  Inner functions have played a prominent role in the function theory of
several domains. In this note, we first show that the closed algebra generated
by them in the symmetrized bidisc is a proper subalgebra of the algebra of all
bounded holomorphic functions. We follow it up by showing, nevertheless,
certain approximation results. Revolving around the theme of a celebrated
classical result of Caratheodory, which states that any holomorphic function
defined on the open unit disc and mapping into it, can be approximated
uniformly on compact subsets by finite Blaschke products, we prove a host of
results on the symmetrized bidisc (many of them extending to the symmetrized
polydisc) culminating in a Caratheodory type theorem for matrix-valued
holomorphic functions.
