Automatic continuity of linear maps on spaces of continuous functions

Beckenstein, Edward; Narici, Lawrence; Todd, Aaron R.

doi:10.1007/bf01246833

Cited by 36 publications

(31 citation statements)

References 6 publications

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“…Separating maps were considered by Beckenstein, Narici and Todd [3] for the algebra of complex-valued continuous functions defined on a compact Hausdorff space. The main goal of studies in the field was to prove automatic continuity for separating maps.…”

Section: Introductionmentioning

confidence: 99%

“…As a result, some topological links between underlying spaces are deduced, and weighted composition type representations for separating maps are obtained. In recent years, considerable attention has been given to separating maps; see for example [1], [2], and [4] on Banach lattices, [3] and [6] on spaces of continuous functions, and [5] on group algebras of locally compact Abelian groups.…”

Section: Introductionmentioning

confidence: 99%

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Separating maps on weighted function algebras on topological groups

Maghsoudi

Nasr-Isfahani

2011

Mathematica Slovaca

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Let G 1 and G 2 be locally compact groups and let ω 1 and ω 2 be weight functions on G 1 and G 2, respectively. For i = 1, 2, let also C 0(G i, 1/ω i) be the algebra of all continuous complex-valued functions f on G i such that f/ω i vanish at infinity, and let H: C 0(G 1, 1/ω 1) → C 0(G 2, 1/ω 2) be a separating map; that is, a linear map such that H(f)H(g) = 0 for all f, g ∈ C 0(G 1, 1/ω 1) with fg = 0. In this paper, we study conditions under which H can be represented as a weighted composition map; i.e., H(f) = φ(f ℴ h) for all f ∈ C 0(G 1, 1/ω 1), where φ: G 2 → ℂ is a non-vanishing continuous function and h: G 2 → G 1 is a topological isomorphism. Finally, we offer a statement equivalent to that h is also a group homomorphism.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Separating maps on weighted function algebras on topological groups

Maghsoudi

Nasr-Isfahani

2011

Mathematica Slovaca

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“…The existence and uniqueness of support points are given in the following theorem, which can be proved, with slight changes, as the corresponding one which appears in [BNT,p. 260].…”

Section: Weakly Biseparating Mapsmentioning

confidence: 99%

$\mathbb N$-compactness and automatic continuity in ultrametric spaces of bounded continuous functions

Araujo

1999

Proc. Amer. Math. Soc.

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Abstract. In this paper (weakly) separating maps between spaces of bounded continuous functions over a nonarchimedean field K are studied. It is proven that the behaviour of these maps when K is not locally compact is very different from the case of real-or complex-valued functions: in general, for Ncompact spaces X and Y , the existence of a (weakly) separating additive map T : C * (X) → C * (Y ) implies that X and Y are homeomorphic, whereas when dealing with real-valued functions, this result is in general false, and we can just deduce the existence of a homeomorphism between the Stone-Čech compactifications of X and Y . Finally, we also describe the general form of bijective weakly separating linear maps and deduce some automatic continuity results.

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“…in [16], [10] and [11] are extended to a wider class of regular Banach function algebras that includes, for instance, Segal algebras [21] or the Banach sequence algebras / P (N), p e (0, °°), in [9]. It is, however, important to remark that a separating map need not be continuous; indeed, K. Jarosz proved [16] that, given two compact spaces X (infinite) and Y, there, always exists a discontinuous separating map defined from C(X) into C(Y); see also [8].…”

Section: Juan J Font Theorem ([11]) a Separating Bijection T:l 1 (mentioning

confidence: 99%

Automatic continuity of certain isomorphisms between regular Banach function algebras

Font¹

1997

Glasgow Math. J.

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Abstract. Let A and B be regular semisimple commutative Banach algebras; that is to say, regular Banach function algebras. A linear map T denned from A into B is said to be separating or disjointness preserving if/.g = 0 implies Tf.Tg = 0, for all f,g e A In this paper we prove that if A satisfies Ditkin's condition then a separating bijection is automatically continuous and its inverse is separating. If also B satisfies Ditkin's condition, then it induces a homeomorphism between the structure spaces of A and B.Finally, we show that linear isometries between regular uniform algebras are separating. As corollaries, a classical theorem of Nagasawa ([19]) and the Banach-Stone theorem (both for regular uniform algebras) are easily inferred. Introduction.The basic problem in the theory of automatic continuity of linear operators consists of giving algebraic conditions on two Banach algebras si and 38 which ensure that every algebra homomorphism !T:s4-*2ft is necessarily continuous. The most important positive result in this context is due to B. E. Johnson ([15]). Every algebra homomorphism of a Banach algebra onto a semisimple Banach algebra is continuous.In this paper we shall study the automatic continuity of a special type of linear mapping which extends the concept of algebra homomorphism: Let A and B be two independently, Koldunov [17] have shown that the inverse of a disjointness preserving bijection is disjointness preserving under certain general conditions. Disjointness preserving maps were also considered in [8] for spaces of real or complex-valued continuous functions denned on a compact Hausdorff space with the name of separating maps. The main goal in this context is to prove automatic continuity results for separating maps between different kinds of function spaces of the following type.

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Automatic continuity of linear maps on spaces of continuous functions

Cited by 36 publications

References 6 publications

Separating maps on weighted function algebras on topological groups

Separating maps on weighted function algebras on topological groups

$\mathbb N$-compactness and automatic continuity in ultrametric spaces of bounded continuous functions

Automatic continuity of certain isomorphisms between regular Banach function algebras

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