In the book "Symmetry breaking" by Strocchi, in the chapter were he introduces the Fock representations for C*-algebra of bounded operators over a Hilbert space (Part II, chapter 2, page 76), he proves that two irreducible Fock representations $\pi_1$ and $\pi_2$ of the same algebra $\mathcal{A}$ are always unitarily equivalent. However, there is one step in his proof that I don't fully understand.
He says that, given $\psi_1$ and $\psi_2$ Fock vectors (also known as vacuum states) for, respectively, the irreducible representations $\pi_1$ and $\pi_2$, it is possible to define an operator $U$ such that
- $U \psi_1 = \psi_2$;
- $U \pi_1 (A) \psi_1 = \pi_2 (A) \psi_2, \, \forall A \in \mathcal{A}$.
He says also that since the representations are irreducible, then $U$ and its inverse are defined on dense sets.
Then the proof continues, but I don't understand why it is possible to say that the operator $U$ is well-defined, and why it is invertible.
Any idea?