Maps preserving numerical radius distance on C<sup>*</sup>-algebras

Bai, Zhaofang; Hou, Jinchuan; Zhao, Xu

doi:10.4064/sm162-2-1

Cited by 22 publications

(10 citation statements)

References 6 publications

(5 reference statements)

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“…Moreover, it has been found from some recent results that additive maps play a more basic role in the study of general preserver problems. For example, to characterize the adjacency preservers, the distance preservers and the numerical radius distance preservers on some operator algebras, a key step is to reduce the questions to ones of characterizing the corresponding additive preservers (see [2][3][4][5]16]). …”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Asymptotic similarity preserving additive maps on B(X)

Hou

2004

Journal of Mathematical Analysis and Applications

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Let X be an infinite dimensional complex Banach space and denote B(X) the algebra of all bounded linear operators acting on X. We show that an additive surjective map on B(X) preserves asymptotic similarity in both directions if and only if there exist a nonzero scalar c, an invertible bounded linear or conjugate linear operator A and an asymptotic similarity invariant additive functional φ on B(X) such that either Φ(T ) = cAT A −1 + φ(T )I for all T or Φ(T ) = cAT * A −1 + φ(T )I for all T . In the case that X has infinite multiplicity, especially if X is an infinite dimensional Hilbert space, above asymptotic similarity invariant additive functional φ is always zero.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Asymptotic similarity preserving additive maps on B(X)

Hou

2004

Journal of Mathematical Analysis and Applications

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“…Note that the conclusion of Theorem 1 also holds if ϕ satisfies equation (2) for each A ∈ A and every operator B ∈ B(X) of finite rank, or even of rank at most two. If B ∈ B 1 (X) then the equation σ p (ϕ(A) • ϕ(B)) = σ p (AB), similar to (2) and considered in [11], where σ p (A) is the point spectrum of A, implies (2), but not vice versa.…”

Section: Proof According To Lemmas 2 and 4 The Map ϕ Is A Bijective mentioning

confidence: 88%

“…in [2,3,12,14,15]). However, for operator algebras other than B(X), and in particular for non-unital algebras or algebras of compact operators, the subject has not been sufficiently studied.…”

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confidence: 99%

Algebra isomorphisms between standard operator algebras

Tonev

Luttman

2009

Studia Math.

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Abstract. If X and Y are Banach spaces, then subalgebras A ⊂ B(X) and B ⊂ B(Y ), not necessarily unital nor complete, are called standard operator algebras if they contain all finite rank operators on X and Y respectively. The peripheral spectrum of A ∈ A is the set σπ(A) = {λ ∈ σ(A) : |λ| = max z∈σ(A) |z|} of spectral values of A of maximum modulus, and a map ϕ : A → B is called peripherally-multiplicative if it satisfies the equation σπ(ϕ(A) • ϕ(B)) = σπ(AB) for all A, B ∈ A. We show that any peripherally-multiplicative and surjective map ϕ : A → B, neither assumed to be linear nor continuous, is a bijective bounded linear operator such that either ϕ or −ϕ is multiplicative or anti-multiplicative. This holds in particular for the algebras of finite rank operators or of compact operators on X and Y and extends earlier results of Molnár. If, in addition, σπ(ϕ(A0)) = −σπ(A0) for some A0 ∈ A then ϕ is either multiplicative, in which case X is isomorphic to Y , or anti-multiplicative, in which case X is isomorphic to Y * . Therefore, if X ∼ = Y * then ϕ is multiplicative, hence an algebra isomorphism, while if X ∼ = Y , then ϕ is anti-multiplicative, hence an algebra anti-isomorphism.

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“…Numerical range of operators is a very important concept and is extensively studied in both theory and applications. Particularly, many authors have studied numerical range preserving maps on various operator algebras; see [1]- [6], [9], [11], [12], [13,Chapter 5]. In this paper, we characterize surjective maps φ : B(H) → B(K) such that (…”

Section: For a Hilbert Space H · · Stands For Its Inner Product B(mentioning

confidence: 99%

“…Then there exist linearly independent vectors x 1 , x 2 ∈ H such that Ax 1 ⊥Ax 2 and Ax 1 = Ax 2 = 1. Let B = αx 1 …”

Section: Lemma 23 Let a ∈ B(h) The Following Conditions Are Equivamentioning

confidence: 99%

Maps preserving numerical ranges of operator products

Hou

2005

Proc. Amer. Math. Soc.

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Maps preserving numerical radius distance on C^*-algebras

Cited by 22 publications

References 6 publications

Asymptotic similarity preserving additive maps on B(X)

Asymptotic similarity preserving additive maps on B(X)

Algebra isomorphisms between standard operator algebras

Maps preserving numerical ranges of operator products

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Maps preserving numerical radius distance on C*-algebras

Cited by 22 publications

References 6 publications

Asymptotic similarity preserving additive maps on B(X)

Asymptotic similarity preserving additive maps on B(X)

Algebra isomorphisms between standard operator algebras

Maps preserving numerical ranges of operator products

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Maps preserving numerical radius distance on C^*-algebras