Cocycle perturbation on Banach algebra

Abstract

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

9 pages