Cluster values of holomorphic functions of bounded type (Q2787974)

The authors use \(B\) to denote the open unit ball of a Banach space \(X\) and study the spectra, \(\mathcal M\), of the Fréchet algebra, \(\mathcal H_b(X)\), of all holomorphic functions on \(X\) which are bounded on bounded subsets of \(X\) and of the Banach algebra, \(\mathcal A_u(B)\), of all functions which are both holomorphic and uniformly continuous on \(B\). Given \(z\) in the bidual, \(X^{\ast\ast}\), of \(X\), \(\mathcal M_z\) denotes all \(\phi\) in \(\mathcal M\) such that \(\phi_{X^\ast}=z\). For \(\phi\) in the spectrum of \(\mathcal H_b(X)\), the radius of \(\phi\), \(R(\phi)\) is defined as \(\inf\{r>0: |\phi(f)|\leq \| f\|_{rB}\), \(f\in \mathcal H_b(X)\}\), while \(\mathcal M_{z,r}\) denotes all \(\phi\) in \(\mathcal M_z\) with \(R(\phi)\leq r\).NEWLINENEWLINEThe authors show that for any Banach space \(X\) with a shrinking \(1\)-unconditional basis and any \(f\) in \(\mathcal H_b(X)\), the cluster set, \(\mathrm{Cl}_r(f,0)\), of all limits of \(f\) along nets in \(rB^{\ast\ast}\) which converge in the \(\mathrm{weak}^\ast\) topology to \(0\) is equal to \(\hat f(\mathcal M_{0,r}):=\{\phi(f):f \in \mathcal M_{0,r}\}\). It is also shown that for any Banach space \(X\) with a shrinking \(K\)-unconditional basis, any \(f\) in \(\mathcal H_b(X)\) and any \(z_o\) in \(rB^{\ast\ast}\), NEWLINE\[NEWLINE\mathrm{Cl}_r(f,z_o)\subseteq\hat f(\mathcal M_{z_o,r})\subseteq\mathrm{Cl}_{K(\| z_o\|+r)}(f,z). NEWLINE\]NEWLINE When \(X\) is a Banach space which admits a homogeneous polynomial which is not weakly continuous on bounded sets at the origin and \(z\) belongs to the open unit ball of \(X^{\ast\ast}\), the authors prove that the cardinality of the spectrum of \(\mathcal A_u(B_X)\) over \(z\) is at least the continuum. In the final section, the authors show that the spectrum of \(\mathcal A_u(B_{\ell_1})\) over \(z\) consists of a point when \(z\) belongs to the unit sphere of \(\ell_1\), contains a copy of \(\beta\mathbf N\) when \(z\) belongs to the unit ball of \(\ell_1^{\ast\ast}\) and that there is a copy of \(\beta\mathbf N\) in the unit sphere of \(\ell_1^{\ast\ast}\) where the spectrum of \(\mathcal A_u(B_{\ell_1})\) over \(z\) has cardinality at least the continuum.