Essentially commuting projections

ANDRUCHOW ESTEBAN; CHIUMIENTO EDUARDO; DI IORIO Y LUCERO, EUGENIA

JOURNAL OF FUNCTIONAL ANALYSIS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 2015 vol. 268 p. 336 - 362

0022-1236

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. PE_+ - E_+P is compact. By means of the projection C onto the Calkin algebra, one sees that these projections P fall into nine classes. Four discrete classes, which correspond to C(P) being 0,1, C(E_+) or C(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 P_cc, being the set of selfadjoint projections of the C*-algebra B_cc 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.