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We give sufficient topological conditions which when satisfied the ${C}^*$-correspondence $\\mathcal{H}(X)$ is proper, that is, the ${C}^*$-algebra ${C}^*(G,\\alpha)$ acts on the Hilbert ${C}^*(H,\\beta)$-module ${H}(X)$ via the comapct operators. Thus a proper topological correspondence produces an element in ${KK}({C}^*(G,\\"},"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":"1709.08938","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2017-09-26T10:55:57Z","cross_cats_sorted":[],"title_canon_sha256":"67ec300a6275e90d7112c0a59fe96985cba26489d810be6425f4d7c2fcf39dd7","abstract_canon_sha256":"6cbfb427694aa7039c1b448ba41b5600c7614123690dfbd3c6865780495f1009"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:34:21.535741Z","signature_b64":"a4NHOpiEy08bAj1rgEQPMIjjPi1ZtW4BzJXRx+RjKS7FuJ438HBmA/N3wtyEn7sRmF4dtLQ+H3nMl8ujbXB8Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"488c677710e775a90f457969bb5e052de9ba50774d40379dd380acdf2f1fb514","last_reissued_at":"2026-05-18T00:34:21.535204Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:34:21.535204Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Locally free actions of groupoids and proper topological correspondences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Rohit Dilip Holkar","submitted_at":"2017-09-26T10:55:57Z","abstract_excerpt":"Let $(G,\\alpha)$ and $(H,\\beta)$ be locally compact Hausdorff groupoids with Haar systems, and let $(X,\\lambda)$ be a topological correspondence from $(G,\\alpha)$ to $(H,\\beta)$ which induce the ${C}^*$-correspondence $\\mathcal{H}(X)\\colon {C}^*(G,\\alpha)\\to {C}^*(H,\\beta)$. We give sufficient topological conditions which when satisfied the ${C}^*$-correspondence $\\mathcal{H}(X)$ is proper, that is, the ${C}^*$-algebra ${C}^*(G,\\alpha)$ acts on the Hilbert ${C}^*(H,\\beta)$-module ${H}(X)$ via the comapct operators. 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