Essentially commuting projections

Abstract

Let H = H+ ⊕ H− be a fixed orthogonal decomposition of a Hilbert space, with both subspaces of infinite dimension, and let E+, E− be the projections onto H+ and H−. We study the set Pcc of orthogonal projections P in H which essentially commute with E+, (or equivalently with E−), i.e. [P, E+] = P E+ − E+P is compact. By means of the projection π onto the Calkin algebra, one sees that these projections P ∈ Pcc fall into nine classes. Four discrete classes, which correspond to π(P) being 0, 1, π(E+) or π(E−), and five essential classes which we describe below. The discrete classes are, respectively, the finite rank projections, finite co-rank projections, the Sato Grassmannian of H+ and the Sato Grassmannian of H−. Thus the connected components of each of these classes are parametrized by the integers (via de rank, the co-rank or the Fredholm index, respectively). The essential classes are shown to be connected. We are interested in the geometric structure of Pcc, being the set of selfadjoint projections of the C∗ -algebra Bcc of operators in B(H) which essentially commute with E+. In particular, we study the problem of existence of minimal geodesics joining two given projections in the same component. We show that the Hopf-Rinow Theorem holds in the discrete classes, but not in the essential classes. Conditions for the existence and uniqueness of geodesics in these latter classes are found.