On large submodules in Hilbert \(C^\ast \)-modules (Q6635193)

A submodule in a Hilbert \(C^*\)-module, or in particular, a one-sided ideal in a \(C^*\)-algebra may be large. There are at least two different ways to state that a closed right ideal \(J\) of a \(C^*\)-algebra \(A\) is large: (i) \(J\) is topologically essential if \(J \cap I \neq 0\) for every nonzero closed right ideal \(I\) of \(A\); (ii) \(J\) is thick if \(aJ=0\) implies that \(a=0\) or \(a\in A\). It is known that (i) implies (ii) but the reverse implication is not necessarily true. The author also compares some properties of a submodule in a Hilbert \(C^*\)-module \(M\) with their counterparts for a specific ideal in the \(C^*\)-algebra of compact operators on \(M\). Furthermore, he introduces similar concepts for closed submodules of a Hilbert \(C^*\)-module. Moreover, he utilizes essential right ideals to expand the inner product on a Hilbert \(C^*\)-module \(M\) to a submodule of the dual module \(M'\).