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Discontinuity of the Fuglede-Kadison determinant on a group von Neumann algebra. (English) Zbl 1334.47042
B. Fuglede and R. V. Kadison [Ann. Math. (2) 55, 520–530 (1952; Zbl 0046.33604)] introduced their determinant for operators in a finite factor. They showed that for regular operators, the new determinant has many algebraic and analytic properties similar to the usual matrix determinant. The author proves that the determinant is not continuous on all isomorphisms in \(\mathscr{N}(\mathbb{Z})\) with respect to the strong topology and in the case of the weak operator topology, the example of discontinuity can be constructed within the class of operators in \(\mathscr{N}(\mathbb{Z})\) given by left multiplication with elements of \(\mathbb{Z}[\mathbb{Z}]\). He considers the operator norm topology and shows that the Fuglede-Kadison determinant can be discontinuous at \(\lambda =0\) on a line \(\{T+\lambda\cdot\operatorname{id}_{l^2(\mathbb{Z})}:\lambda \in \mathbb{R}\}\), \(T \in \mathscr{N}(\mathbb{Z})\), that consists entirely of weak isomorphisms of determinant class.
MSC:
47C15 Linear operators in \(C^*\)- or von Neumann algebras
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[1] Fuglede, B., Kadison, R.V.: Determinant theory in finite factors. Ann. of Math., 55, 2, 1952, 520-530, · Zbl 0046.33604 · doi:10.2307/1969645
[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 · doi:10.1090/S0002-9939-2013-11757-0 · arxiv:1204.6293
[3] Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras II. 1983, Academic Press, ISBN 0-1239-3302-1. · Zbl 0518.46046
[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
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