On separating maps between locally compact spaces

Font, Juan J.; Hernández, Salvador

doi:10.1007/bf01189890

Cited by 67 publications

(56 citation statements)

References 7 publications

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“…• In [10] and [12], the authors prove that linear isometries between C 00 (A r )-spaces and spaces of vector-valued continuous functions, respectively, are separating. The following result shows that the same is true in the context of regular uniform algebras.…”

Section: Isomerries Between Regular Uniform Algebrasmentioning

confidence: 99%

“…cit.) 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%

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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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Section: Isomerries Between Regular Uniform Algebrasmentioning

confidence: 99%

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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“…However, let us recall (see [11]) that a separating bijection from C 0 (Gi) (respectively CQO(GI)) onto Co(G 2 ) (respectively Coo(G 2 )) is automatically continuous and induces a homeomorphism between the locally compact (not necessarily real-compact) spaces G : and G 2 .…”

Section: )) Where V(h(g 2 )) Stands For the Real-compactification Ofmentioning

confidence: 99%

“…https://doi.org/10.1017/S1446788700001099 [11] Characterizing locally compact groups 415 REMARK. If we consider the Bohr compactification bG 2 of G 2 instead of its StoneCech compactification in the remark following Theorem 2, then 7 is also character preserving.…”

Section: Algebraic Characterization Of Locally Compact Abelian Groupsmentioning

confidence: 99%

“…The main goal of these two papers and [20,4,11,12] is to prove automatic continuity results for separating maps when G, (i=l,2) is compact, real-compact, locally compact and a locally compact Abelian group respectively. As a consequence, certain topological links between the underlying spaces are deduced and weighted composition type representations for separating maps are obtained.…”

Section: Introductionmentioning

confidence: 99%

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Algebraic characterizations of locally compact groups

Font

Hernández

1997

J Aust Math Soc A

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Let d , G 2 be locally compact real-compact spaces. A linear map T defined from. I n this paper we prove that both a separating map which preserves non-vanishing functions and a separating bijection which satisfies condition (M) (see Definition 4) are automatically continuous and can be written as weighted composition maps. We also study the effect of separating surjections (respectively injections) on the underlying spaces Gj and G 2 .Next we apply the above results to give an algebraic characterization of locally compact Abelian groups, similar to the one given in [7] for compact Abelian groups in the presence of ring isomorphisms.Finally, locally compact (not necessarily Abelian) groups are considered. We provide a sharpening of a result of Edwards and study the effect of onto (respectively injective) weighted composition maps on the groups G\ and G 2 .

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The structure of compact disjointness preserving operators on continuous functions

Lin

Wong

2009

Mathematische Nachrichten

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The duals of C 0 (a, b) and C[a, b] with respect to disjointness preserving groups are characterized. A. Plessner's result (1929) about the translation group is extended. A Wiener-Young type theorem for disjointness preserving groups is obtained. It is assumed that the reader is familiar with the notions of a Banach lattice, positive operator and disjointness preserving operator. [AB], [LZ] and [MN] are good references for this material. We also assume familiarity with the theory of C 0-semigroups and groups. [Pa] and [Na] should provide all necessary information when needed. Throughout −∞ ≤ a < b ≤ +∞, X is either (a, b) or [a, b], where by [−∞, b] we mean (−∞, b], and by [a, +∞] we mean [a, +∞). Definition. A function f : X → C is called vanishing at infinity if ∀ε > 0 ∃K ⊂ X, K compact such that ∀x ∈ X \ K |f (x)| < ε. We denote the lattice of all continuous functions f : X → C vanishing at infinity by C 0 (X), the lattice of all bounded continuous functions f : X → C by C b (X), the lattice of all continuous functions f : X → C with compact support by C c (X), and the lattice of all regular complex-valued Borel measures on X with finite variation by M (X). Let {T (t)} t∈R be a C 0-group on a Banach space E. Definition. The group dual of E with respect to {T (t)} t∈R , denoted E and pronounced E-sun, is defined in the following way: E = { u * ∈ E * : lim t→0 T * (t)u * − u * = 0 }. [vN] is an excellent source of information about the semigroup and group duals of Banach spaces and related subjects. Definition. A group {T (t)} t∈R on a Banach lattice E is called disjointness preserving if ∀t ∈ R T (t) is a disjointness preserving operator. Let {T (t)} t∈R be a disjointness preserving C 0-group on C 0 (X). It follows from [Na, B-II.3.8] that ∃ϕ : R × X → X a continuous flow ∃q : R → C b (X) a

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On separating maps between locally compact spaces

Cited by 67 publications

References 7 publications

Automatic continuity of certain isomorphisms between regular Banach function algebras

Automatic continuity of certain isomorphisms between regular Banach function algebras

Algebraic characterizations of locally compact groups

The structure of compact disjointness preserving operators on continuous functions

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