In this paper we investigate the properties of minus partial order in unital rings. We generalize several results well known for matrices and bounded linear operators on Banach spaces. We also study linear maps preserving the minus partial order in unital semisimple Banach algebras with essential socle. A ≤ − Bif and only if rank(B − A) = rank(B ) − rank(A). He proved thatwhere A − denotes a {1}-inverse of A. This partial order is usually named the minus partial order.

Abstract. Let A and B be C * -algebras. We investigate linear maps from A to B strongly preserving Moore-Penrose invertibility, where A is unital, and either it is linearly spanned by its projections, or has large socle, or has real rank zero (in this last case the map T is assumed to be bounded).Mathematics subject classification (2010): 47B49, 46L05, 47B48.

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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 , ...

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