Restricting/Extending Operators to/from Thick Hilbert $$C^*$$-Submodules

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 every submodule here is thick, i.e., its orthogonal complement 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 in the $$C^*$$ -algebras of all adjointable operators both of $$M_J$$ and of $$M^J$$ . DOI 10.1134/S1061920824601782