The Riemann sphere of a C*-algebra

TL;DR

This paper introduces the Riemann sphere of a unital C*-algebra as a geometric object with a rich differential structure, exploring its properties, geodesics, and applications to operator theory.

Contribution

It defines the Riemann sphere of a C*-algebra as a homogeneous reductive manifold and studies its geometric properties and applications to operator geometry.

Findings

Riemann sphere is a homogeneous reductive C∞ manifold.

Geodesics and exponential map are characterized.

Applications include bounded deformations of unbounded operators.

Abstract

Given the unital C$^*$-algebra $A$, the unitary orbit of the projector $p_0=\begin{pmatrix}1 & 0 \\ 0 & 0 \end{pmatrix}$ in the C$^*$-algebra $M_2(A)$ of $2\times 2$ matrices with coefficients in $A$ is called in this paper, the Riemann sphere $R$ of $A$. We show that $R$ is a homogeneous reductive C$^\infty$ manifold of the unitary group $U_2(A)\subset M_2(A)$ 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 $H\times H$. 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 an unbounded operator and give some relevant examples.