{"paper":{"title":"Bivariant $K$-theory with $R/Z$-coefficients and rho classes of unitary representations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.KT"],"primary_cat":"math.OA","authors_text":"Georges Skandalis, Paolo Antonini, Sara Azzali","submitted_at":"2015-04-17T12:51:33Z","abstract_excerpt":"We construct equivariant $KK$-theory with coefficients in $\\mathbb{R}$ and $\\mathbb{R}/\\mathbb{Z}$ as suitable inductive limits over ${\\rm II}_1$-factors. We show that the Kasparov product, together with its usual functorial properties, extends to $KK$-theory with real coefficients.\n  Let $\\Gamma$ be a group. We define a $\\Gamma$-algebra $A$ to be $K$-theoretically free and proper (KFP) if the group trace ${\\bf tr}$ of $\\Gamma$ acts as the unit element in $KK^{\\Gamma}_{\\mathbb{R}}(A,A)$. We show that free and proper $\\Gamma$-algebras (in the sense of Kasparov) have the (KFP) property. Moreover"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.04495","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}