TY - JOUR
T1 - Diameter two properties in some vector-valued function spaces
AU - Lee, Han Ju
AU - Tag, Hyung Joon
N1 - Publisher Copyright:
© 2021, The Author(s) under exclusive licence to The Royal Academy of Sciences, Madrid.
PY - 2022/1
Y1 - 2022/1
N2 - 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.
AB - 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.
KW - Diameter two property
KW - Shilov boundary
KW - Uniform algebra
KW - Urysohn-type lemma
UR - http://www.scopus.com/inward/record.url?scp=85117291768&partnerID=8YFLogxK
U2 - 10.1007/s13398-021-01165-6
DO - 10.1007/s13398-021-01165-6
M3 - Article
AN - SCOPUS:85117291768
SN - 1578-7303
VL - 116
JO - Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas
JF - Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas
IS - 1
M1 - 17
ER -