Approximation by inner functions

Abstract

Let L°°(T) denote the complex Banach algebra of (equivalence classes of) bounded measurable functions on the unit circle T, relative to Lebesgue measure m. The norm 11 /11» of an / in L^iT) is the essential supremum of |/| on T. The collection of all bounded holomorphic functions in the open unit disc U forms a Banach algebra which can be identified (via radial limits) with the norm-closed subalgebra H°° of L~(T). A function / in L°°(T) is unimodular if |/| = 1 a.e., on Tm The inner functions are the unimodular members of H°°. It is well known that they play an important role in the study of H°°. The main result (Theorem 1) is that the set of quotients of inner functions is norm-dense in the set of unimodular functions in L°°(T). One consequence of this (Theorem 7) is that the set of radial limits of holomorphic functions of bounded characteristic in U is norm-dense in L°°(T). It is also shown (Theorem 3, 4) that the Gelfand transforms of the inner functions separate points on the Silov boundary of H°°, and this is used to obtain a new proof (and generalization) of a theorem of D. J. Newman (Theorem 4).