Vol. 3, No. 2, 2018
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Algebraic $K\mskip-2mu$-theory and a semifinite Fuglede–Kadison determinant

Peter Hochs, Jens Kaad and André Schemaitat
Vol. 3 (2018), No. 2, 193–206
DOI: 10.2140/akt.2018.3.193
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.

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Keywords
algebraic $K\mskip-2mu$-theory, semifinite von Neumann algebras, determinants
Mathematical Subject Classification 2010
Primary: 46L80
Milestones
Received: 30 August 2016
Revised: 23 May 2017
Accepted: 7 June 2017
Published: 24 March 2018
Authors
Peter Hochs
School of Mathematical Sciences
North Terrace Campus
The University of Adelaide
Adelaide, SA
Australia
Jens Kaad
Department of Mathematics and Computer Science
University of Southern Denmark
Odense
Denmark
André Schemaitat
University of Münster
Münster
Germany