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arXiv · 1608.07395

Algebraic $K$-theory and a semi-finite Fuglede-Kadison determinant

Peter Hochs·Jens Kaad·André Schemaitat
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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.

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Peter Hochs, Jens Kaad, André Schemaitat. 2016-08-26. Algebraic $K$-theory and a semi-finite Fuglede-Kadison determinant. https://doi.org/10.2140/akt.2018.3.193

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