Boundaries for algebras of holomorphic functions.
Let \(A_u(B_E)\) be the Banach algebra of all complex valued functions which are uniformly continuous on the closed unit ball \(B_E\) of a Banach space \(E\) and holomorphic in the interior of \(B_E\). The authors study the boundary and the Shilov boundary of \(A_u(B_E)\) for \(E=G_p\), the Gowers sequence space with characteristic \(p\), and for its \(n\)-dimensional version \(E=G_{p,n}\) (\(p=1\) gives the classical Gowers space). It is shown that the Shilov boundary of \(A_u(B_{G_{p,n}})\) coincides with both, the set of all extreme points of \(B_{G_{p,n}}\) and the set of all of its peak points. An explicit characterization of this set is also given. For the infinite dimensional case \(G_p\), a characterization of the boundaries in terms of its finite dimensional projections to \(G_{p,n}\) is given. Using this, it is shown that \(A_u(G_p)\) does not have a Shilov boundary for all \(p \in \mathbb N\). Finally, examples of ``small'' boundaries are constructed.
scientific article; zbMATH DE number 1956883