论文标题
在无限尺寸JB-Elgebra的结构组上
On the structure group of an infinite dimensional JB-algebra
论文作者
论文摘要
我们将真正的Jordan代数$ V $的结构组扩展到无限的JB-Elgebras的设置。我们证明，结构组$ str（v）$，保存圆锥形的$ g（ω）$和代数$ v $的$ aut（v）$ automorism grout $ aut（v）$嵌入了$ gl（v）$的banach-lie groups $ gl（v）$，以及每个包含$ aut（v）\ aut aut（v）\ subset g（Ω）我们通过锥，同位素和中央预测对$ str（v）$的组件进行完整描述。我们将这些结果应用于$ v = b（h）_ {sa} $在无限的尺寸复杂的希尔伯特（Hilbert Space）上的自我接合运算符的特殊JB-Algebra，描述了$ str（v），g（ω），aut（v）$，他们的Banach-lie代数及其连接的组件及其连接的组件。我们表明，$ h $ $ h $在$ aut（v）$上的统一组的动作具有平稳的本地交叉部分，因此$ aut（v）$是整个统一组的平稳校长捆绑包，circle结构组。
We extend several results for the structure group of a real Jordan algebra $V$, to the setting of infinite dimensional JB-algebras. We prove that the structure group $Str(V)$, the cone preserving group $G(Ω)$ and the automorphism group $Aut(V)$ of the algebra $V$ are embedded Banach-Lie groups of $GL(V)$, and that each of the inclusions $Aut(V)\subset G(Ω)\subset Str(V)$ are of embedded Banach-Lie subgroups. We give a full description of the components of $Str(V)$ via cones, isotopes and central projections. We apply these results to $V=B(H)_{sa}$ the special JB-algebra of self-adjoint operators on an infinite dimensional complex Hilbert space, describing the groups $Str(V), G(Ω), Aut(V)$, their Banach-Lie algebras and their connected components. We show that the action of the unitary group of $H$ on $Aut(V)$ has smooth local cross sections, thus $Aut(V)$ is a smooth principal bundle over the unitary group, with circle structure group.
