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 $\mathfrak{c}$-algebrable for all separable Banach spaces. For specific spaces including ${\ell }_p$ or duals of Lorentz sequence spaces, we have strongly $\mathfrak{c}$-algebrability and spaceability even for the subalgebra of uniformly continous holomorphic functions on the ball.

中文翻译：

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具有大簇集的全纯函数

我们研究球上的有界全纯函数集中的线性和代数结构，这些函数在每个可能的点（即球面上的每个复变量中的每个点以及无穷大中的二元的封闭单位球的每个点）都有大的簇集。维情况）。我们证明这个集合是强$\mathfrak{丙}$- 所有可分 Banach 空间的代数。对于特定空间，包括${\ell }_{磷}$ 或洛伦兹序列空间的对偶，我们有强 $\mathfrak{丙}$- 代数性和空间性，即使对于球上均匀连续的全纯函数的子代数也是如此。