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Let $α$ be a flow on a Banach algebra $\mathcal{B}$, and $t → u_t$ a continuous function from $\boldsymbol{R}$ into the group of invertible elements of $\mathcal{B}$ such that $u_sα_s(u_t) = u_{s+t}, s, t ∈ \boldsymbol{R}$. Then $β_t$ = Ad$u_t ◦ α_t$, $t ∈ \boldsymbol{R}$ is also a flow on $\mathcal{B}$, where Ad$u_t(B) \triangleq u_tBu^{−1}_t$ for any $B ∈ \mathcal{B}$. $β$ is said to be a cocycle perturbation of $α$. We show that if $α$, $β$ are two flows on a nest algebra (or quasi-triangular algebra), then $β$ is a cocycle perturbation of $α$. And the flows on a nest algebra (or quasi-triangular algebra) are all uniformly continuous.
}, issn = {2707-8523}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cmr/18982.html} }Let $α$ be a flow on a Banach algebra $\mathcal{B}$, and $t → u_t$ a continuous function from $\boldsymbol{R}$ into the group of invertible elements of $\mathcal{B}$ such that $u_sα_s(u_t) = u_{s+t}, s, t ∈ \boldsymbol{R}$. Then $β_t$ = Ad$u_t ◦ α_t$, $t ∈ \boldsymbol{R}$ is also a flow on $\mathcal{B}$, where Ad$u_t(B) \triangleq u_tBu^{−1}_t$ for any $B ∈ \mathcal{B}$. $β$ is said to be a cocycle perturbation of $α$. We show that if $α$, $β$ are two flows on a nest algebra (or quasi-triangular algebra), then $β$ is a cocycle perturbation of $α$. And the flows on a nest algebra (or quasi-triangular algebra) are all uniformly continuous.