TY - JOUR
T1 - Cocycle Perturbation on Banach Algebras
AU - Shi , Luoyi
AU - Wu , Yujing
JO - Communications in Mathematical Research 
VL - 1
SP - 1
EP - 10
PY - 2021
DA - 2021/05
SN - 30
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/cmr/18982.html
KW - cocycle perturbation, inner perturbation, nest algebra, quasi-triangular
algebra.
AB - <p style="text-align: justify;">Let $α$ be a flow on a Banach algebra&nbsp;$\mathcal{B}$, and $t → u_t$ a continuous function
from&nbsp;$\boldsymbol{R}$&nbsp;into the group of invertible elements of&nbsp;$\mathcal{B}$&nbsp;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.</p>