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2-local Lie isomorphisms of operator algebras on Banach spaces

Lin Chen, Lizhong Huang, Fangyan Lu (2015)

Studia Mathematica

Let X and Y be complex Banach spaces of dimension greater than 2. We show that every 2-local Lie isomorphism ϕ of B(X) onto B(Y) has the form ϕ = φ + τ, where φ is an isomorphism or the negative of an anti-isomorphism of B(X) onto B(Y), and τ is a homogeneous map from B(X) into ℂI vanishing on all finite sums of commutators.

A generalization of peripherally-multiplicative surjections between standard operator algebras

Takeshi Miura, Dai Honma (2009)

Open Mathematics

Let A and B be standard operator algebras on Banach spaces X and Y, respectively. The peripheral spectrum σπ (T) of T is defined by σπ (T) = z ∈ σ(T): |z| = maxw∈σ(T) |w|. If surjective (not necessarily linear nor continuous) maps φ, ϕ: A → B satisfy σπ (φ(S)ϕ(T)) = σπ (ST) for all S; T ∈ A, then φ and ϕ are either of the form φ(T) = A 1 TA 2 −1 and ϕ(T) = A 2 TA 1 −1 for some bijective bounded linear operators A 1; A 2 of X onto Y, or of the form φ(T) = B 1 T*B 2 −1 and ϕ(T) = B 2 T*B −1 for some...

A metric on the space of projections admitting nice isometries

Lajos Molnár, Werner Timmermann (2009)

Studia Mathematica

Motivated by the concept of separation between propositions in quantum logic, we introduce the so-called separation metric or Santos metric on the space of all projections in a Hilbert space. We show that the resulting metric space has only "nice" surjective isometries. On the nontrivial projections they are all unitarily or antiunitarily equivalent to the identity or to taking the orthogonal complement. We relate this result to Wigner's classical theorem on the form of quantum mechanical symmetry...

Algebra isomorphisms between standard operator algebras

Thomas Tonev, Aaron Luttman (2009)

Studia Mathematica

If X and Y are Banach spaces, then subalgebras ⊂ B(X) and ⊂ 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 ∈ is the set σ π ( A ) = λ σ ( A ) : | λ | = m a x z σ ( A ) | z | of spectral values of A of maximum modulus, and a map φ: → is called peripherally-multiplicative if it satisfies the equation σ π ( φ ( A ) φ ( B ) ) = σ π ( A B ) for all A,B ∈ . We show that any peripherally-multiplicative and surjective map φ: → , neither assumed to be linear nor...

(E,F)-Schur multipliers and applications

Fedor Sukochev, Anna Tomskova (2013)

Studia Mathematica

For two given symmetric sequence spaces E and F we study the (E,F)-multiplier space, that is, the space of all matrices M for which the Schur product M ∗ A maps E into F boundedly whenever A does. We obtain several results asserting continuous embedding of the (E,F)-multiplier space into the classical (p,q)-multiplier space (that is, when E = l p , F = l q ). Furthermore, we present many examples of symmetric sequence spaces E and F whose projective and injective tensor products are not isomorphic to any subspace...

From geometry to invertibility preservers

Hans Havlicek, Peter Šemrl (2006)

Studia Mathematica

We characterize bijections on matrix spaces (operator algebras) preserving full rank (invertibility) of differences of matrix (operator) pairs in both directions.

Isometries of the unitary groups in C*-algebras

Osamu Hatori (2014)

Studia Mathematica

We give a complete description of the structure of surjective isometries between the unitary groups of unital C*-algebras. While any surjective isometry between the unitary groups of von Neumann algebras can be extended to a real-linear Jordan *-isomorphism between the relevant von Neumann algebras, this is not the case for general unital C*-algebras. We show that the unitary groups of two C*-algebras are isomorphic as metric groups if and only if the C*-algebras are isomorphic in the sense that...

Jordan isomorphisms and maps preserving spectra of certain operator products

Jinchuan Hou, Chi-Kwong Li, Ngai-Ching Wong (2008)

Studia Mathematica

Let ₁, ₂ be (not necessarily unital or closed) standard operator algebras on locally convex spaces X₁, X₂, respectively. For k ≥ 2, consider different products T T k on elements in i , which covers the usual product T T k = T T k and the Jordan triple product T₁ ∗ T₂ = T₂T₁T₂. Let Φ: ₁ → ₂ be a (not necessarily linear) map satisfying σ ( Φ ( A ) Φ ( A k ) ) = σ ( A A k ) whenever any one of A i ’s has rank at most one. It is shown that if the range of Φ contains all rank one and rank two operators then Φ must be a Jordan isomorphism multiplied by a root...

Linear mappings preserving similarity on B(H)

Tatjana Petek (2004)

Studia Mathematica

Let H be an infinite-dimensional complex Hilbert space. We give a characterization of surjective linear mappings on B(H) that preserve similarity in both directions.

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

Abdellatif Bourhim, Vivien G. Miller (2008)

Studia Mathematica

Let x₀ be a nonzero vector in ℂⁿ. We show that a linear map Φ: Mₙ(ℂ) → Mₙ(ℂ) preserves the local spectral radius at x₀ if and only if there is α ∈ ℂ of modulus one and an invertible matrix A ∈ Mₙ(ℂ) such that Ax₀ = x₀ and Φ ( T ) = α A T A - 1 for all T ∈ Mₙ(ℂ).

Linear maps preserving quasi-commutativity

Heydar Radjavi, Peter Šemrl (2008)

Studia Mathematica

Let X and Y be Banach spaces and ℬ(X) and ℬ(Y) the algebras of all bounded linear operators on X and Y, respectively. We say that A,B ∈ ℬ(X) quasi-commute if there exists a nonzero scalar ω such that AB = ωBA. We characterize bijective linear maps ϕ : ℬ(X) → ℬ(Y) preserving quasi-commutativity. In fact, such a characterization can be proved for much more general algebras. In the finite-dimensional case the same result can be obtained without the bijectivity assumption.

Local spectrum and local spectral radius of an operator at a fixed vector

Janko Bračič, Vladimír Müller (2009)

Studia Mathematica

Let be a complex Banach space and e ∈ a nonzero vector. Then the set of all operators T ∈ ℒ() with σ T ( e ) = σ δ ( T ) , respectively r T ( e ) = r ( T ) , is residual. This is an analogy to the well known result for a fixed operator and variable vector. The results are then used to characterize linear mappings preserving the local spectrum (or local spectral radius) at a fixed vector e.

Locally spectrally bounded linear maps

M. Bendaoud, M. Sarih (2011)

Mathematica Bohemica

Let ( ) be the algebra of all bounded linear operators on a complex Hilbert space . We characterize locally spectrally bounded linear maps from ( ) onto itself. As a consequence, we describe linear maps from ( ) onto itself that compress the local spectrum.

Maps on idempotent operators

Peter Šemrl (2007)

Banach Center Publications

The set of all bounded linear idempotent operators on a Banach space X is a poset with the partial order defined by P ≤ Q if PQ = QP = P. Another natural relation on the set of idempotent operators is the orthogonality relation defined by P ⊥ Q ⇔ PQ = QP = 0. We briefly survey known theorems on maps on idempotents preserving order or orthogonality. We discuss some related results and open problems. The connections with physics, geometry, theory of automorphisms, and linear preserver problems will...

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