2026/07/20 by Zhujun Yang, You Zhou, Liguang Wang
#math.OA
Let H be an infinite dimensional separable Hilbert space and N a nest of projections on H with at least four projections. Let ξ be a separating vector of N'' and Pξ the orthogonal projection from H onto the one-dimensional subspace of H generated by ξ. Let L be the lattice generated by N and Pξ, and \rmAlgL the corresponding Kadison-Singer algebra. In this note, we show that every Lie automorphism ψ on \rmAlgL can be decomposed as ψ=ε+τ when I-N\vee Pξ<I, where ε is an automorphism and τ is a linear functional τ on \rmAlgL vanishing on each commutator. For the complementary case, where I-N<I with I-N\vee Pξ=I, we also give a construction of the Lie automorphism.