p-Schatten commutators of projections

Abstract

Let H= H+⊕ H- be a fixed orthogonal decomposition of the complex separable Hilbert space H in two infinite-dimensional subspaces. We study the geometry of the set Pp of selfadjoint projections in the Banach algebra Ap= { A∈ B(H) : [A, E+] ∈ Bp(H) } , where E+ is the projection onto H+ and Bp(H) is the Schatten ideal of p-summable operators (1 ≤ p< ∞). The norm in Ap is defined in terms of the norms of the matrix entries of the operators given by the above decomposition. The space Pp is shown to be a differentiable C∞ submanifold of Ap, and a homogeneous space of the group of unitary operators in Ap. The connected components of Pp are characterized, by means of a partition of Pp 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.