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Using these, we identify all free products $C \\ast D$, where $C,D$ are of the form $A_1 \\oplus A_2$ or $M_2(B)$; $A_1,A_2,B$ are finite von Neumann algebras, as is $A_1 \\oplus A_2$ with the 'uniform trace' given by $tr(a_1, a_2) = 1/2 (tr(a_1) + tr(a_2))\\}$ and $M_2(B)$ with the normalized trace given by $tr((b_{i,j}))=1/2(tr(b_{1,1}) + tr(b_{2,2}))$. Those results are then used to compute various"},"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":"1111.6183","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2011-11-26T18:46:28Z","cross_cats_sorted":[],"title_canon_sha256":"f2a3c467c8ee4a3e61a37c3996d350531eac9f5a7f61d04b80a49c646b6a0173","abstract_canon_sha256":"35f681e571ed745f2faadeae57c8f13a1382371b38f4fce2da1a73c75e052329"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:07:31.626912Z","signature_b64":"3M1XQuBaTx8BgAxVYHsPEoYLJZJ9ptrxwIRJ/MhVEIfP8iM//mi2DbrZdxbgBPtY95FzoR7wuWx/qnFX8jHXBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7593fb1e842aa95b990eca7fb8e3074718d39c17cc9aaf57764a89b3eb1dca3d","last_reissued_at":"2026-05-18T04:07:31.626401Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:07:31.626401Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some explicit computations and models of free products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Madhushree Basu","submitted_at":"2011-11-26T18:46:28Z","abstract_excerpt":"In this note, we first work out some `bare hands' computations of the most elementary possible free products involving $\\mathbb{C}^2 ~(=\\mathbb{C} \\oplus \\mathbb{C} $) and $M_2 ~(= M_2(\\mathbb{C}))$. Using these, we identify all free products $C \\ast D$, where $C,D$ are of the form $A_1 \\oplus A_2$ or $M_2(B)$; $A_1,A_2,B$ are finite von Neumann algebras, as is $A_1 \\oplus A_2$ with the 'uniform trace' given by $tr(a_1, a_2) = 1/2 (tr(a_1) + tr(a_2))\\}$ and $M_2(B)$ with the normalized trace given by $tr((b_{i,j}))=1/2(tr(b_{1,1}) + tr(b_{2,2}))$. 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