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Geometry and analytic boundaries of Marcinkiewicz sequence spaces. (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 $\parallel ·\parallel$ is rearrangement invariant and ${m}_{{\Psi }}^{0}$ is a predual of a Lorentz space. Given a complex Banach space $E$, denote by ${𝒜}_{b}\left({B}_{E}\right)$ 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 ${𝒜}_{u}\left({B}_{E}\right)$ denote the Banach algebra of functions in ${𝒜}_{b}\left({B}_{E}\right)$ which are uniformly continuous on ${B}_{E}$. A subset $B$ of ${B}_{E}$ is said to be a boundary for ${𝒜}_{u}\left({B}_{E}\right)$ if $\parallel f\parallel ={sup}_{z\in B}|f\left(z\right)|$ for all $f\in {𝒜}_{u}\left({B}_{E}\right)$. The Šilov boundary of ${𝒜}_{u}\left({B}_{E}\right)$ is the minimal closed boundary. A point $x\in {B}_{E}$ is a peak point of ${𝒜}_{u}\left({B}_{E}\right)$ if there is $f\in {𝒜}_{u}\left({B}_{E}\right)$ such that $|f\left(y\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 ${𝒜}_{u}\left({B}_{{m}_{{\Psi }}^{0}}\right)$. 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 ${𝒜}_{u}\left({B}_{{m}_{{\Psi }}^{0}}\right)$. It is also shown that it is possible that a set of peak points of ${𝒜}_{u}\left({B}_{{m}_{{\Psi }}^{0}}\right)$ is a boundary for ${𝒜}_{u}\left({B}_{{m}_{{\Psi }}^{0}}\right)$, yet it is not the Šilov boundary for ${𝒜}_{u}\left({B}_{{m}_{{\Psi }}^{0}}\right)$.

##### 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