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It was proved by Cuntz in \\cite{Cu81} that if $A$ is a \\textit{purely infinite} simple \\cst-algebra, then the kernel of the natural map for the unitary group $\\U(A)$ to the $K$-theory group $K_1(A)$ is reduced to the connected component $\\U^0(A)$, i.e. $A$ is \\textit{$K_1$-injective} (see \\S 3). We study in this note a finitely generated \\cst-algebra, the $K_1$-injectivity of which would imply the $K_1$-injectivity of all unital \\textit{properly infinite} \\cst-algebras."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0804.4624","kind":"arxiv","version":13},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2008-04-29T14:33:46Z","cross_cats_sorted":[],"title_canon_sha256":"6ffea2c3f8a0403e3af54983ae969a181ca82a37e8f6c2607fc6f5d236db4b9e","abstract_canon_sha256":"4b4d3c7d9141683ebc6b76305e75914d803bbd8a8be32379b3ef7c8ae067eeb8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:28.991068Z","signature_b64":"mtyLd/foViSWJ6ghIYljsPULiN1jL21csvtXJuNCYHk6six57lfZ28xMp6LcOS+iFQNNUWssJ3wnFo65XvByCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"74beca3eac1d514678c6ffbb238d18a76663bd9687b45d4b64c2eaa136849bb2","last_reissued_at":"2026-05-18T04:39:28.990405Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:28.990405Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"K_1-injectivity for properly infinite C*-algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Etienne Blanchard","submitted_at":"2008-04-29T14:33:46Z","abstract_excerpt":"One of the main tools to classify \\cst-algebras is the study of its projections and its unitaries. 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