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Geometry and analytic boundaries of Marcinkiewicz sequence spaces. (English) Zbl 1211.46009
Let $\Psi = (\Psi(k))_{k=0}^\infty$ be an increasing sequence of non-negative real numbers with $\Psi(0)=0$ and $\Psi(k) > 0$ if $k\ge 1$. The Marcinkiewicz sequence space $m_\Psi$ is the space of all bounded sequences $z=(z_k)_k$ such that $\|z\|=\sup_k \sum_{j=1}^k z_j^*/ \Psi(k)<\infty$, and its subspace $m^0_\Psi$ contains all elements $z$ such that $\lim_{k\to\infty} \sum_{j=1}^k z_j^*/ \Psi(k)=0$. The space $m_\Psi$ equipped with the norm $\|\cdot\|$ is rearrangement invariant and $m_\Psi^0$ is a predual of a Lorentz space. Given a complex Banach space $E$, denote by $\mathcal{A}_b(B_E)$ 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 $\mathcal{A}_u(B_E)$ denote the Banach algebra of functions in $\mathcal{A}_b(B_E)$ which are uniformly continuous on $B_E$. A subset $B$ of $B_E$ is said to be a boundary for $\mathcal{A}_u(B_E)$ if $\|f\| = \sup_{z\in B} |f(z)|$ for all $f\in \mathcal{A}_u(B_E)$. The Šilov boundary of $\mathcal{A}_u(B_E)$ is the minimal closed boundary. A point $x\in B_E$ is a peak point of $\mathcal{A}_u(B_E)$ if there is $f \in \mathcal{A}_u(B_E)$ such that $|f(y)| < f(x)$ for all $y\in B_E\setminus \{x\}$. Characterizations of extreme, complex extreme and exposed points of $m^0_\Psi$ are given. For instance, it is proved that $z\in B_{m^0_\Psi}$ is complex extreme if and only if $z$ is a peak point of $\mathcal{A}_u(B_{m^0_\Psi})$. Applying those characterizations, a condition is found which is necessary and sufficient for a subset of $B_{m^0_\Psi}$ to be a boundary for $\mathcal{A}_u(B_{m^0_\Psi})$. It is also shown that it is possible that a set of peak points of $\mathcal{A}_u(B_{m^0_\Psi})$ is a boundary for $\mathcal{A}_u(B_{m^0_\Psi})$, yet it is not the Šilov boundary for $\mathcal{A}_u(B_{m^0_\Psi})$.

46B20Geometry and structure of normed linear spaces
46J15Banach algebras of differentiable or analytic functions, $H^p$-spaces
46E50Spaces of differentiable or holomorphic functions on infinite-dimensional spaces
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