Volume 30, Issue 1
Cocycle Perturbation on Banach Algebras

Commun. Math. Res., 30 (2014), pp. 1-10.

Published online: 2021-05

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• Abstract

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.

• Keywords

cocycle perturbation, inner perturbation, nest algebra, quasi-triangular algebra.

47D03, 46H99, 46K50, 46L57

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@Article{CMR-30-1, author = {Luoyi and Shi and and 18772 and and Luoyi Shi and Yujing and Wu and and 18774 and and Yujing Wu}, title = {Cocycle Perturbation on Banach Algebras}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {30}, number = {1}, pages = {1--10}, abstract = {

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} }
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 -

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.

Luoyi Shi & Yujing Wu. (2021). Cocycle Perturbation on Banach Algebras. Communications in Mathematical Research . 30 (1). 1-10. doi:
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