Inner divisors and composition operators

This characterization is stated and proved in much greater generality, beginning with subspaces of arbitrary L^∞ spaces and using a notion of inner divisors. Among the consequences are a variant of Wermer's theorem on embedding disks in maximal ideal spaces and a result on linear isometrics of H^∞. The ranges of composition operators C_I when I is not necessarily inner are characterized. In particular, relatively closed sets E ?cΔ of zero logarithmic and zero analytic capacity are characterized in terms of the algebras of bounded analytic functions on A invariant under corresponding Fuchsian groups. The paper concludes with an example of a uniformly closed subalgebra of H^∞ which contains the constants and the inner factors of its members, but is not of the form H ∞ I.