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 semifinite 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 semifinite case since the first topological $K$-group of the trace ideal in a semifinite von Neumann algebra is trivial. Along the 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.