(untitled) (Communications in Mathematical Research, 2021)
Let α be a flow on a Banach algebra B, and t → u_t a continuous function from R into the group of invertible elements of B such that u_sα_s(u_t) = u_s+t, s, t ∈ R. Then β_t = Adu_t ◦ α_t, t ∈ R is also a flow on B, where Adu_t(B) riangleq u_tBu^−1_t for any B ∈ 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.