The decoherence-free subalgebra of Gaussian Quantum Markov Semigroups

Gaussian Quantum Markov semigroups (QMSs) have been used in the literature also under the name of quasi-free semigroups. They are semigroups on the set of bounded operators on the symmetric Fock space $\mathcal{H} = \Gamma_s(\mathbb{C}^d)$. Notable operators on this set are annihilation and creation operators $a_j, a_j^\dagger$ for $j=1, \dots, d$, which are not bounded but are used in many applications and in the very definition of gaussian QMSs. Indeed we introduce gaussian QMSs by their generator in the GKLS form
$$ \mathcal{L}(x) = i [ H, x] -\frac{1}{2} \sum_{\ell \geq 1} \left( L_\ell^* L_\ell x - 2 L_\ell^* x L_\ell + x L_\ell^* L_\ell \right), $$
with $H$ a quadratic polynomial in $a_j,a_j^\dagger$ and $L_\ell$ a linear polynomial in $a_j, a_j^\dagger$. We show that the decoherence-free subalgebra, i.e. the biggest von Neumann subalgebra of $\mathcal{B}(\mathcal{H})$ on which the semigroup acts as a *-homomorphism, of this class of semigroups is always unitarily equivalent to
$$ L^\infty(\mathbb{R}^{d_c}; \mathbb{C}) \overline{\otimes} \mathcal{B}(\mathbb{C}^{d_f}), $$
for some $d_c, d_f \geq 0$ with $d_c+d_f \geq d$.