Cocycle perturbation on Banach algebra

Abstract

Let $α$ be a flow on a Banach algebra $\mathfrak{B}$, and $t\longmapsto u_t$ a continuous function on $\mathbb{R}$ into the group of invertible elements of $\mathfrak{B}$ such that $u_sα_s(u_t )=u_{s+t}, s, t \in \mathbb{R}$. Then $β_t=$Ad$u_t\circα_t, t\in \mathbb{R}$ is also a flow on $\mathfrak{B}$. $β$ is said to be a cocycle perturbation of $α$. We show that if $α,β$ are two flows on nest algebra (or quasi-triangular algebra), then $β$ is a cocycle perturbation of $α$. And the flows on nest algebra (or quasi-triangular algebra) are all uniformly continuous.