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