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Keywords:
Fuglede-Kadison determinant; group von Neumann algebra
Fuglede-Kadison determinant; group von Neumann algebra
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 $\lim _{\lambda \to 0}\det (T+\lambda \cdot {\mathrm{id}} _{l^{2}(\mathbb {Z})})\neq \det (T)$.
References:
[1] Fuglede, B., Kadison, R.V.: Determinant theory in finite factors. Ann. of Math., 55, 2, 1952, 520-530, DOI 10.2307/1969645 | MR 0052696 | Zbl 0046.33604
[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, DOI 10.1090/S0002-9939-2013-11757-0 | MR 3119192 | Zbl 1282.47061
[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
[4] Lück, W.: $L^2$-Invariants: Theory and Applications to Geometry and K-Theory. 2002, Springer Verlag (Heidelberg), ISBN 978-3-540-43566-2. MR 1926649 | Zbl 1009.55001
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