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Then $X = C({\\mathcal G})$ is a Hilbert bimodule over $A = C(K)$. We associate a $C^*$-algebra ${\\mathcal O}_{\\gamma}(K)$ with them as a Cuntz-Pimsner algebra ${\\mathcal O}_X$. We show that if a system of proper contractions satisfies the open set condition in $K$, then the $C^*$-algebra ${\\math"},"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":"math/0312481","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.OA","submitted_at":"2003-12-29T07:29:27Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"6d8e08aab45013ed86d75fe4ec6b6d6fe88de9529d6f6bf3b899d80d2d5c539b","abstract_canon_sha256":"c5bdfdf5cf68cbb93f3162b5e479a75a60faf1f1dcb734a96ac528cd9c36cc4d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:38:07.533819Z","signature_b64":"3+L9z4C2UKNRshzecN29j1DRDURDYIIFGTfCjbmdh9PTHgZmldRI8e85UOf+fUks07YGje8qTazSoc+iZS1eDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0e68ede3cd8b6ed2c84d034ba87554d37e92ae626bf128288ec9e837f3d732e6","last_reissued_at":"2026-07-04T14:38:07.533435Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:38:07.533435Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"C^*-algebras associated with self-similar sets","license":"","headline":"","cross_cats":["math.DS"],"primary_cat":"math.OA","authors_text":"Tsuyoshi Kajiwara, Yasuo Watatani","submitted_at":"2003-12-29T07:29:27Z","abstract_excerpt":"Let $\\gamma = (\\gamma_1,...,\\gamma_N)$, $N \\geq 2$, be a system of proper contractions on a complete metric space. Then there exists a unique self-similar non-empty compact subset $K$. We consider the union ${\\mathcal G} = \\cup_{i=1}^N \\{(x,y) \\in K^2 ; x = \\gamma_i(y)\\}$ of the cographs of \\gamma _i$. Then $X = C({\\mathcal G})$ is a Hilbert bimodule over $A = C(K)$. We associate a $C^*$-algebra ${\\mathcal O}_{\\gamma}(K)$ with them as a Cuntz-Pimsner algebra ${\\mathcal O}_X$. We show that if a system of proper contractions satisfies the open set condition in $K$, then the $C^*$-algebra ${\\math"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0312481","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0312481/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"math/0312481","created_at":"2026-07-04T14:38:07.533502+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0312481v2","created_at":"2026-07-04T14:38:07.533502+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0312481","created_at":"2026-07-04T14:38:07.533502+00:00"},{"alias_kind":"pith_short_12","alias_value":"BZUO3Y6NRNXN","created_at":"2026-07-04T14:38:07.533502+00:00"},{"alias_kind":"pith_short_16","alias_value":"BZUO3Y6NRNXNFSCN","created_at":"2026-07-04T14:38:07.533502+00:00"},{"alias_kind":"pith_short_8","alias_value":"BZUO3Y6N","created_at":"2026-07-04T14:38:07.533502+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/BZUO3Y6NRNXNFSCNANF2Q5KU2N","json":"https://pith.science/pith/BZUO3Y6NRNXNFSCNANF2Q5KU2N.json","graph_json":"https://pith.science/api/pith-number/BZUO3Y6NRNXNFSCNANF2Q5KU2N/graph.json","events_json":"https://pith.science/api/pith-number/BZUO3Y6NRNXNFSCNANF2Q5KU2N/events.json","paper":"https://pith.science/paper/BZUO3Y6N"},"agent_actions":{"view_html":"https://pith.science/pith/BZUO3Y6NRNXNFSCNANF2Q5KU2N","download_json":"https://pith.science/pith/BZUO3Y6NRNXNFSCNANF2Q5KU2N.json","view_paper":"https://pith.science/paper/BZUO3Y6N","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0312481&json=true","fetch_graph":"https://pith.science/api/pith-number/BZUO3Y6NRNXNFSCNANF2Q5KU2N/graph.json","fetch_events":"https://pith.science/api/pith-number/BZUO3Y6NRNXNFSCNANF2Q5KU2N/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BZUO3Y6NRNXNFSCNANF2Q5KU2N/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BZUO3Y6NRNXNFSCNANF2Q5KU2N/action/storage_attestation","attest_author":"https://pith.science/pith/BZUO3Y6NRNXNFSCNANF2Q5KU2N/action/author_attestation","sign_citation":"https://pith.science/pith/BZUO3Y6NRNXNFSCNANF2Q5KU2N/action/citation_signature","submit_replication":"https://pith.science/pith/BZUO3Y6NRNXNFSCNANF2Q5KU2N/action/replication_record"}},"created_at":"2026-07-04T14:38:07.533502+00:00","updated_at":"2026-07-04T14:38:07.533502+00:00"}