Cyclomorphy (Q2778020)

The paper is motivated by the free probability theory and is devoted to studies of cyclic derivatives in von Neumann algebras. The author begins by recalling some properties of noncommutative polynomials, cyclic gradients, free difference quotients and semicircular systems. Then the exponentiation of noncommutative vector fields is decribed, giving estimates that prove that certain derivations can be exponentiated to automorphism groups. These estimates are then used in the case of cyclic gradients. NEWLINENEWLINENEWLINENext, the author describes endomorphic orbits and their tangent spaces and studies real and complex cyclomorphic maps. Lie algebras of noncommutative vector fields with particular emphasis on the semicircular case are described. Finally, the author sketches how the notion of cyclomorphic maps can be generalised to the context of B-von Neumann algebras.