The Riemann sphere of a C*-algebra
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913829506514944 |
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| author | Andruchow, Esteban Corach, Gustavo Recht, Lázaro Varela, Alejandro |
| author_facet | Andruchow, Esteban Corach, Gustavo Recht, Lázaro Varela, Alejandro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06434 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Riemann sphere of a C*-algebra Andruchow, Esteban Corach, Gustavo Recht, Lázaro Varela, Alejandro Operator Algebras Differential Geometry Functional Analysis 58B20, 46L05, 46L08, 47A05, 14M15 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. |
| title | The Riemann sphere of a C*-algebra |
| topic | Operator Algebras Differential Geometry Functional Analysis 58B20, 46L05, 46L08, 47A05, 14M15 |
| url | https://arxiv.org/abs/2505.06434 |