Cluster values of analytic functions on a Banach space

CARANDO, DANIEL

Oldenburg

Seminario; Coloquio del Institut für Mathematik, Universidad de Oldenburg; 2011

We investigate uniform algebras of bounded analytic functions on the unit ball of a complex Banach space. We prove several cluster value theorems, relating cluster sets of a function to its range on the fibers of the spectrum of the algebra. These lead to weak versions of the corona theorem for $\ell_2$ and for $c_0$. In the case of the open unit ball of $c_0$, we solve the corona problem whenever all but one of the functions comprising the corona data are uniformly approximable by polynomials in functions in $c_0^*$.