On the Jordan-Chevalley-Dunford decomposition of operators in type $I$ Murray-von Neumann algebras

We show that, for $n \ge 3$, the mapping on $M_n(\mathbb{C})$ which sends a matrix to its diagonalizable part in its Jordan-Chevalley decomposition, is {\bf norm-unbounded} on any neighbourhood of the zero matrix. Let $X$ be a Stonean space, and $\mathcal{N}(X)$ denote the $*$-algebra of (unbounded) normal functions on $X$, containing $C(X)$ as a $*$-subalgebra. We show that every element of $M_n\big(\mathcal{N}(X)\big)$ has a unique Jordan-Chevalley decomposition. Furthermore, when $n \ge 3$ and $X$ has infinitely many points, using the unboundedness of the Jordan-Chevalley decomposition, we show that there is an element of $M_n\big(C(X)\big)$ whose diagonalizable and nilpotent parts are not bounded, that is, do not lie in $M_n\big(C(X)\big)$. Using these results in the context of a type $I$ finite von Neumann algebra $\mathscr{N}$, we prove a canonical Jordan-Chevalley-Dunford decomposition for densely-defined closed operators affiliated with $\mathscr{N}$, expressing each such operator as the strong-sum of a unique commuting pair consisting of (what we call) a $\mathfrak{u}$-scalar-type affiliated operator and an $\mathfrak{m}$-quasinilpotent affiliated operator. The functorial nature of Murray-von Neumann algebras, coupled with the above observations, indicates that considering unbounded affiliated operators is both necessary and natural in the quest for a Jordan-Chevalley-Dunford decomposition for bounded operators in type $II_1$ von Neumann algebras.