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If $A$ is maximal, then it is essentially pervasive and proper. We explore the gap between these two concepts. We show the following: (1) If $A$ is pervasive and proper, and has a nonconstant unimodular element, then $A$ contains an infinite descending chain of pervasive subalgebras on $X$. (2) It is possibl"},"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":"1005.0719","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-05-05T11:36:05Z","cross_cats_sorted":[],"title_canon_sha256":"c3ec9a80c593a44c153f0c8e37d6d4daab85e76f5fadb04e82d82ade7cc350c4","abstract_canon_sha256":"d41e4b0ebf79a1d92fd06b75a37cdab1249e5bf4aaeebe7abcb524f151e6d3bf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:59:45.648599Z","signature_b64":"Irt/p0GTUIW5l31CqU8F412Ce/v8xsn8NjV8D4biNdPDQDzcUs8WI0kcTvHMLKx/R7fxZRpITIi6PwTNygRWBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0e2acb27c40fb7f9c6cd42c381e485d5d0be715ab6b73fe2d2eccabbcf54af5b","last_reissued_at":"2026-05-18T02:59:45.647817Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:59:45.647817Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Pervasive Algebras and Maximal Subalgebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Anthony G. O'Farrell, Pamela Gorkin","submitted_at":"2010-05-05T11:36:05Z","abstract_excerpt":"A uniform algebra $A$ on its Shilov boundary $X$ is {\\em maximal} if $A$ is not $C(X)$ and there is no uniform algebra properly contained between $A$ and $C(X)$. It is {\\em essentially pervasive} if $A$ is dense in $C(F)$ whenever $F$ is a proper closed subset of the essential set of $A$.  If $A$ is maximal, then it is essentially pervasive and proper. We explore the gap between these two concepts. We show the following: (1) If $A$ is pervasive and proper, and has a nonconstant unimodular element, then $A$ contains an infinite descending chain of pervasive subalgebras on $X$. 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