A note on invertibility preservers on Banach algebras

Abstract

Abstract. Let A be B be semisimple Banach algebras and let φ : A → B be a unital bijective linear operator that preserves invertibility. If the socle of A is an essential ideal of A, then φ is a Jordan isomorphism.

Additive Maps Preserving Local Spectrum

“…In [6], Brešar andŠemrl showed that a surjective, linear map Φ : L(X) → L(X) preserves the spectral radius if and only if Φ is, up to a multiplicative constant of modulus one, an algebra automorphism or anti-automorphism. These results have been extended in various directions (see [5], [7], [8], [13]). …”

Noninvertibility preservers on Banach algebras

“…The present note was motivated by Sourour's question in [4]: Is a linear, unital surjection Φ : B(X) → B(Y ), which preserves invertibility, necessarily injective? We show below, with help of [3], that the answer is affirmative when 'invertibility' is replaced by 'noninvertibility'.…”

“…In a recent result, Brešar, Fošner, and Šemrl [3] extended Sourour's result [4] on the form of linear bijection, which preserve invertibility, from B(X) to arbitrary semisimple Banach algebras with 'large socle' (see also Aupetit and Mouton [2]). The present note was motivated by Sourour's question in [4]: Is a linear, unital surjection Φ : B(X) → B(Y ), which preserves invertibility, necessarily injective?…”

Linear maps on Mn(C) preserving the local spectral radius

“…Concluding remarks. In [2], M. Brešar, A. Fošner, and P.Šemrl extended Sourour's result which describes linear bijective maps from B(X) to B(Y ) that preserve invertibility [17]. They characterized linear bijective invertibility preserving maps from an arbitrary semisimple Banach algebra with large socle to another one.…”

“…The second part of this theorem is a consequence of the bijectivity of Φ and the main result of [2]. To establish the injectivity of Φ, B. Kuzma made use of the scarcity lemma; see [1,Theorem 3.4.25,and Corollary 3.4.18].…”