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 ${\mathcal{N}}^{{}^{\mathrm{\prime }\mathrm{\prime }}}$ 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 $\mathrm{A}\mathrm{l}\mathrm{g}\mathcal{L}$ the corresponding Kadison-Singer algebra. In this note, we show that every Lie automorphism $\psi$ on $\mathrm{A}\mathrm{l}\mathrm{g}\mathcal{L}$ 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 $\mathrm{A}\mathrm{l}\mathrm{g}\mathcal{L}$ 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.