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1
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Topologies and bornologies determined by operator ideals, II

100%
Wong N.-C.
Studia Mathematica
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1994
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tom 111
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nr 2
153-162
EN
Let 𝔄 be an operator ideal on LCS's. A continuous seminorm p of a LCS X is said to be 𝔄- continuous if $Q̃_p ∈ 𝔄^{inj}(X,X̃_p)$, where $X̃_p$ is the completion of the normed space $X_p = X/p^{-1}(0)$ and $Q̃_p$ is the canonical map. p is said to be a Groth(𝔄)- seminorm if there is a continuous seminorm q of X such that p ≤ q and the canonical map $Q̃_{pq} : X̃_q → X̃_p$ belongs to $𝔄(X̃_q,X̃_p)$. It is well known that when 𝔄 is the ideal of absolutely summing (resp. precompact, weakly compact) operators, a LCS X is a nuclear (resp. Schwartz, infra-Schwartz) space if and only if every continuous seminorm p of X is 𝔄-continuous if and only if every continuous seminorm p of X is a Groth(𝔄)-seminorm. In this paper, we extend this equivalence to arbitrary operator ideals 𝔄 and discuss several aspects of these constructions which were initiated by A. Grothendieck and D. Randtke, respectively. A bornological version of the theory is also obtained.
2
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Left quotients of a C*-algebra, III: Operators on left quotients

64%
Brown L., Wong N.-C.
Studia Mathematica
|
2013
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tom 218
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nr 3
189-217
EN
Let L be a norm closed left ideal of a C*-algebra A. Then the left quotient A/L is a left A-module. In this paper, we shall implement Tomita's idea about representing elements of A as left multiplications: $π_{p}(a)(b + L) = ab + L$. A complete characterization of bounded endomorphisms of the A-module A/L is given. The double commutant $π_{p}(A)''$ of $π_{p}(A)$ in B(A/L) is described. Density theorems of von Neumann and Kaplansky type are obtained. Finally, a comprehensive study of relative multipliers of A is carried out.
3
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On the Banach-Stone problem

64%
Jeang J.-S., Wong N.-C.
Studia Mathematica
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2003
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tom 155
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nr 2
95-105
EN
Let X and Y be locally compact Hausdorff spaces, let E and F be Banach spaces, and let T be a linear isometry from C₀(X,E) into C₀(Y,F). We provide three new answers to the Banach-Stone problem: (1) T can always be written as a generalized weighted composition operator if and only if F is strictly convex; (2) if T is onto then T can be written as a weighted composition operator in a weak sense; and (3) if T is onto and F does not contain a copy of $ℓ₂^{∞}$ then T can be written as a weighted composition operator in the classical sense.
4
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An abstract Cauchy problem for higher order functional differential inclusions with infinite delay

51%
Ke T., Obukhovskii V., Wong N.-C., Yao J.-C.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2011
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tom 31
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nr 2
199-229
EN
The existence results for an abstract Cauchy problem involving a higher order differential inclusion with infinite delay in a Banach space are obtained. We use the concept of the existence family to express the mild solutions and impose the suitable conditions on the nonlinearity via the measure of noncompactness in order to apply the theory of condensing multimaps for the demonstration of our results. An application to some classes of partial differential equations is given.
5
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Generalized n-circular projections on JB*-triples and Hilbert C0(Ω)-modules

51%
Ilišević D., Liu C.-N., Wong N.-C.
Concrete Operators
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2017
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tom 4
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nr 1
109-120
EN
Being expected as a Banach space substitute of the orthogonal projections on Hilbert spaces, generalized n-circular projections also extend the notion of generalized bicontractive projections on JB*-triples. In this paper, we study some geometric properties of JB*-triples related to them. In particular, we provide some structure theorems of generalized n-circular projections on an often mentioned special case of JB*-triples, i.e., Hilbert C*-modules over abelian C*-algebras C0(Ω).
6
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Jordan isomorphisms and maps preserving spectra of certain operator products

51%
Hou J., Li C.-K., Wong N.-C.
Studia Mathematica
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2008
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tom 184
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nr 1
31-47
EN
Let 𝓐₁, 𝓐₂ be (not necessarily unital or closed) standard operator algebras on locally convex spaces X₁, X₂, respectively. For k ≥ 2, consider different products $T₁ ∗ ⋯ ∗ T_{k}$ on elements in $𝓐_{i}$, which covers the usual product $T₁ ∗ ⋯ ∗ T_{k} = T₁ ⋯ T_{k}$ and the Jordan triple product T₁ ∗ T₂ = T₂T₁T₂. Let Φ: 𝓐₁ → 𝓐₂ be a (not necessarily linear) map satisfying $σ(Φ(A₁) ∗ ⋯ ∗ Φ(A_{k})) = σ(A₁ ∗ ⋯ ∗ A_{k})$ 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 of unity. Similar results for self-adjoint operators acting on Hilbert spaces are obtained.
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