preserve extreme points of the dual ball.The above results of Mascioni and L. Molnár are the most conclusive answers we know about linear maps between unital C * -algebras preserving extreme points. In this note we shall revisit the problem in full generality. We present several counter-examples to illustrate that the conclusions proved by Mascioni and Molnár for von Neumann factors need not be true for linear mappings preserving extreme points between unital C * -algebras (compare Remarks 5.8 and 5.9). It seems natural to ask whether a different class of linear preservers satisfies the same conclusions found by Mascioni and Molnár.Every unital Jordan homomorphism between Banach algebras strongly preserves invertibility, that is, T (a −1 ) = T (a) −1 , ...

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

“…If C*-algebras, and are considered as JB*-triples in Proposition 3.1, then [4,Proposition 5.5] elements, obviously strongly preserves BP-quasi invertible elements. Moreover, it was shown in (Theorem 5.11 [4]) that this operator between JB*-triples with is indeed a triple homomorphism which means 𝑒𝑥𝑡 (𝐽 1 ) ≠ ∅, that it preserves triple products. Since the class of extreme points of a JB*-triple is included in the class of BP-quasi invertible elements of JB*-triples.…”

“…In [4], Burgos et al studied linear operators strongly preserving Brown-Pedersen quasi invertibility between C*-algebras considered as JB*-triples and they proved that it is a triple homomorphism. They discussed a consequence of this result that concerns only C*-algebras; if is a linear operator strongly preserving 𝐺 Brown-Pedersen-quasi invertible elements (BP-quasi invertible, for short) between two unital C*-algebras 𝐴 and , authors proved that there is a Jordan * -homomorphism that satisfies for 𝐵 ϕ: 𝐴 → 𝐵 𝐺(𝑎) = 𝐺(𝑒)ϕ(𝑎) every where is the unit of .…”

“…They discussed a consequence of this result that concerns only C*-algebras; if is a linear operator strongly preserving 𝐺 Brown-Pedersen-quasi invertible elements (BP-quasi invertible, for short) between two unital C*-algebras 𝐴 and , authors proved that there is a Jordan * -homomorphism that satisfies for 𝐵 ϕ: 𝐴 → 𝐵 𝐺(𝑎) = 𝐺(𝑒)ϕ(𝑎) every where is the unit of . They also explored other types of linear operators between some Jordan 𝑎 ∈ 𝐴 𝑒 𝐴 algebra structures that preserve; Bergmann-zero pairs, BP-quasi invertible elements and extreme points [4].…”

“…Burgos et al in[4] studied some new linear preservers between JB*-triples. If is a linear operator 𝐺: 𝐽 → 𝐻 between JB*-triples and satisfies that , then preserves extreme points [4,. Definition 5.4].…”