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A deep theorem by Freed-Hopkins-Teleman showed that the twisted K-homology $KK^{G}(C_{\\A^{k+h}}(G), \\C)$ is isomorphic to the level k Verlinde ring R_{k}(G). By the construction of crossed product, we define a C*-algebra $C^{*}(G,C_{\\A^{k+h}}(G))$. We show that the K-homology KK(C^{*}(G,C_{\\A^{k+h}}(G)),\\C) is isomorphic to the "},"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":"1404.4850","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-04-18T18:20:43Z","cross_cats_sorted":["math.KT","math.SG"],"title_canon_sha256":"cbcd966673e4f35d7e35f7d5f75a0e2eb04bfa3182bc27592129343b8e0f9218","abstract_canon_sha256":"3ca0f0afe1a162d5a7f63ec4cf0a51311899357b5cbad83548cb4e4d808e7a59"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:53:54.730894Z","signature_b64":"KQROW967YcyQhDS5qIHUmAlS34Uqbaqxrp65QLzy+ehu9F7SwxIrLtZCbIYUAWSQvC6clixI+0e7+/spqKuDBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"02390cd4cce3b7ff2bc608f44c47503bffa8a69fa336d90e0d504181a3c9c191","last_reissued_at":"2026-05-18T02:53:54.730252Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:53:54.730252Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Formal Verlinde Module","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.KT","math.SG"],"primary_cat":"math.DG","authors_text":"Yanli Song","submitted_at":"2014-04-18T18:20:43Z","abstract_excerpt":"Let G be a compact, simple and simply connected Lie group and $\\A$ be an equivariant Dixmier-Douady bundle over G. 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