Daugavet property of Banach algebras of holomorphic functions and norm-attaining holomorphic functions

TL;DR

This paper investigates the Daugavet property in Banach algebras of holomorphic functions, showing duals lack certain exposed points, and proves density of norm-attaining functions under specific conditions, advancing understanding of reflexivity and polynomial properties.

Contribution

It establishes the Daugavet property for certain Banach algebras of holomorphic functions and introduces new density and reflexivity results related to vector-valued holomorphic functions.

Findings

Duals of Banach algebras lack weak*-strongly exposed points.

Some Banach algebras of holomorphic functions have the Daugavet property.

The set of norm-attaining vector-valued holomorphic functions is dense under the metric π-property.

Abstract

We show that the duals of Banach algebras of scalar-valued bounded holomorphic functions on the open unit ball $B_E$ of a Banach space $E$ lack weak$^*$-strongly exposed points. Consequently, we obtain that some Banach algebras of holomorphic functions on an arbitrary Banach space have the Daugavet property which extends the observation of P. Wojtaszczyk. Moreover, we present a new denseness result by proving that the set of norm-attaining vector-valued holomorphic functions on the open unit ball of a dual Banach space is dense provided that its predual space has the metric $\pi$-property. Besides, we obtain several equivalent statements for the Banach space of vector-valued homogeneous polynomials to be reflexive, which improves the result of J. Mujica, J. A. Jaramillo and L. A. Moraes. As a byproduct, we generalize some results on polynomial reflexivity due to J. Farmer.