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

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 -