Cyclomorphy

Abstract

A main result is that, roughly, a dense set of the infinitesimal trace-preserving deformations of a semicircular system $s_1,..., s_n$ arise from one-parameter groups of automorphisms of the free-group factor $L(F(n))$ generated by $s_1,..., s_n$. More generally the paper studies cyclic gradients in von Neumann algebras, Lie algebras of noncommutative trace-preserving vector fields and the class of cyclomorphic maps which preserve the orthogonals of spaces of cyclic gradients.