2015/04/17 by Paolo Antonini, Antonini, Paolo, Sara Azzali +3
Mathematics · #19K35 #46L80 #FOS: Mathematics #K-Theory and Homology (math.KT) #Operator Algebras (math.OA) #math.KT #math.OA #msc:19K35 #msc:46L80
paper · pdf · doi:10.48550/arxiv.1504.04495
arxiv created 2015/04/17 · arxiv updated 2015/04/20
We construct equivariant KK-theory with coefficients in ℝ and ℝ/ℤ as suitable inductive limits over \rm II1-factors. We show that the Kasparov product, together with its usual functorial properties, extends to KK-theory with real coefficients. Let Γ be a group. We define a Γ-algebra A to be K-theoretically free and proper (KFP) if the group trace \bf tr of Γ acts as the unit element in KKΓℝ(A,A). We show that free and proper Γ-algebras (in the sense of Kasparov) have the (KFP) property. Moreover, if Γ is torsion free and satisfies the KKΓ-form of the Baum-Connes conjecture, then every Γ-algebra satisfies (KFP). If α:Γ→ Un is a unitary representation and A satisfies property (KFP), we construct in a canonical way a rho class ραA∈ KKℝ/ℤ1,Γ(A,A). This construction generalizes the Atiyah-Patodi-Singer K-theory class with ℝ/ℤ coefficients associated to α.