Linear biseparating maps between spaces of vector-valued differentiable functions and automatic continuity

Araujo, Jesús

doi:10.1016/j.aim.2003.09.007

Cited by 35 publications

(28 citation statements)

References 17 publications

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“…With "disjointness preserving" property at hand, we establish a general description of all order-preserving injective isometries T as above, showing that every such isometry is generated by a Jordan * -isomorphism from M 1 onto a weakly closed * -subalgebra of M 2 . The description of disjointness preserving operators on Banach lattices has been well-studied (see [1,3], see also [4,7,54]). In particular, Abramovich [1] obtained the general description of order-continuous (or normal) disjointness-preserving operators on banach lattices.…”

Section: Introductionmentioning

confidence: 99%

Logarithmic submajorisation and order-preserving linear isometries

Huang

Sukochev

Zanin

2020

Journal of Functional Analysis

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Let E and F be noncommutative operator spaces affiliated with semifinite von Neumann algebras M 1 and M 2 , respectively. We establish a noncommutative version of Abramovich's theorem [1], which provides the general form of normal order-preserving linear operators T : E into −→ F having the disjointness preserving property. As an application, we obtain a noncommutative Huijsmans-Wickstead theorem [43].By establishing the disjointness preserving property for an order-preserving isometry T : E → F from a noncommutative symmetrically ∆-normed (in particular, quasi-normed) space into another, we obtain the existence of a Jordan *monomorphism from M 1 into M 2 and the general form of this isometry, which extends and complements a number of existing results such as [12, Theorem 1], [64, Corollary 1], [69, Theorem 2] and [15, Theorem 3.1]. In particular, we fully resolve the case when F is the predual of M 2 and other untreated cases in [75].2010 Mathematics Subject Classification. 46B04, 46L52, 46A16.

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Section: Introductionmentioning

confidence: 99%

Logarithmic submajorisation and order-preserving linear isometries

Huang

Sukochev

Zanin

2020

Journal of Functional Analysis

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“…For example, a theorem of Peetre [19] states that local linear maps of the space of smooth functions defined on a manifold modeled on R n are exactly the linear differential operators (see [18]). This was extended to the case of vector-valued differentiable functions defined on a finite-dimensional manifold by Kantrowitz and Neumann [14] and Araujo [3], and to the Banach C 1…”

Section: Terminology and Notationmentioning

confidence: 99%

“…We also call E a Hilbert A-module in this case. A complex linear map θ : E → F between two Hilbert A-modules is called an A-module homomorphism if θ (xa) = θ (x)a [3] Linear orthogonality preservers 247 for all a ∈ A and x ∈ E. See, for example, [15] or [20] for a general introduction to the theory of Hilbert C * -modules. In this paper, we are interested in the case where the underlying C * -algebra A is abelian, that is, the space A = C 0 ( ) of all continuous complex-valued functions vanishing at infinity on a locally compact Hausdorff space .…”

Section: Terminology and Notationmentioning

confidence: 99%

Linear Orthogonality Preservers of Hilbert Bundles

Leung

Wong

2010

J. Aust. Math. Soc.

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A C-linear map θ (not necessarily bounded) between two Hilbert C * -modules is said to be 'orthogonality preserving' if θ(x), θ(y) = 0 whenever x, y = 0. We prove that if θ is an orthogonality preserving map from a full Hilbert C 0 ( )-module E into another Hilbert C 0 ( )-module F that satisfies a weaker notion of C 0 ( )-linearity (called 'localness'), then θ is bounded and there exists φ ∈ C b ( ) + such that θ (x), θ (y) = φ · x, y for all x, y ∈ E.2010 Mathematics subject classification: primary 46L08; secondary 46M20, 46H40, 46E40.

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“…Part of its proof depends on an application of the Closed Graph Theorem (Proposition 6). For the use of the Closed Graph Theorem in this context, see [1,3,4].…”

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confidence: 99%

“…(Actually, the more general case of disjointness preserving operators was studied there.) See also [2,4] for related results. The example below shows that the direct carry-over of Theorem 1 is no longer true.…”

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confidence: 99%