Nilpotents in Finite Algebras

We study the set of nilpotents t(t^n = 0) of a type II1 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 (p1,…, pn) of mutually orthogonal projections of the algebra which sum 1, via the map (phi)(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.