*(English)*Zbl 1211.46009

Let ${\Psi}={\left({\Psi}\left(k\right)\right)}_{k=0}^{\infty}$ be an increasing sequence of non-negative real numbers with ${\Psi}\left(0\right)=0$ and ${\Psi}\left(k\right)>0$ if $k\ge 1$. The Marcinkiewicz sequence space ${m}_{{\Psi}}$ is the space of all bounded sequences $z={\left({z}_{k}\right)}_{k}$ such that $\parallel z\parallel ={sup}_{k}{\sum}_{j=1}^{k}{z}_{j}^{*}/{\Psi}\left(k\right)<\infty $, and its subspace ${m}_{{\Psi}}^{0}$ contains all elements $z$ such that ${lim}_{k\to \infty}{\sum}_{j=1}^{k}{z}_{j}^{*}/{\Psi}\left(k\right)=0$. The space ${m}_{{\Psi}}$ equipped with the norm $\parallel \xb7\parallel $ is rearrangement invariant and ${m}_{{\Psi}}^{0}$ is a predual of a Lorentz space. Given a complex Banach space $E$, denote by ${\mathcal{A}}_{b}\left({B}_{E}\right)$ the Banach algebra of all functions which are continuous and bounded on ${B}_{E}$, the closed unit ball of $E$, and holomorphic on the interior of ${B}_{E}$. By ${\mathcal{A}}_{u}\left({B}_{E}\right)$ denote the Banach algebra of functions in ${\mathcal{A}}_{b}\left({B}_{E}\right)$ which are uniformly continuous on ${B}_{E}$. A subset $B$ of ${B}_{E}$ is said to be a boundary for ${\mathcal{A}}_{u}\left({B}_{E}\right)$ if $\parallel f\parallel ={sup}_{z\in B}\left|f\left(z\right)\right|$ for all $f\in {\mathcal{A}}_{u}\left({B}_{E}\right)$. The Šilov boundary of ${\mathcal{A}}_{u}\left({B}_{E}\right)$ is the minimal closed boundary. A point $x\in {B}_{E}$ is a peak point of ${\mathcal{A}}_{u}\left({B}_{E}\right)$ if there is $f\in {\mathcal{A}}_{u}\left({B}_{E}\right)$ such that $\left|f\right(y\left)\right|<f\left(x\right)$ for all $y\in {B}_{E}\setminus \left\{x\right\}$.

Characterizations of extreme, complex extreme and exposed points of ${m}_{{\Psi}}^{0}$ are given. For instance, it is proved that $z\in {B}_{{m}_{{\Psi}}^{0}}$ is complex extreme if and only if $z$ is a peak point of ${\mathcal{A}}_{u}\left({B}_{{m}_{{\Psi}}^{0}}\right)$. Applying those characterizations, a condition is found which is necessary and sufficient for a subset of ${B}_{{m}_{{\Psi}}^{0}}$ to be a boundary for ${\mathcal{A}}_{u}\left({B}_{{m}_{{\Psi}}^{0}}\right)$. It is also shown that it is possible that a set of peak points of ${\mathcal{A}}_{u}\left({B}_{{m}_{{\Psi}}^{0}}\right)$ is a boundary for ${\mathcal{A}}_{u}\left({B}_{{m}_{{\Psi}}^{0}}\right)$, yet it is not the Šilov boundary for ${\mathcal{A}}_{u}\left({B}_{{m}_{{\Psi}}^{0}}\right)$.

##### MSC:

46B20 | Geometry and structure of normed linear spaces |

46J15 | Banach algebras of differentiable or analytic functions, ${H}^{p}$-spaces |

46E50 | Spaces of differentiable or holomorphic functions on infinite-dimensional spaces |