| Title:
|
Discontinuity of the Fuglede-Kadison determinant on a group von Neumann algebra (English) |
| Author:
|
Küter, Benjamin |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 |
| Volume:
|
22 |
| Issue:
|
2 |
| Year:
|
2014 |
| Pages:
|
141-149 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $\lim _{\lambda \to 0}\det (T+\lambda \cdot {\mathrm{id}} _{l^{2}(\mathbb {Z})})\neq \det (T)$. (English) |
| Keyword:
|
Fuglede-Kadison determinant |
| Keyword:
|
group von Neumann algebra |
| MSC:
|
47C15 |
| idZBL:
|
Zbl 06410231 |
| idMR:
|
MR3303135 |
| . |
| Date available:
|
2015-01-27T09:37:37Z |
| Last updated:
|
2020-01-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144127 |
| . |
| Reference:
|
[1] Fuglede, B., Kadison, R.V.: Determinant theory in finite factors.Ann. of Math., 55, 2, 1952, 520-530, Zbl 0046.33604, MR 0052696, 10.2307/1969645 |
| Reference:
|
[2] Georgescu, C., Picioroaga, G.: Fuglede-Kadison determinants for operators in the von Neumann algebra of an equivalence relation.Proc. Amer. Math. Soc., 142, 2014, 173-180, Zbl 1282.47061, MR 3119192, 10.1090/S0002-9939-2013-11757-0 |
| Reference:
|
[3] Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras II.1983, Academic Press, ISBN 0-1239-3302-1. MR 0719020 |
| Reference:
|
[4] Lück, W.: $L^2$-Invariants: Theory and Applications to Geometry and K-Theory.2002, Springer Verlag (Heidelberg), ISBN 978-3-540-43566-2. Zbl 1009.55001, MR 1926649 |
| . |
