Discontinuity of the Fuglede-Kadison determinant on a group von Neumann algebra

Benjamin Küter

Communications in Mathematics (2014)

  • Volume: 22, Issue: 2, page 141-149
  • ISSN: 1804-1388

Abstract

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We show that in contrast to the case of the operator norm topology on the set of regular operators, the Fuglede-Kadison determinant is not continuous on isomorphisms in the group von Neumann algebra 𝒩 ( ) with respect to the strong operator topology. Moreover, in the weak operator topology the determinant is not even continuous on isomorphisms given by multiplication with elements of [ ] . Finally, we define T 𝒩 ( ) such that for each λ the operator T + λ · id l 2 ( ) is a self-adjoint weak isomorphism of determinant class but lim λ 0 det ( T + λ · id l 2 ( ) ) det ( T ) .

How to cite

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Küter, Benjamin. "Discontinuity of the Fuglede-Kadison determinant on a group von Neumann algebra." Communications in Mathematics 22.2 (2014): 141-149. <http://eudml.org/doc/269865>.

@article{Küter2014,
abstract = {We show that in contrast to the case of the operator norm topology on the set of regular operators, the Fuglede-Kadison determinant is not continuous on isomorphisms in the group von Neumann algebra $\mathcal \{N\}(\mathbb \{Z\})$ with respect to the strong operator topology. Moreover, in the weak operator topology the determinant is not even continuous on isomorphisms given by multiplication with elements of $\mathbb \{Z\}[\mathbb \{Z\}]$. Finally, we define $T\in \mathcal \{N\}(\mathbb \{Z\})$ such that for each $\lambda \in \mathbb \{R\}$ the operator $T+\lambda \cdot \{\mathrm \{id\}\} _\{l^\{2\}(\mathbb \{Z\})\}$ is a self-adjoint weak isomorphism of determinant class but $\lim _\{\lambda \rightarrow 0\}\det (T+\lambda \cdot \{\mathrm \{id\}\} _\{l^\{2\}(\mathbb \{Z\})\})\ne \det (T)$.},
author = {Küter, Benjamin},
journal = {Communications in Mathematics},
keywords = {Fuglede-Kadison determinant; group von Neumann algebra; Fuglede-Kadison determinant; group von Neumann algebra},
language = {eng},
number = {2},
pages = {141-149},
publisher = {University of Ostrava},
title = {Discontinuity of the Fuglede-Kadison determinant on a group von Neumann algebra},
url = {http://eudml.org/doc/269865},
volume = {22},
year = {2014},
}

TY - JOUR
AU - Küter, Benjamin
TI - Discontinuity of the Fuglede-Kadison determinant on a group von Neumann algebra
JO - Communications in Mathematics
PY - 2014
PB - University of Ostrava
VL - 22
IS - 2
SP - 141
EP - 149
AB - We show that in contrast to the case of the operator norm topology on the set of regular operators, the Fuglede-Kadison determinant is not continuous on isomorphisms in the group von Neumann algebra $\mathcal {N}(\mathbb {Z})$ with respect to the strong operator topology. Moreover, in the weak operator topology the determinant is not even continuous on isomorphisms given by multiplication with elements of $\mathbb {Z}[\mathbb {Z}]$. Finally, we define $T\in \mathcal {N}(\mathbb {Z})$ such that for each $\lambda \in \mathbb {R}$ the operator $T+\lambda \cdot {\mathrm {id}} _{l^{2}(\mathbb {Z})}$ is a self-adjoint weak isomorphism of determinant class but $\lim _{\lambda \rightarrow 0}\det (T+\lambda \cdot {\mathrm {id}} _{l^{2}(\mathbb {Z})})\ne \det (T)$.
LA - eng
KW - Fuglede-Kadison determinant; group von Neumann algebra; Fuglede-Kadison determinant; group von Neumann algebra
UR - http://eudml.org/doc/269865
ER -

References

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  1. Fuglede, B., Kadison, R.V., 10.2307/1969645, Ann. of Math., 55, 2, 1952, 520-530, (1952) Zbl0046.33604MR0052696DOI10.2307/1969645
  2. Georgescu, C., Picioroaga, G., 10.1090/S0002-9939-2013-11757-0, Proc. Amer. Math. Soc., 142, 2014, 173-180, (2014) Zbl1282.47061MR3119192DOI10.1090/S0002-9939-2013-11757-0
  3. Kadison, R.V., Ringrose, J.R., Fundamentals of the Theory of Operator Algebras II, 1983, Academic Press, ISBN 0-1239-3302-1. (1983) MR0719020
  4. Lück, W., L 2 -Invariants: Theory and Applications to Geometry and K-Theory, 2002, Springer Verlag (Heidelberg), ISBN 978-3-540-43566-2. (2002) Zbl1009.55001MR1926649

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