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Let  B  be a closed subalgebra of      L ∞  {L^\infty }       of the unit circle which contains      H ∞  {H^\infty }       properly. Let      C B {C_B}       be the      C ∗  {C^\ast }      -algebra generated by the inner functions that are invertible in  B . It is shown that the linear span      H ∞  + C B {H^\infty } + {C_B}       is equal to  B . Also, a closed subspace (called      V M O B VM{O_B}      ) of  BMO  (space of functions of bounded mean oscillation) is identified to which  B  bears the same relation as      L ∞  {L^\infty }       does to  BMO .

Structure of subalgebras between 𝐿^{∞} and 𝐻^{∞}