Nilpotents in Finite Algebras

Abstract

We study the set of nilpotents \( t\,(t^{n} = 0) \) of a type \( II_{1} \) von Neumann algebra \( \mathcal{A} \) which verify that \( t^{n-1} + t^{\ast} \) is invertible. These are shown to be all similar in \( \mathcal{A} \). The set of all such operators, named by D.A. Herrero very nice Jordan nilpotents, forms a simply connected smooth submanifold of \( \mathcal{A} \) in the norm topology. Nilpotents are related to systems of projectors, i.e., n-tuples\( (p_{1},\ldots, p_{n}) \) of mutually orthogonal projections of the algebra which sum 1, via the map¶\( \varphi(t) = (P_{\textrm{ker}\,t}, \,P_{\textrm{ker}\,t^{2}}-P_{\textrm{ker}\,t},\ldots,P_{\textrm{ker}\,t^{n-\textrm{1}}}- P_{\textrm{ker}\,t^{n-\textrm{2}}},1-P_{\textrm{ker}\,t^{n-\textrm{1}}}). \)¶The properties of this map, called the canonical decomposition of nilpotents in the literature, are examined.