Boundaries for algebras of holomorphic functions on Banach spaces

TL;DR

This paper explores the relationship between boundaries of holomorphic function algebras on Banach spaces and their geometric properties, establishing conditions under which the unit sphere serves as the Shilov boundary.

Contribution

It demonstrates that the Shilov boundary coincides with the unit sphere in certain Banach spaces, such as locally c-convex sequence spaces and Orlicz-Lorentz spaces under specific conditions.

Findings

Shilov boundary equals the unit sphere in locally c-convex sequence spaces.

In Orlicz-Lorentz spaces satisfying the δ₂-condition, the unit sphere is the Shilov boundary.

Links between complex convexity and boundary properties of holomorphic function algebras.

Abstract

We study the relations between boundaries for algebras of holomorphic functions on Banach spaces and complex convexity of their balls. In addition, we show that the Shilov boundary for algebras of holomorphic functions on an order continuous sequence space $X$ is the unit sphere $S_X$ if $X$ is locally c-convex. In particular, it is shown that the unit sphere of the Orlicz-Lorentz sequence space $\lambda_{\phi, w}$ is the Shilov boundary for algebras of holomorphic functions on $\lambda_{\phi, w}$ if $\phi$ satisfies the $\delta_2$-condition.