Proposition: Scalar centralizer is not a finite-code assumption

A finite-dimensional von Neumann algebra with a faithful state and scalar modular centralizer is $\mathbb{C}1$.

Proof. The center is fixed by every modular automorphism. A scalar centralizer therefore forces the finite algebra to be a matrix factor $M_n$. Write the faithful state with positive density matrix $\rho$. Its modular group is ${\mathrm{Ad}}_{{\rho }^{it}}$. If $n\ge 2$, the commutant of $\rho$ contains nontrivial spectral projections, or all of $M_n$ if $\rho$ is scalar. The centralizer cannot be scalar. Thus $n=1$.