2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/64070We generalize some facts about function algebras to operator algebras, using the `noncommutative Shilov boundary' or $C^*$-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.24 pagesOperator AlgebrasFunctional Analysistext