Cocycle perturbation on Banach algebra

TL;DR

This paper studies cocycle perturbations of flows on Banach algebras, demonstrating that all flows on nest and quasi-triangular algebras are cocycle perturbations of each other and are uniformly continuous.

Contribution

It proves that any two flows on nest or quasi-triangular algebras are related by cocycle perturbation and are uniformly continuous.

Findings

All flows on nest algebras are cocycle perturbations of each other.

Flows on these algebras are uniformly continuous.

The concept of cocycle perturbation is extended to Banach algebra flows.

Abstract

Let $\alpha$ 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 $u_s\alpha_s(u_t )=u_{s+t}, s, t \in \mathbb{R}$. 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.