SUBALGEBRAS OF H('INFINITY) AND THE CORONA PROPERTY

This dissertation is an investigation of subalgebras of H('(INFIN)), the Banach algebra of bounded analytic functions on the open unit disk, with respect to the corona property. That is, if B is a closed subalgebra of H('(INFIN)), is the open unit disk w*-dense in the maximal ideal space of B? Recent research has shown that for some subalgebras this is true while for others it is not. A major achievement of the dissertation is a theorem which relates the corona property to the polynomial convexity of cluster sets of functions of the subalgebra in question. The proof of this theorem depends upon certain facts concerning the fibers and fiber algebras of a subalgebra of H('(INFIN)). These are established and independently are of some interest.

Finally some examples of subalgebras with and without the corona property are produced. Previously known examples are investigated in light of the characterization mentioned above. New examples are exhibited which show that the corona property is not related to familiar concepts used in dealing with subalgebras of H('(INFIN)).