Logmodularity and isometries of operator algebras

We generalize some facts about function algebras to operator algebras, using the `noncommutative Shilov boundary’ or $C^{\ast }$-envelope first considered by Arveson. In the first part we study and characterize complete isometries between operator algebras. In the second part we introduce and study a notion of logmodularity for operator algebras. We also give a result on conditional expectations. Many miscellaneous applications are given.