Algebraic K-theory and a semifinite Fuglede-Kadison determinant

Peter Hochs, Jens Kaad, André Schemaitat

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Resumé

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.
OriginalsprogEngelsk
TidsskriftAnnals of K-Theory
Vol/bind3
Udgave nummer2
Sider (fra-til)193–206
ISSN2379-1683
DOI
StatusUdgivet - mar. 2018

Fingeraftryk

Algebraic K-theory
K-group
Determinant
Von Neumann Algebra
Trace
Banach algebra
Homomorphism
Trivial

Citer dette

Hochs, Peter ; Kaad, Jens ; Schemaitat, André. / Algebraic K-theory and a semifinite Fuglede-Kadison determinant. I: Annals of K-Theory. 2018 ; Bind 3, Nr. 2. s. 193–206.
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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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Hochs, P, Kaad, J & Schemaitat, A 2018, 'Algebraic K-theory and a semifinite Fuglede-Kadison determinant', Annals of K-Theory, bind 3, nr. 2, s. 193–206. https://doi.org/10.2140/akt.2018.3.193

Algebraic K-theory and a semifinite Fuglede-Kadison determinant. / Hochs, Peter; Kaad, Jens; Schemaitat, André.

I: Annals of K-Theory, Bind 3, Nr. 2, 03.2018, s. 193–206.

Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › peer review

TY - JOUR

T1 - Algebraic K-theory and a semifinite Fuglede-Kadison determinant

AU - Hochs, Peter

AU - Kaad, Jens

AU - Schemaitat, André

PY - 2018/3

Y1 - 2018/3

N2 - 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.

AB - 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.

U2 - 10.2140/akt.2018.3.193

DO - 10.2140/akt.2018.3.193

M3 - Journal article

VL - 3

SP - 193

EP - 206

JO - Annals of K-Theory

JF - Annals of K-Theory

SN - 2379-1683

IS - 2

ER -

Hochs P, Kaad J, Schemaitat A. Algebraic K-theory and a semifinite Fuglede-Kadison determinant. Annals of K-Theory. 2018 mar;3(2):193–206. https://doi.org/10.2140/akt.2018.3.193

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