The Riemann sphere of a C∗-algebra

Given the unital C∗-algebra A, the unitary orbit of the projector p˜=(1000) in the C∗-algebra M2(A) of 2×2 matrices with coefficients in A is called in this paper the Riemann sphere R of A. We show that R is a reductive homogeneous C∞ manifold of the unitary group U2(A)⊂M2(A) and carries the differential geometry deduced from this structure (including an invariant Finsler metric). Special attention is paid to the properties of geodesics and the exponential map. If the algebra A is represented in a Hilbert space H, in terms of local charts of R, elements of the Riemann sphere may be identified with (graphs of) closed operators on H (bounded or unbounded). In the first part of the paper, we develop several geometric aspects of R including a relation between the exponential map of the reductive connection and the cross-ratio of subspaces of H×H. In the last section we show some applications of the geometry of R to the geometry of operators on a Hilbert space. In particular, we define the notion of bounded deformation of a closed operator and give some relevant examples.