Bivariant $K$-theory with $R/Z$-coefficients and rho classes of unitary representations

Abstract

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. 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 $KK^Γ_{\mathbb{R}}(A,A)$. 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 $KK^Γ$-form of the Baum-Connes conjecture, then every $Γ$-algebra satisfies (KFP). If $α:Γ\to U_n$ is a unitary representation and $A$ satisfies property (KFP), we construct in a canonical way a rho class $ρ_α^A\in KK_{\mathbb{R}/\mathbb{Z}}^{1,Γ}(A,A)$. This construction generalizes the Atiyah-Patodi-Singer $K$-theory class with $\mathbb{R}/\mathbb{Z}$ coefficients associated to $α$.