Trivial Gleason parts and the topological stable rank of H∞

We show that the subset of the maximal ideal space of H^∞ formed by the trivial Gleason parts is totally disconnected. A pair (ƒ, g) ∈ H^∞ × H^∞ is called a corona pair if [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /][inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] > 0, where [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] is the open unit disc. We prove that the set U₂(H^∞) of corona pairs is dense in H^∞ × H^∞.