Linear maps preserving Fredholm and Atkinson elements of<i>C</i><b>*</b>-algebras

Bendaoud, M.; Bourhim, Abdellatif; Burgos, María; Sarih, M.

doi:10.1080/03081080802253698

Cited by 9 publications

(5 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark 2.4. As in [3,Section 5] and [6,Section 4], the main result of this paper and its corollaries can be stated in a more general situation where one considers essentially spectrally bounded linear maps from a unital purely infinite * -algebra with real rank zero onto a semisimple Banach algebra.…”

Section: Corollary 23 Let : ℒ(ℋ) → ℒ(ℋ) Be a Linear Map Surjective mentioning

confidence: 92%

Essentially spectrally bounded linear maps

Bendaoud

Bourhim

2009

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. Let ℒ(ℋ) be the algebra of all bounded linear operators on an infinite dimensional complex Hilbert space ℋ. We characterize essentially spectrally bounded linear maps from ℒ(ℋ) onto ℒ(ℋ) itself. As a consequence, we characterize linear maps from ℒ(ℋ) onto ℒ(ℋ) itself that compress different essential spectral sets such as the the essential spectrum, the (left, right) essential spectrum, and the semi-Fredholm spectrum.

show abstract

Section: Corollary 23 Let : ℒ(ℋ) → ℒ(ℋ) Be a Linear Map Surjective mentioning

confidence: 92%

Essentially spectrally bounded linear maps

Bendaoud

Bourhim

2009

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…By Theorem 3.6, θ is unital and bijective. θ preserves left invertibility in both directions and so preserves invertibility in both directions by [4,Theorem 2.2]. Therefore θ(Inv(C R (A))) = Inv(C R (B)).…”

Section: Proof (I) ⇒ (Iii)mentioning

confidence: 98%

“…When A is a semisimple Banach algebra, as a special case in Definition 2.2, we may assume F to be the socle of A. In this case the relative Calkin algebra coincides with the generalized Calkin algebra as in [4]. For x, y ∈ A let x • y = x + y − xy denote the quasi product of x and y.…”

Section: Preliminariesmentioning

confidence: 99%

“…In the context of operator algebras, some of the most well known preserving problems deal with characterizing linear maps on the algebra of all bounded linear operators acting on a Banach space which preserve Fredholm, semi-Fredholm or generalized invertible operators. New contributions to the study of linear preserver problems in L(H), the algebra of all bounded linear operators on an infinite dimensional complex Hilbert space H, have been recently made by several authors, see [3,4,5,12,13,14]. In [11], Kim and Park investigated linear maps φ on a unital C * -algebra A of real rank zero that are surjective up to some fixed closed ideal I and π(a) is invertible in A/I if and only if π(φ(A)) is invertible in A/I, where π : A → A/I is the canonical quotient map.…”

Section: Ams Mathematics Subject Classification: 47b49 47a10mentioning

confidence: 99%

See 1 more Smart Citation

Maps preserving Fredholm or semi-Fredholm elements relative to some ideal

Rostamani

Hejazian

2015

Cubo

View full text Add to dashboard Cite

We consider the Calkin algebra C R (A) and the Fredholm theory in a Banach algebra A, relative to some fixed ideal F of A. Our aim is to study linear maps between unital Banach algebras A and B which are surjective up to the inessential elements relative to F, and preserve Fredholm or semi-Fredholm elements in both directions or equivalently different relatively essential spectral sets such as essential spectrum, left or right essential spectrum, the boundary of essential spectrum or the full essential spectrum. We characterize such mappings when one of C R (A) or C R (B) is commutative and also investigate similar problems when A is assumed to be a unital C * -algebra of real rank zero and B is an arbitrary Banach algebra. RESUMENConsideramos elálgebra de Calkin C R (A) y la teoría de Fredholm en unálgebra de Banach A relativa a algún ideal fijo F de A. Nuestra meta es estudiar aplicaciones lineales entreálgebras de Banach unitales A y B las cuales son sobrejectivas salvo los elementos no esenciales relativos a F y preservan los elementos de Fredholm o semi-Fredholm en ambas direcciones o equivalentemente conjuntos espectrales esenciales relativos diferentes tales como el espectro esencial izquierdo o derecho, la frontera del espectro esencial o el espectro esencial completo. Caracterizamos dichas aplicaciones cuando uno de los C R (A) o C R (B) es conmutativo e investigamos problemas similares cuando A se asume que es una C * -álgebra unital de rango real cero y B es unaálgebra de Banach cualquiera.

show abstract

“…This problem was solved in some special cases of semi-simple Banach algebras ( [9], [12], [16], [22]). Several authors have studied linear maps which preserve the classes of semi-Fredholm operators, Fredholm operators, and the related operators in both directions ( [1], [4], [7]). It has been shown that such maps preserve the ideal of compact operators in both directions, and the maps induced by them on the Calkin algebra are Jordan automorphisms.…”

Section: Introductionmentioning

confidence: 99%