Fibers over the sphere of a uniformly convex Banach space
Let \(B\) be the open unit ball of an infinite-dimensional complex Banach space \(X\). In this paper the author studies the boundary behavior of bounded analytic functions on \(B\) when \(X\) is uniformly convex. Several interesting results are obtained regarding the structure of the spectrum of \(H^\infty (B)\). Among them: If \(X\) is uniformly convex, and \((\varphi_i)\) lie in fibers over the boundary and converge to \(\varphi\) in the Gelfand topology, then all but finitely many \(\varphi_i\) lie in the same fiber as \(\varphi\). If \(X=\ell_p\), there are uniformly homeomorphic copies of balls of ultrapowers of \(\ell_p\) in all fibers.
scientific article; zbMATH DE number 1548271