On some Lie automorphisms of a class of Kadison-Singer algebras

Let $\mathcal{H}$ be an infinite dimensional separable Hilbert space and $\mathcal{N}$ a nest of projections on $\mathcal{H}$ with at least four projections.

Let $\xi$ be a separating vector of $\mathcal{N}^{''}$ and $P_{\xi}$ the orthogonal projection from $\mathcal{H}$ onto the one-dimensional subspace of $\mathcal{H}$ generated by $\xi$.

Let $\mathcal{L}$ be the lattice generated by $\mathcal{N}$ and $P_{\xi}$, and ${\rm{Alg}}\mathcal{L}$ the corresponding Kadison-Singer algebra.

In this note, we show that every Lie automorphism $\psi$ on ${\rm{Alg}}\mathcal{L}$ can be decomposed as $\psi=\epsilon+\tau$ when $I_{-}^{\mathcal{N}}\vee P_{\xi}<I$, where $\epsilon$ is an automorphism and $\tau$ is a linear functional $\tau$ on ${\rm{Alg}}\mathcal{L}$ vanishing on each commutator.

For the complementary case, where $I_{-}^{\mathcal{N}}<I$ with $I_{-}^{\mathcal{N}}\vee P_{\xi}=I$, we also give a construction of the Lie automorphism.