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 $${\mathcal {A}}^p=\{A\in {\mathcal {B}}({\mathcal {H}}): [A,E_+]\in {\mathcal {B}}_p({\mathcal {H}})\},$$ 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.