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 ∥f∥∞ of an f in
L∞(T) is the essential supremum of |f| 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 f in L∞(T) is unimodular if |f| = 1 a.e., on T. 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 Šilov boundary of
H∞, and this is used to obtain a new proof (and generalization) of a theorem of D. J.
Newman (Theorem 4).
