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Fuglede-Kadison determinant; group von Neumann algebra
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 $\lim _{\lambda \to 0}\det (T+\lambda \cdot {\mathrm{id}} _{l^{2}(\mathbb {Z})})\neq \det (T)$.
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