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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","authors_text":"Madhushree Basu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2011-11-26T18:46:28Z","title":"Some explicit computations and models of free products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.6183","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:8c04ac359d43a01e1dd7107638c9428382dfb7749838f33dc9c5e7d6c1882ba4","target":"record","created_at":"2026-05-18T04:07:31Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"35f681e571ed745f2faadeae57c8f13a1382371b38f4fce2da1a73c75e052329","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2011-11-26T18:46:28Z","title_canon_sha256":"f2a3c467c8ee4a3e61a37c3996d350531eac9f5a7f61d04b80a49c646b6a0173"},"schema_version":"1.0","source":{"id":"1111.6183","kind":"arxiv","version":1}},"canonical_sha256":"7593fb1e842aa95b990eca7fb8e3074718d39c17cc9aaf57764a89b3eb1dca3d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7593fb1e842aa95b990eca7fb8e3074718d39c17cc9aaf57764a89b3eb1dca3d","first_computed_at":"2026-05-18T04:07:31.626401Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:07:31.626401Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3M1XQuBaTx8BgAxVYHsPEoYLJZJ9ptrxwIRJ/MhVEIfP8iM//mi2DbrZdxbgBPtY95FzoR7wuWx/qnFX8jHXBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:07:31.626912Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.6183","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8c04ac359d43a01e1dd7107638c9428382dfb7749838f33dc9c5e7d6c1882ba4","sha256:cbed127d59c4991cbb4a0c0c8fd6925f1c8d9810e1be7b7d8d276315e2b449f5"],"state_sha256":"77cc984d4af8ab636b420f2649e281ffb262b6eae60dd545f52ecd27381ce9d8"}