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The Riemann sphere of a C*-algebra

Esteban Andruchow, Gustavo Corach, Lázaro Recht, Alejandro Varela

TL;DR

This work introduces the Riemann sphere $\mathcal{R}$ of a unital C*-algebra $\mathcal{A}$ as the $\mathcal{U}_2$-orbit of the basic projection $\tilde{p}_0$ in $M_2(\mathcal{A})$, and develops its differential geometry as a homogeneous reductive space with an invariant Finsler metric. It constructs a full smooth structure via a Hopf fibration $\mathfrak{h}:\mathcal{K}\to\mathcal{R}$, describes explicit geodesics and the exponential map, and provides a geometric interpretation of the logarithm in terms of cross-ratio-like invariants. The paper also connects the Riemann sphere geometry to the theory of (bounded and unbounded) operator graphs on a Hilbert space, giving concrete formulas for minimal geodesics between graphs and for deformations of unbounded operators. Applications include canonical deformations to graphs of differential operators and explicit analysis in both finite and infinite-dimensional settings, along with density phenomena for geodesic neighborhoods. Overall, the work offers a robust noncommutative geometric framework for operator graphs and their deformations, with potential impacts on spectral theory and noncommutative geometry.

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.

The Riemann sphere of a C*-algebra

TL;DR

This work introduces the Riemann sphere of a unital C*-algebra as the -orbit of the basic projection in , and develops its differential geometry as a homogeneous reductive space with an invariant Finsler metric. It constructs a full smooth structure via a Hopf fibration , describes explicit geodesics and the exponential map, and provides a geometric interpretation of the logarithm in terms of cross-ratio-like invariants. The paper also connects the Riemann sphere geometry to the theory of (bounded and unbounded) operator graphs on a Hilbert space, giving concrete formulas for minimal geodesics between graphs and for deformations of unbounded operators. Applications include canonical deformations to graphs of differential operators and explicit analysis in both finite and infinite-dimensional settings, along with density phenomena for geodesic neighborhoods. Overall, the work offers a robust noncommutative geometric framework for operator graphs and their deformations, with potential impacts on spectral theory and noncommutative geometry.

Abstract

Given the unital C-algebra , the unitary orbit of the projector in the C-algebra of matrices with coefficients in is called in this paper, the Riemann sphere of . We show that is a homogeneous reductive C manifold of the unitary group 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 is represented in a Hilbert space , in terms of local charts of , elements of the Riemann sphere may be identified with (graphs of) closed operators on (bounded or unbounded). In the first part of the paper, we develop several geometric aspects of including a relation between the exponential map of the reductive connection and the cross-ratio of subspaces of . In the last section we show some applications of the geometry of , 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.
Paper Structure (24 sections, 23 theorems, 138 equations, 2 figures)

This paper contains 24 sections, 23 theorems, 138 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The smooth structure of ${\mathcal{R}}$
  4. The C$^\infty$ structure on ${\mathcal{K}}$
  5. The C$^\infty$ structure of ${\mathcal{R}}$
  6. The C$^\infty$ atlas $\mathfrak{A}_ {\mathcal{R}}$ of ${\mathcal{R}}$
  7. The tangent map of the principal chart
  8. Riemann sphere projectors in C$^*$-algebras
  9. The Riemann sphere ${\mathcal{R}}$ as a homogeneous reductive space
  10. The inverse of the exponential map in ${\mathcal{R}}$
  11. Geometric interpretation of the logarithm
  12. The Hopf fibration
  13. Relation between geodesic and homogeneous coordinates in ${\mathcal{R}}$
  14. The canonical connection on the Hopf fibration
  15. The structure morphism $\kappa: \mathfrak{R}\to T {\mathcal{K}}$
  16. ...and 9 more sections

Key Result

Proposition 2.6

The Hopf fibration is equivariant under the action of ${\mathcal{U}} _2$. More explicitly,

Figures (2)

  • Figure 1: Cross ratio and complementary cross ratio.
  • Figure 2: Unoriented angle.

Theorems & Definitions (78)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.8
  • proof
  • Proposition 2.9
  • ...and 68 more