The Riemann sphere of a $C^*$-algebra
Abstract
Given the unital $C^*$-algebra $ {\mathcal A}$, the unitary orbit of the projector $\tilde{p}=\bigl(\begin{smallmatrix} 1&0\\0&0\end{smallmatrix}\bigr)$ in the $C^*$-algebra $M_2( {\mathcal A})$ of $2\times 2$ matrices with coefficients in $ {\mathcal A}$ is called in this paper the Riemann sphere $ {\mathcal R}$ of $ {\mathcal A}$.
We show that $ {\mathcal R}$ is a reductive homogeneous C$^\infty $ manifold of the unitary group $ \mathcal {U}_2( {\mathcal A})\subset M_2( {\mathcal 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 $ {\mathcal A}$ is represented in a Hilbert space $H$, in terms of local charts of $ {\mathcal 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 $ {\mathcal R}$ including a relation between the exponential map of the reductive connection and the cross-ratio of subspaces of $H\times H$.
In the last section we show some applications of the geometry of $ {\mathcal 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.
