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We prove that $A$ has this property provided that any of the following three conditions holds: (i) $A$ is a weakly amenable Banach algebra with property $\\mathbb{B}$ and having a bounded approximate identity, (ii) every continuous cyclic Jordan derivation from $A$ into $A^*$ is an inner derivation, (iii) $A$ is the algebra of all $n\\times "},"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":"1709.09432","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-09-27T10:17:25Z","cross_cats_sorted":[],"title_canon_sha256":"a51b4c017816d6cb02cb66ccfab0e9a2cf8c1371e1db45153f52b04fc451f8da","abstract_canon_sha256":"91a5549aaa87830a2733662ae68f5a53fac8833f1e378aa4d49560bda7611323"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:34:11.772800Z","signature_b64":"FrP7gQ1VhyGQtXBtpj6shOycXH6p9iLWzSszJE0y3+C6zZnE0GvlpB+L6COQP85e0/1ATiEEhVOY6EVQtXWKCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cb718deda1960b7a32d0a505889ffdc6103e4f1b1439be57dcc390d53c304d84","last_reissued_at":"2026-05-18T00:34:11.772233Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:34:11.772233Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Zero Lie product determined Banach algebras, II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"A. 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