Let \({\mathcal {H}}={\mathcal {H}}_+\oplus {\mathcal {H}}_-\) be a fixed orthogonal decomposition of the complex separable Hilbert space \({\mathcal {H}}\) in two infinite-dimensional subspaces. We study the geometry of the set \({\mathcal {P}}^p\) of selfadjoint projections in the Banach algebra

where \(E_+\) is the projection onto \({\mathcal {H}}_+\) and \({\mathcal {B}}_p({\mathcal {H}})\) is the Schatten ideal of p-summable operators (\(1\le p <\infty\)). The norm in \({\mathcal {A}}^p\) is defined in terms of the norms of the matrix entries of the operators given by the above decomposition. The space \({\mathcal {P}}^p\) is shown to be a differentiable \(C^\infty\) submanifold of \({\mathcal {A}}^p\), and a homogeneous space of the group of unitary operators in \({\mathcal {A}}^p\). The connected components of \({\mathcal {P}}^p\) are characterized, by means of a partition of \({\mathcal {P}}^p\) in nine classes, four discrete classes, and five essential classes: (1) the first two corresponding to finite rank or co-rank, with the connected components parametrized by these ranks; (2) the next two discrete classes carrying a Fredholm index, which parametrizes their components; (3) the remaining essential classes, which are connected.

令\（{\ mathcal {H}} = {\ mathcal {H}} _ + \ oplus {\ mathcal {H}} _- \\）是复数可分离Hilbert空间\（{\ mathcal { H}} \）在两个无限维子空间中。我们研究Banach代数中自伴投影的集合\（{\ mathcal {P}} ^ p \）的几何

其中\（E _ + \）是\（{\ mathcal {H}} _ + \）的投影，而\（{\ mathcal {B}} _ p（{\ mathcal {H}}）\）是Schatten理想的p -summable运营商（\（1 \文件p <\ infty \） ）。在常态\（{\ mathcal {A}} ^ P \）中的由上述分解给定运营商的矩阵条目的规范来定义。空间\ {{\ mathcal {P}} ^ p \）被显示为\ {{\ mathcal {A}} ^ p \）的可微分\（C ^ \ infty \）子流形和齐次空间\（{\ mathcal {A}} ^ p \）中的of运算符组。\（{\ mathcal {P}} ^ p \）的连接组件通过将\（{\ mathcal {P}} ^ p \）划分为9个类，4个离散类和5个基本类来进行表征：（1）前两个对应于有限秩或同秩，这些等级已将连接的组件参数化；（2）带有弗雷德霍尔姆索引的接下来的两个离散类，将它们的组成参数化；（3）其余必不可少的类别，它们相互关联。