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Every $E_0$-semigroup that possesses a strongly continuous intertwining semigroup of isometries is cocycle conjugate to an $E_0$-semigroup induced by the Bhat induction of a $CP$-flow over a separable Hilbert space $K$. We say an $E_0$-semigroup $\\alpha$ is $q$-pure if the $CP$-subordinates $\\beta$ of norm one (i.e. $\\Vert\\beta_t(I)\\Vert = 1$ and $\\alpha_t-\\beta_t$ is completely positive for all $t \\geq 0$) are totally ordered in the sense that if $\\beta$ and"},"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":"1807.09824","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2018-07-25T18:49:45Z","cross_cats_sorted":[],"title_canon_sha256":"4da2ed18439874eebaa4c791cd443788a109a35094e3f92cfe17f21617538bea","abstract_canon_sha256":"f6065b836c801225f3aa972665f2556afbb5c204686bc7bd096864a5346a54c9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:46.514347Z","signature_b64":"/kg44vKQGtJ4k5w4XmTLuARHgX17ll2bfx864Tpfb/H8oFUbZV3Q/YecFG/VVoKPz895jtF+IXG/F4qpJqIXBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"478d6c5c6afa09df94e84fd63dc786dd7027a8406f530416b316919c317f819a","last_reissued_at":"2026-05-18T00:09:46.513703Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:46.513703Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Classification of q-pure q-weight maps over finite dimensional Hilbert spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Christopher Jankowski, Daniel Markiewicz, Robert T. Powers","submitted_at":"2018-07-25T18:49:45Z","abstract_excerpt":"An $E_0$-semigroup of $B(H)$ is a one parameter strongly continuous semigroup of $*$-endomorphisms of $B(H)$ that preserve the identity. Every $E_0$-semigroup that possesses a strongly continuous intertwining semigroup of isometries is cocycle conjugate to an $E_0$-semigroup induced by the Bhat induction of a $CP$-flow over a separable Hilbert space $K$. We say an $E_0$-semigroup $\\alpha$ is $q$-pure if the $CP$-subordinates $\\beta$ of norm one (i.e. $\\Vert\\beta_t(I)\\Vert = 1$ and $\\alpha_t-\\beta_t$ is completely positive for all $t \\geq 0$) are totally ordered in the sense that if $\\beta$ and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.09824","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1807.09824","created_at":"2026-05-18T00:09:46.513799+00:00"},{"alias_kind":"arxiv_version","alias_value":"1807.09824v1","created_at":"2026-05-18T00:09:46.513799+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.09824","created_at":"2026-05-18T00:09:46.513799+00:00"},{"alias_kind":"pith_short_12","alias_value":"I6GWYXDK7IE5","created_at":"2026-05-18T12:32:28.185984+00:00"},{"alias_kind":"pith_short_16","alias_value":"I6GWYXDK7IE57FHI","created_at":"2026-05-18T12:32:28.185984+00:00"},{"alias_kind":"pith_short_8","alias_value":"I6GWYXDK","created_at":"2026-05-18T12:32:28.185984+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/I6GWYXDK7IE57FHIJ7LD3R4G3V","json":"https://pith.science/pith/I6GWYXDK7IE57FHIJ7LD3R4G3V.json","graph_json":"https://pith.science/api/pith-number/I6GWYXDK7IE57FHIJ7LD3R4G3V/graph.json","events_json":"https://pith.science/api/pith-number/I6GWYXDK7IE57FHIJ7LD3R4G3V/events.json","paper":"https://pith.science/paper/I6GWYXDK"},"agent_actions":{"view_html":"https://pith.science/pith/I6GWYXDK7IE57FHIJ7LD3R4G3V","download_json":"https://pith.science/pith/I6GWYXDK7IE57FHIJ7LD3R4G3V.json","view_paper":"https://pith.science/paper/I6GWYXDK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1807.09824&json=true","fetch_graph":"https://pith.science/api/pith-number/I6GWYXDK7IE57FHIJ7LD3R4G3V/graph.json","fetch_events":"https://pith.science/api/pith-number/I6GWYXDK7IE57FHIJ7LD3R4G3V/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/I6GWYXDK7IE57FHIJ7LD3R4G3V/action/timestamp_anchor","attest_storage":"https://pith.science/pith/I6GWYXDK7IE57FHIJ7LD3R4G3V/action/storage_attestation","attest_author":"https://pith.science/pith/I6GWYXDK7IE57FHIJ7LD3R4G3V/action/author_attestation","sign_citation":"https://pith.science/pith/I6GWYXDK7IE57FHIJ7LD3R4G3V/action/citation_signature","submit_replication":"https://pith.science/pith/I6GWYXDK7IE57FHIJ7LD3R4G3V/action/replication_record"}},"created_at":"2026-05-18T00:09:46.513799+00:00","updated_at":"2026-05-18T00:09:46.513799+00:00"}