# Holomorphic functions with large cluster sets

## TL;DR

This paper investigates the algebraic and linear structures of bounded holomorphic functions with large cluster sets on the ball, demonstrating strong c-algebrability and spaceability in various Banach spaces.

## Contribution

It establishes the strong c-algebrability and spaceability of sets of holomorphic functions with large cluster sets across different Banach spaces, including specific classical spaces.

## Key findings

- Set is strongly c-algebrable for all separable Banach spaces.
- Strong c-algebrability and spaceability hold for lp and dual Lorentz spaces.
- Results include subalgebras of uniformly continuous holomorphic functions.

## Abstract

We study linear and algebraic structures in sets of bounded holomorphic functions on the ball which have large cluster sets at every possible point (i.e., every point on the sphere in several complex variables and every point of the closed unit ball of the bidual in the infinite dimensional case). We show that this set is strongly c-algebrable for all separable Banach spaces. For specific spaces including lp or duals of Lorentz sequence spaces, we have strongly c-algebrability and spaceability even for the subalgebra of uniformly continuous holomorphic functions on the ball.