Nilpotents in finite algebras

Abstract

We study the set of nilpotents t (t^n=0) of a type $II_1 von Neumann algebra A which verify that t^{n-1}+t* is invertible. These are shown to be all similar in A. The set of all such operators, named by D.A. Herrero very nice Jordan nilpotents, forms a simply connected smooth submanifold of A in the norm topology. Nilpotents are related to systems of projectors, i.e. n-tuples (p_1,...,p_n) of mutually orthogonal projections of the algebra which sum 1, via the map φ(t)=(P_{ker t},P_{ker t^2}-P_{ker t},...,P_{ker t^{n-1}}-P_{ker t^{n-2}},1-P_{ker t^{n-1}}). The properties of this map, called the canonical decomposition of nilpotents in the literature, are examined.