Diameter two properties in some vector-valued function spaces

Han Ju Lee, Hyung Joon Tag

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

We introduce a vector-valued version of a uniform algebra, called the vector-valued function space over a uniform algebra. The diameter two properties of the vector-valued function space over a uniform algebra on an infinite compact Hausdorff space are investigated. Every nonempty relatively weakly open subset of the unit ball of a vector-valued function space A(K, (X, τ)) over an infinite dimensional uniform algebra has diameter two, where τ is a locally convex Hausdorff topology on a Banach space X compatible to a dual pair. Under the assumption of X equipped with the norm topology being uniformly convex and the additional condition that A⊗ X⊂ A(K, X) , it is shown that Daugavet points and Δ -points on A(K, X) over a uniform algebra A are the same, and they are characterized by the norm-attainment at a limit point of the Shilov boundary of A. In addition, a sufficient condition for the convex diametral local diameter two property of A(K, X) is also provided. Similar results also hold for an infinite dimensional uniform algebra.

Original languageEnglish
Article number17
JournalRevista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas
Volume116
Issue number1
DOIs
StatePublished - Jan 2022

Keywords

  • Diameter two property
  • Shilov boundary
  • Uniform algebra
  • Urysohn-type lemma

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