On Hilbert $C^*$-modules with Hilbert dual and on extending functionals

A submodule in a Hilbert $C^*$-module is said to be thick if its orthogonal complement is zero. Recently, Kaad and Skeide found examples of non-trivial functionals whose restriction to a thick submodule is zero. These examples work for $C^*$-algebras that are far from $W^*$-algebras (for example, for $C[0,1]$). For a long time it was assumed that such examples are impossible for $W^*$-algebras. We still do not know this, but we have proved it for commutative $W^*$-algebras and for $B(H)$. For Hilbert modules with a Hilbert dual (and the mentioned cases are included here), we propose a tool to test this conjecture. It can be used to prove that for modules over the algebra of Borel functions on $[0,1]$ (over this algebra, all Hilbert modules have a Hilbert dual) functionals of Kaad-Skeide type exist.