# Article

 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 .

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