Factorization of Blaschke products and ideal theory in H ∞

Let H ∞ be the Banach algebra of bounded analytic functions on the open unit disk D. Let G be the union set of all nontrivial Gleason parts in the maximal ideal space of H ∞. Let E be a nonvoid compact and totally disconnected subset of G and nE be a bounded numbering function on E. We characterize nE for which there is a closed ideal I in H ∞ such that Z(I ) = E and ord(I, x) = nE(x) for every x ∈ E. Let I1, I2, . . . , Ik be closed ideals in H ∞ satisfying Z(Ii ) ⊂ G for 1 i k. We prove that k i=1 Ii = { k i=1 fi : fi ∈ Ii , 1 i k} is a closed ideal. A local ideal theory in H ∞ plays an important role to prove our results. © 2010 Elsevier Inc. All rights reserved