I'm having trouble understanding the final step in the proof of the Stone-von Neumann theorem.
So we have a Hilbert space $H$, $k \in \mathbb{N}$ and weakly continuous unitary representations $U(\cdot)$ and $V(\cdot)$ of the additive group $\mathbb{R}^k$ on $H$ satisfying the Weyl's canonical commutation relations for a unitary $W(\cdot, \cdot)$.
I was able to show that there exists a Hilbert space $K$, unique in cardinality and a unitary isomorphism $\Phi$ between $H$ and $L^2(Leb^k) \otimes K$ such that $\Phi W(t,s) \Phi^{-1} = W^k(t,s) \otimes I_K,$ where $W^k$ is from a $k$-dimensional Schrödinger representation of the canonical commutation relations.
I can follow the proof until here, but I can't show that if $W$ acts irreducibly on $H$, then $K$ is one-dimensional.
I was able to show that if $W$ is irreducible then $vNa(W) = \mathcal{B}(H)$ and I also know that $vNa(W^k) = \mathcal{B}(L^2(Leb^k))$. Hence, I have a unitary equivalence between von Neumann algebras $\mathcal{B}(H)$ and $\mathcal{B}(L^2(Leb^k)) \otimes \mathbb{C}I_K$ given by $\Phi$, but I just don't see how it follows rigorously that $dim K =1$, even though intuitively it seems reasonable.
I know my question is probably trivial, but I would really appreciate the help.
Because $W$ acts irreducibly on $H$ we get (via the unitary equivalence) that $W^k\otimes I_K$ acts irreducibly on $L^2(Leb^k)\otimes K$. If $\dim K\geq2$, we can get a non-trivial projection $p$ acting on $K$ (for instance, fix any nonzero vector $x$ an let $p$ be the orthogonal projection onto $\mathbb C x$). Then $I\otimes p$ commutes with $W^k\otimes I_K$, contradicting its irreducibility.