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Separating and Bi-separating Maps and Their Relation to Banach-Stone Theorem

Rajaei, Reza | 2019

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 52290 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Ranjbar Motlagh, Alireza
  7. Abstract:
  8. Let C(S) and C(T) denote the sup-normed Banach spaces of real or complex-valued continuous functions on the compact Hausdorff spaces S and T, respectively. A linear map H:C(T)→C(S) is called separating if, for x,y∈C(T), xy≡0 implies HxHy≡0. In the second chapter of this thesis, we will show that any continuous separating map is a continuous multiple of a composition map. Moreover, it will be proved that any linear separating isomorphism of C(T) onto C(S) is continuous. We will also define separating and biseparating maps on the rings of continuous functions equipped with the compact-open topology. In addition, vector-valued separating maps will be investigated. For example, assume that C(T) and C(S) denote the rings of real-valued continuous functions on the Tihonov spaces T and S endowed with their respective compact-open topologies; a linear bijection H:C(T)→C(S) is called biseparating if and only if both H and H^(-1) are separating. In the fourth chapter, we will see how the separating property implies that the Banach-Stone theorem holds in non-Archimedean spaces although it fails in general. Investigating the properties of separating and biseparating maps and their relation to homeomorphisms between T and S will lead to some theorems which resemble the well-known Banach-Stone theorem. One of these theorems asserts that if H:C(T)→C(S) is biseparating then the realcompactifications of T and S are homeomorphic. In the last chapter, we impose constraints upon T and S resulting in a few propositions which guarantee that a separating map is biseparating; in other words, we will find conditions under which the biseparating property is obtained
  9. Keywords:
  10. Separating Map ; Bi-separating Map ; Banach-Stone Theorem ; Stone-Cech Compactification ; Compact-open Topology ; Real Compactification

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