Trace class operators, regulators, and assembly maps in \(K\)-theory (Q2439240)

This paper addresses a contingency result for the homotopy algebraic \(K\)-theory (\(KH\)) version of the Novikov conjecture. Let \(G\) be any group, let \({\mathbf{Fin}}\) denote the family of its finite subgroups, and let \({\mathcal{E}}(G, \, {\mathbf{Fin}})\) be the classifying space. Let \(H^{G}_*\) denote equivariant homology and let \(L^1\) be the algebra of trace-class operators in an infinite dimensional, simple, complex Hilbert space. The rational \(KH\)-isomorphism conjecture \textit{A. Bartels} and \textit{W. Lück} [J. Pure Appl. Algebra 205, No. 3, 660--696 (2006; Zbl 1093.19002)] states that the (rational) assembly map \[ \varphi : H^G_p( {\mathcal{E}}(G, \, {\mathbf{Fin}}), KH(L^1)) \otimes {\mathbb{Q}} \rightarrow KH_p( L^1[G]) \otimes {\mathbb{Q}} \] is an isomorphism. By the work of \textit{G. Yu} [``The algebraic \(K\)-theory Novikov conjecture for group algebras over the ring of Schatten class operators, Preprint, \url{arXiv:1106.3796}] and the previous work of the authors [\textit{G. Cortiñas} and \textit{G. Tartaglia}, Proc. Am. Math. Soc. 142, No. 4, 1089--1099 (2014; Zbl 1328.19004)], the map \(\varphi\) is known to be injective. Proven in the paper under review, is that if \(\varphi\) is surjective and \(n \equiv (p+1)\mod(2)\), then \[ H^G_n ( {\mathcal{E}}(G, \, \{ 1 \}), \, K({\mathbb{Z}}) ) \otimes{\mathbb{Q}} \rightarrow K_n( {\mathbb{Z}}[G]) \otimes {\mathbb{Q}} \] is injective, where \(\{ 1 \}\) is the trivial family. Also, under the above hypotheses, for any number field \(F\), \[ H^G_n ( {\mathcal{E}}(G, \, {\mathbf{Fin}}), \, K(F)) \otimes {\mathbb{Q}}\rightarrow K_n( F[G]) \otimes {\mathbb{Q}} \] is injective. This second result is equivalent to the rational injectivity of the the \(K\)-theory Farrell-Jones conjecture for number fields. The key ingredient of the proof is an algebraic, equivariant version of Karoubi's multiplicative \(K\)-theory.