Perturbations of flows on Banach spaces and operator algebras

For automorphism groups of operator algebras we show how properties of the difference {norm of matrix}α_t - α'_t{norm of matrix} are reflected in relations between the generators δ_α, δ′_α. Indeed for a von Neumann algebra M with separable predual we show that if {norm of matrix}α_t - α'_t{norm of matrix} ≦ 0.28 for small t, then δ_α = γ0(δ′_α+δ′)°γ^-1 where γ is an inner automorphism of M and δ is a bounded derivation of M. If the difference {norm of matrix}α_t - α'_t{norm of matrix}=O(t) as t →; 0, then δ_α = δ′_α + δ and if {norm of matrix}α_t - α'_t{norm of matrix} ≦ 0.28 for all t then δ_α=. We prove analogous results for unitary groups on a Hilbert space and C₀, C₀^* groups on a Banach space.