It is known that every operator on a (separable) Hilbert space is the direct integral of irreducible operators, but not every one is the direct sum of irreducible ones. We show that an operator can have either finitely or uncountably many reducing subspaces, and the former holds if and only if the operator is the direct sum of finitely many irreducible operators no two of which are unitarily equivalent. We also characterize operators T which are direct sums of irreducible operators in terms of the...

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 $𝒩\left(\mathbb{Z}\right)$ 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 $\mathbb{Z}\left[\mathbb{Z}\right]$. Finally, we define $T\in 𝒩\left(\mathbb{Z}\right)$ such that for each $\lambda \in ℝ$ the operator $T+\lambda ·{\mathrm{id}}_{l^2\left(\mathbb{Z}\right)}$ is a self-adjoint weak isomorphism of determinant class but...