On Certain Equations of Automorphisms of C*--Algebras.

In this paper we study some properties of automorphisms of C*-algebras satisfying certain equations. Among other results we show that if u, v are invertible elements of a C*-algebra A such that uvu-1 + cu-1vu = (1 + c)v, and uv*u-1 + cu-1v*u = (1 + c)v*, where c ∈ C with c ≠ -1 and if u is normal, then u commutes with v and v*. Further, if α, β are inner automorphisms of a W*-algebra M implemented by normal operators u and v, respectively, then α + α-1 = β + β-1 if and only if there is a central projection P ∈ M such that Puv and (1 - P)uv-1 are in the center of M and consequently M admits a decomposition via a central projection q = (1 - P) such that α = β on Mq and α = β-1 on M(1 - q). [ABSTRACT FROM AUTHOR]