Restricting operators to thick Hilbert C*-submodules

Abstract: Given an essential ideal $J\subset A$ of a C*-algebra $A$, and a Hilbert C*-module $M$ over $A$, we place $M$ between two other Hilbert C*-modules over $A$, $M_J\subset M\subset M^J$, in such a way that each submodule here is thick, i.e. its orthogonal conmplement in the greater module is trivial. We introduce the class $\mathbb B_J(M)$ of $J$-adjointable operators on a Hilbert C*-module $M$ over $A$, and prove that this class isometrically embeds into the C*-algebras of all adjointable operators both of $M_J$ and of $M^J$.