Extremely Strong Boundary Points and Real-linear Isometries

Jamshidi, Arya; Sady, Fereshteh

doi:10.3836/tjm/1452806051

Cited by 7 publications

(3 citation statements)

References 14 publications

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“…It is well known that Ch(A) is a boundary for A, that is, for each f ∈ A, there exists a point x ∈ Ch(A) such that |f (x)| = f X , see [16,Page 184]. In general, δ(A) ⊆ Ch(A) (see [8,Lemma 3.1]) and if A is a function algebra, then δ(A) = Ch(A) (see [10,Theorem 4.7.22] for compact case and [15, Theorem 2.1] for general case).…”

Section: Preliminariesmentioning

confidence: 99%

Composition in Modulus Maps on Semigroups of Continuous Functions

Jafarzadeh¹,

Sady²

2019

Preprint

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For locally compact Hausdorff spaces X and Y , and function algebras A and B on X and Y , respectively, surjections T : A −→ B satisfying norm multiplicative condition T f T g Y = f g X , f, g ∈ A, with respect to the supremum norms, and those satisfying |T f | + |T g| Y = |f | + |g| X have been extensively studied. Motivated by this, we consider certain (multiplicative or additive) subsemigroups A and B of C 0 (X) and C 0 (Y ), respectively, and study surjections T : A −→ B satisfying the norm condition ρ(T f, T g) = ρ(f, g), f, g ∈ A, for some class of two variable positive functions ρ. It is shown that T is also a composition in modulus map.

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Section: Preliminariesmentioning

confidence: 99%

Composition in Modulus Maps on Semigroups of Continuous Functions

Jafarzadeh¹,

Sady²

2019

Preprint

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“…The goal of this work is to give a complete characterization of surjective isometries T : A → B between completely regular subspaces. It is worth noting that a similar problem was recently investigated by Jamshidi and Sady [5]; however, our approach is significantly different. Instead of using the mapping T to induce a mapping T * : B * → A * between the dual spaces and then investigating the extreme points of the unit ball thereof, we adapt Eilenberg's [3,Theorem 7.2] proof of the Banach-Stone theorem, whose arguments hinge on the fact that the maximal convex subsets of the unit sphere of C(X) are essentially in a one-to-one correspondence with X.…”

Section: Introductionmentioning

confidence: 98%

Nonlinear isometries between function spaces

Roberts¹,

Lee²

2017

Ann. Funct. Anal.

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We demonstrate that any surjective isometry T : A → B not assumed to be linear between unital, completely regular subspaces of complexvalued, continuous functions on compact Hausdorff spaces is of the formwhere µ and ν are continuous and unimodular, there exists a clopen set K with ν = µ on K and ν = −µ on K c , and τ and ρ are homeomorphisms.where |µ(y)| = 1 for all y ∈ Y and where τ : Y → X is a homeomorphism. This classic result has been extended to mappings between subspaces of C(X) and C(Y ), and a general survey of such results can be found in [4]. We note one in

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“…The study of isometries between subspaces of continuous functions originally dates back to the classical Banach-Stone theorem. The theorem has various generalizations (in scalar valued case) based on different techniques, see for example [6,9,12,13,16,15].…”

Section: Introductionmentioning

confidence: 99%

A note on nonlinear isometries between vector-valued function spaces

Jamshidi

Sady

2018

Linear and Multilinear Algebra

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Let X, Y be compact Hausdorff spaces and E, F be Banach spaces over R or C. In this paper, we investigate the general form of surjective (not necessarily linear) isometries T : A −→ B between subspaces A and B of C(X, E) and C(Y, F ), respectively. In the case that F is strictly convex, it is shown that there exist a subset Y 0 of Y , a continuous function Φ : Y 0 −→ X onto the set of strong boundary points of A and a family {V y } y∈Y0 of real-linearIn particular, we get some generalizations of the vector-valued Banach-Stone theorem and a generalization of Cambern's result. We also give a similar result in the case that F is not strictly convex, but its unit sphere contains a maximal convex subset which is singleton.2010 Mathematics Subject Classification. Primary 46J10, 46J20; Secondary 47B33. Key words and phrases. isometries, vector-valued continuous functions, strong boundary points, Choquet boundary . 1 strong operator topology, such that T f (t) = V t (f (Φ(t)) for all f ∈ C(X, E) and t ∈ Y .This result has been generalized by Cambern [4] for into linear isometries and by Font [8] for certain vector-valued subspaces of continuous functions. In [1] Al-Halees and Fleming relaxed the strict convexity condition on E, and by considering T -sets in companion with another condition on E, called condition (P), they obtained more general results. We also refer the reader to the nice books [6,7] including many earlier results.More recently, surjective isometries between certain subspaces of vector-valued continuous functions have been studied in [3] and [11]. We should note that the method used in these papers is based on extreme point technique. By [3] (see also [2]), for a compact Hausdorff space X and a reflexive real Banach space F whose dual is strictly convex, if A is a subspace of C(X, F ) which separates X in the sense of [3, Definition 3.1], and T : A −→ A is a surjective isometry preserving constant functions, then there exist a surjective isometry V : F −→ F and a homeomorphism τ : X −→ X such that

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Extremely Strong Boundary Points and Real-linear Isometries

Cited by 7 publications

References 14 publications

Composition in Modulus Maps on Semigroups of Continuous Functions

Composition in Modulus Maps on Semigroups of Continuous Functions

Nonlinear isometries between function spaces

A note on nonlinear isometries between vector-valued function spaces

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