Algebraic K-theory and a semifinite Fuglede–Kadison determinant

Abstract

In this paper we apply algebraic K-theory techniques to construct a Fuglede-Kadison type determinant for a semi-finite von Neumann algebra equipped with a fixed trace. Our construction is based on the approach to determinants for Banach algebras developed by Skandalis and de la Harpe. This approach can be extended to the semi-finite case since the first topological K-group of the trace ideal in a semi-finite von Neumann algebra is trivial. On our way we also improve the methods of Skandalis and de la Harpe by considering relative K-groups with respect to an ideal instead of the usual absolute K-groups. Our construction recovers the determinant homomorphism introduced by Brown, but all the relevant algebraic properties are automatic due to the algebraic K-theory framework.in these (in general) rather complicated abelian groups. On the other hand, basing the construction of determinants purely on functional analytic methods, requires a substantial amount of work for proving the main algebraic properties and the more conceptual framework provided by algebraic K-theory is entirely lost.The key property that we investigate in this text is the relationship between the operator trace, the logarithm and the determinant as expressed by the identity: log(det(g)) = Tr(log(g)).In order to expand on this basic relationship in a K-theoretic context one considers a unital Banach algebra A together with the homomorphismwhere GL(A) denotes the general linear group (over A) equipped with the discrete topology and GL top (A) is the same algebraic group but with the topology coming from the unital Banach algebra A. Passing to classifying spaces and applying Quillen's plus construction, [Qui73], one obtains a continuous map (which is unique up to homotopy)