Nilpotents in finite algebras

E. ANDRUCHOW; D. STOJANOFF

INTEGRAL EQUATIONS AND OPERATOR THEORY

BIRKHAUSER VERLAG AG

Lugar: BASEL; Año: 2003 vol. 45 p. 251 - 267

0378-620X

We study the set of nilpotents $t$ ($t^n=0$) of a type $II_1$ von Neumann  algebra $cala$ which verify that $t^{n-1}+t^*$ is invertible. These are shown to be all similar in $cala$. The set of all such operators, named by D.A. Herrero {it very nice Jordan} nilpotents, forms a simply connected smooth submanifold of $cala$ 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$$(t)=(P_{ker t},P_{ker t^2}-P_{ker t},...,P_{ker t^{n-1}}-P_{ker t^{n-2}},1-P_{ker t^{n-1}}).$$The properties of this map, called the {it canonical decomposition} of nilpotents in the literature, are examined.