Characterizations of essential ideals as operator modules over C*-algebras

In this paper we give characterizations of essential left ideals of a C*-algebra $A$ in terms of their properties as operator $A$-modules. Conversely, we seek C*-algebraic characterizations of those ideals $J$ in $A$ such that $A$ is an essential extension of $J$ in various categories of operator modules. In the case of two-sided ideals, we prove that all the above concepts coincide. We obtain results, analogous to {M. Hamana's} results, which characterize the injective envelope of a C*-algebra as a maximal essential extension of the C*-algebra, but with completely positive maps replaced by completely bounded module maps. By restricting to one-sided ideals, module actions reveal clear differences which do not show up in the two-sided case. Throughout this paper, module actions are crucial.