Proposition: A standard modular pair with many invariant vectors

Cyclicity and separation of $\Omega$ for a von Neumann algebra do not imply that the modular unitary group has a unique invariant vector.

Proof. Let $\mathcal{M}=M_2(\mathbb{C})$ act by left multiplication on the Hilbert–Schmidt space $H=M_2(\mathbb{C})$, and choose

where $\rho$ is a strictly positive density matrix. Invertibility makes $\Omega$ cyclic and separating. The modular operator acts by

Every matrix commuting with $\rho$ is fixed by ${\Delta }^{it}$. For diagonal $\rho$, the diagonal subspace already has dimension two.