Shilov boundary for holomorphic functions on some classical Banach spaces
Volume 179 / 2007
Abstract
Let $ {\mathcal{A}}_\infty (B_X)$ be the Banach space of all bounded and continuous functions on the closed unit ball $B_X$ of a complex Banach space $X$ and holomorphic on the open unit ball, with sup norm, and let $ {\mathcal{A}}_{\rm u} (B_X)$ be the subspace of $ {\mathcal{A}}_\infty (B_X)$ of those functions which are uniformly continuous on $B_X.$ A subset $B \subset B_X$ is a boundary for $ {\mathcal{A}}_\infty (B_X)$ if $\Vert f \Vert = \sup _{ x \in B} \vert f(x) \vert $ for every $f \in {\mathcal{A}}_\infty (B_X)$. We prove that for $X= d(w,1) $ (the Lorentz sequence space) and $X= C_1(H)$, the trace class operators, there is a minimal closed boundary for $ {\mathcal{A}}_\infty (B_X)$. On the other hand, for $X=\mathcal{S}$, the Schreier space, and $X= K(\ell_p, \ell_q ) $ ($1 \le p \le q < \infty$), there is no minimal closed boundary for the corresponding spaces of holomorphic functions.
