{"paper":{"title":"Projective geometry in the Poincar\\'e disk of a $C^*$-algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.FA"],"primary_cat":"math.OA","authors_text":"Esteban Andruchow, Gustavo Corach, L\\'azaro Recht","submitted_at":"2018-06-21T22:02:55Z","abstract_excerpt":"We study the Poincar\\'e disk ${\\cal D}=\\{a\\in {\\cal A}: \\|a\\|<1\\}$ of a C$^*$-algebra ${\\cal A}$ from a projective point of view: ${\\cal D}$ is regarded as an open subset of the projective line $\\mathbb{P}_1{\\cal A}$, the space of complemented rank one submodules of ${\\cal A}^2$. We introduce the concept of cross ratio of four points in $\\mathbb{P}_1{\\cal A}$. Our main result establishes the relation between the exponential map $Exp_{z_0}(z_1)$ of ${\\cal D}$ ($z_0,z_1\\in {\\cal D}$) and the cross ratio of the four-tuple $$ \\delta(-\\infty), \\delta(0)=z_0, \\delta(1)=z_1 , \\delta(+\\infty), $$ wher"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.08786","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}