A Filter Description for the Homomorphisms of the Algebra of Bounded Analytic Functions on the Unit Disc

Abstract

For any filter F defined on the unit disc D, F* is the filter generated by ∈-neighbourhoods of the sets of F, using hyperbolic distance. Any complex homomorphism I of β, the algebra of bounded analytic functions on D, is given by I(g) = lim g(F*) for some maximal closed filter F. The homomorphisms can be classified according to the direction of approach to the boundary of the corresponding filters. For those which are obtained by oricycle or non-tangential approach, the *-filters are in 1-1 correspondence with the homomorphisms; and into these subsets, one can analytically embed discs. On the Silov boundary of β, the above correspondence fails to be 1-1, and smaller filters are considered.