Perturbations of flows on Banach spaces and operator algebras

Abstract

For automorphism groups of operator algebras we show how properties of the difference {norm of matrix}αt - α't{norm of matrix} are reflected in relations between the generators δα, δ′α. Indeed for a von Neumann algebra M with separable predual we show that if {norm of matrix}αt - α't{norm of matrix} ≦ 0.28 for small t, then δα = 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}αt - α't{norm of matrix}=O(t) as t →; 0, then δα = δ′α + δ and if {norm of matrix}αt - α't{norm of matrix} ≦ 0.28 for all t then δα=. We prove analogous results for unitary groups on a Hilbert space and C0, C0* groups on a Banach space. Show more