关于一类卡迪逊 - 辛格代数的一些李自同构
On some Lie automorphisms of a class of Kadison-Singer algebras
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中文总结 AI 辅助
研究一类卡迪逊 - 辛格代数的李自同构，当\(I_{-}^{\mathcal{N}}\vee P_{\xi}<I\)时，李自同构\(\psi\)可分解为自同构\(\epsilon\)与特定线性泛函\(\tau\)之和，对互补情况也给出了李自同构构造。
中文摘要 AI 辅助
设\(\mathcal{H}\)为无限维可分希尔伯特空间，\(\mathcal{N}\)为\(\mathcal{H}\)上具有至少四个投影的投影巢。设\(\xi\)为\(\mathcal{N}^{''}\)的分离向量，\(P_{\xi}\)为由\(\xi\)生成的\(\mathcal{H}\)的一维子空间上的正交投影。设\(\mathcal{L}\)为由\(\mathcal{N}\)和\(P_{\xi}\)生成的格，\({\rm{Alg}}\mathcal{L}\)为相应的卡迪逊 - 辛格代数。本文表明，当\(I_{-}^{\mathcal{N}}\vee P_{\xi}<I\)时，\({\rm{Alg}}\mathcal{L}\)上的每个李自同构\(\psi\)可分解为\(\psi=\epsilon+\tau\)，其中\(\epsilon\)为自同构，\(\tau\)为\({\rm{Alg}}\mathcal{L}\)上在每个换位子上消失的线性泛函。对于\(I_{-}^{\mathcal{N}}<I\)且\(I_{-}^{\mathcal{N}}\vee P_{\xi}=I\)的互补情况，我们也给出了李自同构的一种构造。
英文摘要
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 $ξ$ be a separating vector of $\mathcal{N}^{''}$ and $P_ξ$ the orthogonal projection from $\mathcal{H}$ onto the one-dimensional subspace of $\mathcal{H}$ generated by $ξ$. Let $\mathcal{L}$ be the lattice generated by $\mathcal{N}$ and $P_ξ$, and ${\rm{Alg}}\mathcal{L}$ the corresponding Kadison-Singer algebra. In this note, we show that every Lie automorphism $ψ$ on ${\rm{Alg}}\mathcal{L}$ can be decomposed as $ψ=ε+τ$ when $I_{-}^{\mathcal{N}}\vee P_ξ<I$, where $ε$ is an automorphism and $τ$ is a linear functional $τ$ on ${\rm{Alg}}\mathcal{L}$ vanishing on each commutator. For the complementary case, where $I_{-}^{\mathcal{N}}<I$ with $I_{-}^{\mathcal{N}}\vee P_ξ=I$, we also give a construction of the Lie automorphism.
