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 ${\mathrm{II}}_1$-factors. We show that the Kasparov product, together with its usual functorial properties, extends to $KK$-theory with real coefficients. Let $\Gamma$ be a group. We define a $\Gamma$-algebra $A$ to be $K$-theoretically free and proper (KFP) if the group trace $\mathbf{tr}$ of $\Gamma$ acts as the unit element in $K{K}_{\mathbb{R}}^{\Gamma }(A,A)$. We show that free and proper $\Gamma$-algebras (in the sense of Kasparov) have the (KFP) property. Moreover, if $\Gamma$ is torsion free and satisfies the $K{K}^{\Gamma }$-form of the Baum-Connes conjecture, then every $\Gamma$-algebra satisfies (KFP). If $\alpha :\Gamma \to U_n$ is a unitary representation and $A$ satisfies property (KFP), we construct in a canonical way a rho class ${\rho }_{\alpha }^A\in K{K}_{\mathbb{R}/\mathbb{Z}}^{1,\Gamma }(A,A)$. This construction generalizes the Atiyah-Patodi-Singer $K$-theory class with $\mathbb{R}/\mathbb{Z}$ coefficients associated to $\alpha$.