Unimodular approximation in function algebras (Q1075560)

Consider the algebra C(X) of continuous complex-valued functions on the compact Hausdorff space X. Let A be a subalgebra which contains constants and is closed under the sup-norm. I(A) denotes the set of inner functions in A and Q the set of quotients of such inner functions. So \(U=I(C(X))\) is the group of continuous unimodular functions on X under pointwise multiplication and log U the subgroup of members with continuous logarithms. The natural quotient map from U to U/log U is P. Theorem. Q is dense in U iff (1) \(P(Q)=U/\log U\) and (2) I(A) separates the points of X. A consequence is a result of Douglas and Rudin concerning quotients of Blaschke products. Applications, including one on compact Abelian groups, are discussed. A note of interest: the author remarks that he has been unable to find a function algebra satisfying (2) but not (1) of the theorem.