Trace class operators, regulators, and assembly maps in $K$-theory

Abstract

Let $G$ be a group and let $KH$ be homotopy algebraic $K$-theory. We prove that if $G$ satisfies the rational $KH$ isomorphism conjecture for the group algebra $L^1[G]$ with coefficients in the algebra of trace-class operators in Hilbert space, then it also satisfies the $K$-theoretic Novikov conjecture for the group algebra over the integers, and the rational injectivity part of the Farrell-Jones conjecture with coefficients in any number field.