Homogeneous spaces of operators

Abstract

Let A be a unital C^∗-algebra. In this chapter (and in the next one) we study the Poincaré half-space H of A H:={h ∈ A : Im(h) is positive and invertible}. In the case of theC^*-algebra B(L) of bounded linear operators on the Hilbert space L, the space H is isomorphic to the tangent bundle of the set of inner products on L. Also consider the following situation. Let E be a complex vector bundle over a compact space M provided with a Hermitian structure. Consider the C^*-algebra Γ(End(E)) of continuous sections of the endomorphism bundle of E. The Poincaré space of Γ(End(E)) consists of the elements of the form X+ ia, where a is a Hermitian structure on E and X is infinitesimal deformation of a. The space H is naturally isomorphic to the following spaces: 1. the Poincaré disk D of A: D:={z ∈ A}: ||z||<1}; 2. the space Qρ of A-module projections q acting in A × A which decompose the form θ = ρ ·, · induced by a fixed symmetry\rho ρ in A × A, in the following sense: θ is positive in R(q) and negative in N(q).