It's true that the operators are adjoints of each other, but I don't see how to use that to find y, given x.

To me this looks a bit similar to the polar decomposition of an element of a von Neumann algebra, as done in Gert Pedersen's bookC*-algebras and their automorphism groups. In fact, if then . Of course, that construction only works if is invertible. But you can avoid that snag as follows. First note that we may assume that A is unital (if not, then adjoin an identity e to it and extend the action of A on X to an action of the unitised algebra on X). Now define a sequence of elements and prove, (as in Pedersen's Proposition 2.2.9) that is Cauchy.

I don't see how to prove the uniqueness part of the problem. If then , from which you get and hence . But that's as far as I can get.