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Moreover, when $\\Gamma_1$ and $\\Gamma_2$ are discrete groups containing copies of noncommutative free groups, then $C^*_r(\\Gamma_1)\\otimes C^*_r(\\Gamma_2)$ and $C^*(\\Gamma_1)\\otimes C_r^*(\\Gamma_2)$ admit $2^{\\aleph_0}$ $C^*$-norms. Analogues of these results continue to hold when these familiar group $C^*$-algebras are replaced by appropriate intermediate group $C^*$-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":"1406.2654","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2014-06-10T18:14:20Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"2920893256da39e72be82dba126c677296c4184c868797fcc36763f414552f0d","abstract_canon_sha256":"7b5de6f42cedec80c3acd8efd17c8a0d8b25299669d9679d4129808ebbddf95d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:53:09.321882Z","signature_b64":"3v5OI/e3TtagQzdbI0Q5eLLoEufMr/bBAGcxFC34UIN9ukPPTZ9Jz985u6CnzSQJKzNRXS3x2PDsmmKK6R+9AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"436086bec493428ea3eda84e30f1bf9a388ba096641ecd98d76282bdffe787c2","last_reissued_at":"2026-05-18T01:53:09.321208Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:53:09.321208Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"C*-norms for tensor products of discrete group C*-algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.OA","authors_text":"Matthew Wiersma","submitted_at":"2014-06-10T18:14:20Z","abstract_excerpt":"Let $\\Gamma$ be a discrete group. We show that if $\\Gamma$ is nonamenable, then the algebraic tensor products $C^*_r(\\Gamma)\\otimes C^*_r(\\Gamma)$ and $C^*(\\Gamma)\\otimes C^*_r(\\Gamma)$ do not admit unique $C^*$-norms. Moreover, when $\\Gamma_1$ and $\\Gamma_2$ are discrete groups containing copies of noncommutative free groups, then $C^*_r(\\Gamma_1)\\otimes C^*_r(\\Gamma_2)$ and $C^*(\\Gamma_1)\\otimes C_r^*(\\Gamma_2)$ admit $2^{\\aleph_0}$ $C^*$-norms. 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