Local derivations and local automorphisms of ℬ(𝒳)

Larson, David R.; Sourour, A. R.

doi:10.1090/pspum/051.2/1077437

Cited by 219 publications

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“…With this notion the set of all surjective isometries of X is algebraically reflexive if and only if every locally surjective isometry of X is a surjective isometry. Note that in [8,11,4,5] a similar terminology is used concerning the algebraic reflexivity of derivation algebras and isomorphism groups of operator algebras.…”

Section: T E B(x) Tx G £X(vx E X) => T E £ [T € B(x) Tx E £X(vx E Xmentioning

confidence: 99%

Reflexivity of the group of surjective isometries on some Banach spaces

Molnár¹,

Zalar²

1999

Proceedings of the Edinburgh Mathematical Society

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To the memory of M. M.In this paper we study the problem of algebraic reflexivity of the isometry group of some important Banach spaces. Because of the previous work in similar topics, our main interest lies in the von Neumann -Schatten p-classes of compact operators. The ideas developed there can be used in £ p -spaces, Banach spaces of continuous functions and spin factors as well. Moreover, we attempt to attract the attention to this problem from general Banach spaces geometry view-point. This study, we believe, would provide nice geometrical results. T e B(X), Tx G £x(Vx e X) => T e £ [T € B(X), Tx e £x(Vx e X) => T e £]holds true. This concept of reflexivity is very useful in the analysis of operator algebras (see [10] and the references therein). Most of the published work is about the reflexivity of derivation algebras and automorphism groups. For results on the algebraic reflexivity of derivation algebras see papers by Bresar, Kadison, Larson, Sourour and Semrl [3,4,8,11]. A theorem, due to Shul'man, on the topological reflexivity of a derivation algebra acting on a C*-algebra can be found in [17]. For results on algebraic reflexivity of the automorphism group see papers by Bresar and Semrl [4] and [5], In a recent article [13] of the first author it was proved that the automorphism group of

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Section: T E B(x) Tx G £X(vx E X) => T E £ [T € B(x) Tx E £X(vx E Xmentioning

confidence: 99%

Reflexivity of the group of surjective isometries on some Banach spaces

Molnár¹,

Zalar²

1999

Proceedings of the Edinburgh Mathematical Society

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“…We recall that a linear mapping T on a C * -algebra A is said to be a local * -automorphism if for every a in A, there exists a * -automorphism π a : A → A, depending on a, such that T (a) = π a,ξ (a) (compare [9] and [3]). Clearly, every local * -automorphism on B(H) is a bilocal * -automorphism.…”

Section: Introductionmentioning

confidence: 99%

Bilocal *-automorphisms of B(H) satisfying the 3-local property

2015

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“…This investigation was motivated by the study of the Hochschild cohomology of operator algebras. Independently, Larson and Sourour [13] studied local derivations of B(X); they proved that every local derivation of B(X) is a derivation. Since then, a considerable amount of work has been done concerning local derivations of various algebras.…”

Section: Introductionmentioning

confidence: 99%

Topological reflexivity of the spaces of (α, β)-derivations on operator algebras

Jing

2003

Studia Math.

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Abstract. We prove that the spaces of (α, β)-derivations on certain operator algebras are topologically reflexive in the weak operator topology. Characterizations of automorphisms and (α, β)-derivations on reflexive algebras are also given. Introduction and preliminaries.The study of reflexive linear subspaces of the algebra B(X) of all bounded linear operators on the Banach space X represents a very active research area in operator theory (see [8] for a beautiful general view). The originators of this research direction are Kadison and Larson. In [11], Kadison studied local derivations from a von Neumann algebra R into a dual R-bimodule M. A linear map from R into M is called a local derivation if it agrees with some derivation at each point in the algebra (the derivations possibly varying from point to point). This investigation was motivated by the study of the Hochschild cohomology of operator algebras. Independently, Larson and Sourour [13] studied local derivations of B(X); they proved that every local derivation of B(X) is a derivation. Since then, a considerable amount of work has been done concerning local derivations of various algebras. See, for example, [4,5,9,10,18]. In this paper, we will extend this research to a more general setting.Let us now define our concept of topological reflexivity. Let X be a Banach space. We denote by L(X) and B(X) the algebras of all linear and all bounded linear operators on X respectively. Let A be a subalgebra of B(X). Given two subsets E ⊆ L(A) and F ⊆ B(A), if τ is a vector topology on A, we define

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Local derivations and local automorphisms of ℬ(𝒳)

Cited by 219 publications

References 0 publications

Reflexivity of the group of surjective isometries on some Banach spaces

Reflexivity of the group of surjective isometries on some Banach spaces

Bilocal *-automorphisms of B(H) satisfying the 3-local property

Topological reflexivity of the spaces of (α, β)-derivations on operator algebras

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