Compact and weakly compact disjointness preserving operators on spaces of differentiable functions

Leung, Denny H.; Wang, Ya-Shu

doi:10.1090/s0002-9947-2012-05831-4

Cited by 5 publications

(9 citation statements)

References 19 publications

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“…As in Section 1, let AX be the compactification of X constructed using A(X) = C p (X). By [LW,Proposition 4], as in Proposition 1.2, there is a continuous function β :…”

Section: Spaces Of Differentiable Functionsmentioning

confidence: 99%

“…Using the fact that y 0 ∈ Y r , one may verify by direct computation that T f (y 0 ) = p k=0 Φ k (y 0 , D k f (β(y 0 ))) for all f ∈ C p (X, E). Refer to the proof of [LW,Theorem 10] for details.…”

Section: Spaces Of Differentiable Functionsmentioning

confidence: 99%

“…As in the Remark after Corollary 6.4, it follows that T is a continuous linear operator. By [LW,Theorem 10], Φ k (y, S) is a continuous function of y for any fixed k and S.…”

Section: Spaces Of Differentiable Functionsmentioning

confidence: 99%

“…Proposition 1.1 ( [LW,Proposition 3]). Let X be a Hausdorff topological space and let A(X) be a subspace of C(X) that separates points from closed sets and is closed under C ∞ * operations.…”

mentioning

confidence: 99%

“…Proposition 1.2 ( [LW,Proposition 4]). Let X and Y be Hausdorff topological spaces and let A(X) and A(Y ) be vector subspaces of C(X) and C(Y ) respectively.…”

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confidence: 99%

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Inverses of disjointness preserving operators

Leung

Wang

2016

Studia Math.

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A linear operator between (possibly vector-valued) function spaces is disjointness preserving if it maps disjoint functions to disjoint functions. Here, two functions are said to be disjoint if at each point at least one of them vanishes. In this paper, we study linear disjointness preserving operators between various types of function spaces, including spaces of (little) Lipschitz functions, uniformly continuous functions and differentiable functions. It is shown that a disjointness preserving linear isomorphism whose domain is one of these types of spaces (scalar-valued) has a disjointness preserving inverse, subject to some topological conditions on the range space. A representation for a general linear disjointness preserving operator on a space of vector-valued C p functions is also given.

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“…As in Section 1, let AX be the compactification of X constructed using A(X) = C p (X). By [LW,Proposition 4], as in Proposition 1.2, there is a continuous function β :…”

Section: Spaces Of Differentiable Functionsmentioning

confidence: 99%

Section: Spaces Of Differentiable Functionsmentioning

confidence: 99%

“…As in the Remark after Corollary 6.4, it follows that T is a continuous linear operator. By [LW,Theorem 10], Φ k (y, S) is a continuous function of y for any fixed k and S.…”

Section: Spaces Of Differentiable Functionsmentioning

confidence: 99%

mentioning

confidence: 99%

“…Proposition 1.2 ( [LW,Proposition 4]). Let X and Y be Hausdorff topological spaces and let A(X) and A(Y ) be vector subspaces of C(X) and C(Y ) respectively.…”

mentioning

confidence: 99%

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Inverses of disjointness preserving operators

Leung

Wang

2016

Studia Math.

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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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Orthogonally additive and orthogonally multiplicative holomorphic functions of matrices

Bu¹,

Liao²,

Wong³

2014

Ann. Funct. Anal.

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Abstract. Let H : M m → M m be a holomorphic function of the algebra M m of complex m × m matrices. Suppose that H is orthogonally additive and orthogonally multiplicative on self-adjoint elements. We show that either the range of H consists of zero trace elements, or there is a scalar sequence {λ n } and an invertible S in M m such thatHere, x t is the transpose of the matrix x. In the latter case, we always have the first representation form when H also preserves zero products. We also discuss the cases where the domain and the range carry different dimensions.

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Compact and weakly compact disjointness preserving operators on spaces of differentiable functions

Cited by 5 publications

References 19 publications

Inverses of disjointness preserving operators

Inverses of disjointness preserving operators

Logarithmic submajorisation and order-preserving linear isometries

Orthogonally additive and orthogonally multiplicative holomorphic functions of matrices

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